CHAPTER 2 CHARACTERISTIC CLASSES Introduction In the second chapter, we develop the theory of characteristic classes. In section 2.1, we discuss the Chern classes of a complex vector bundle and in section 2.2 we discuss the Pontrjagin and Euler classes of a real vector bundle. We shall define the Todd class, the Hirzebruch L-polynomial, and the A-roof genus which will play a central role in our discussion of the index theorem. We also discuss the total Chern and Pontrjagin classes as well as the Chern character. In section 2.3, we apply these ideas to the tangent space of a real mani- fold and to the holomorphic tangent space of a holomorphic manifold. We compute several examples defined by Clifford matrices and compute the Chern classes of complex projective space. We show that suitable products of projective spaces form a dual basis to the space of characteristic classes. Such products will be used in chapter three to find the normalizing con- stants which appear in the formula for the index theorem. In section 2.4 and in the first part of section 2.5, we give a heat equa- tion proof of the Gauss-Bonnet theorem. This is based on first giving an abstract characterization of the Euler form in terms of invariance theory. This permis us to identify the integrand of the heat equation with the Euler integrand. This gives a more direct proof of Theorem 2.4.8 which was first proved by Patodi using a complicated cancellation lemma. (The theorem for dimension m = 2 is due to McKean and Singer). In the remainder of section 2.5, we develop a similar characterization of the Pontrjagin forms of the real tangent space. We shall wait until the third chapter to apply these results to obtain the Hirzebruch signature theorem. There are two different approaches to this result. We have presented both our original approach and also one modeled on an approach due to Atiyah, Patodi and Bott. This approach uses H. Weyl’s theorem on the invariants of the orthogonal group and is not self-contained as it also uses facts about geodesic normal coordinates we shall not develop. The other approach is more combinatorial in flavor but is self-contained. It also generalizes to deal with the holomorphic case for a Kaehler metric. The signature complex with coefficients in a bundle V gives rise to in- variants which depend upon both the metric on the tangent space of M and on the connection 1-form of V. In section 2.6, we extend the results of section 2.5 to cover more general invariants of this type. We also construct dual bases for these more general invariants similar to those constructed in section 2.3 using products of projective spaces and suitable bundles over these spaces.

88 Chapter 2 The material of sections 2.1 through 2.3 is standard and reviews the theory of characteristic classes in the context we shall need. Sections 2.4 through 2.6 deal with less standard material. The chapter is entirely self- contained with the exception of some material in section 2.5 as noted above. We have postponed a discussion of similar material for the holomorphic Kaehler case until sections 3.6 and 3.7 of chapter three since this material is not needed to discuss the signature and spin complexes.

2.1. Characteristic Classes Of a Complex Vector Bundle. The characteristic classes are topological invariants of a vector bundle which are represented by differential forms. They are defined in terms of the curvature of a connection. Let M be a smooth compact manifold and let V be a smooth complex vector bundle of dimension k over M. A connection ∇ on V is a first order partial differential operator ∇: C^{∞}(V) → C^{∞}(T^{∗}M ⊗ V) such that: ∇(fs) = df ⊗ s + f∇s  for f ∈ C^{∞}(M) and s ∈ C^{∞}(V). Let (s_{1},…, s_{k}) be a local frame for V. We can decompose any section s ∈ C^{∞}(V) locally in the form s(x) = f_{i}(x)s_{i}(x) for f_{i} ∈ C^{∞}(M). We adopt the convention of summing over repeated indices unless otherwise specified in this subsection. We compute: ∇s = df_{i} ⊗ s_{i} + f_{i}∇s_{i} = df_{i} ⊗ s_{i} + f_{i}ω_{ij} ⊗ s_{j}  where ∇s_{i} = ω_{ij} ⊗ sj. The connection 1-form ω defined by ω = ωij is a matrix of 1-forms. The connection ∇ is uniquely determined by the derivation property and by the connection 1-form. If we specify ω arbitrar- ily locally, we can define ∇ locally. The convex combination of connections is again a connection, so using a partition of unity we can always construct connections on a bundle V. If we choose another frame for V, we can express s_{′i} = h_{ij}s_{j}. If h_{−1 ij}s_{′j} = s_{i} represents the inverse matrix, then we compute: ∇s′i = ω_{′ij} ⊗ s_{′j} = ∇(h_{ik}s_{k}) = dh_{ik} ⊗ s_{k} + h_{ik}ω_{kl} ⊗ sl = (dh_{ik}h_{−1 kj} + h_{ik}ω_{kl}h_{−1 lj}) ⊗ s_{′j}. This shows ω′ obeys the transformation law: ω^{′ij} = dh^{ik}h^{−1 kj} + h^{ik}ω^{kl}h^{−1 lj}  i.e., ω′ = dh · h^{−1} + hωh−1 in matrix notation. This is, of course, the manner in which the 0^{th} order symbol of a first order operator transforms. We extend ∇ to be a derivation mapping C^{∞}(Λ^{p}T^{∗}M ⊗ V) → C^{∞}(Λ^{p+1}T^{∗}M ⊗ V) so that:  ∇(θ_{p} ⊗ s) = dθ_{p} ⊗ s + (−1)^{p}θ_{p} ∧ ∇s.

We compute that: ∇^{2}(fs) = ∇( df ⊗ s + f∇s) = d^{2}f ⊗ s − df ∧ ∇s + df ∧ ∇s + f∇^{2}s = f∇^{2}s so instead of being a second order operator, ∇^{2} is a 0^{th} order operator. We may therefore express ∇^{2}(s)(x_{0}) = Ω(x_{0})s(x_{0}) where the curvature Ω is a section to the bundle Λ^{2}(T^{∗}M) ⊗ END(V) is a 2-form valued endomorphism of V. In local coordinates, we compute: Ω(s_{i}) = Ω_{ij} ⊗ s_{j} = ∇(ω_{ij} ⊗ s_{j}) = dω_{ij} ⊗ s_{j} − ω_{ij} ∧ ω_{jk} ⊗ sk so that:  Ω_{ij} = dω_{ij} − ω_{ik} ∧ ω_{kj}  i.e., Ω = dω − ω ∧ ω. Since ∇^{2} is a 0^{th} order operator, Ω must transform like a tensor: Ω_{′ij} = h_{ik}Ω_{kl}h_{−1 lj}  i.e., Ω′ = hΩh^{−1}. This can also be verified directly from the transition law for ω′. The reader should note that in some references, the curvature is defined by Ω = dω + ω ∧ ω. This sign convention results from writing V ⊗ T^{∗}M instead of T^{∗}M ⊗ V and corresponds to studying left invariant rather than right invariant vector fields on GL(k, C). It is often convenient to normalize the choice of frame. Lemma 2.1.1. Let ∇ be a connection on a vector bundle V. We can always choose a frame s so that at a given point x_{0} we have ω(x_{0}) = 0  and dΩ(x_{0}) = 0. Proof: We find a matrix h(x) defined near x_{0} so that h(x_{0}) = I and dh(x_{0}) = −ω(x_{0}). If s_{′i} = h_{ij}s_{j}, then it is immediate that ω′(x_{0}) = 0. Similarly we compute dΩ′(x_{0}) = d(dω′ − ω′ ∧ ω′)(x_{0}) = ω′(x_{0}) ∧ dω′(x_{0}) − dω′(x_{0}) ∧ ω′(x_{0}) = 0. We note that as the curvature is invariantly defined, we cannot in general find a parallel frame s so ω vanishes in a neighborhood of x_{0} since this would imply ∇^{2} = 0 near x_{0} which need not be true. Let A_{ij} denote the components of a matrix in END(C^{k}) and let P (A) = P (A_{ij}) be a polynomial mapping END(C^{k}) → C. We assume that P is invariant—i.e., P (hAh^{−1}) = P (A) for any h ∈ GL(k, C). We define P (Ω) as an even differential form on M by substitution. Since P is invariant

Of a Complex Vector Bundle 91

and since the curvature transforms tensorially, P (Ω) ^{def} = P (∇) is defined independently of the particular local frame which is chosen. There are two examples which are of particular interest and which will play an important role in the statement of the Atiyah-Singer index theorem. We define: c(A) = det (I + i 2π A) = 1 + c_{1}(A) + ⋯ + c_{k}(A) (the total Chern form) ch(A) = Tr e^{iA/2π} = _{∑ j} Tr (iA 2π )^{j}/j! (the total Chern character) The c_{j}(A) represent the portion of c(A) which is homogeneous of order j. c_{j}(∇) ∈ Λ^{2j}(M). In a similar fashion, we decompose ch(A) = ∑_{j} ch_{j}(A) for ch_{j}(A) = Tr (iA/2π)^{j}/j! Strictly speaking, ch(A) is not a polynomial. However, when we sub- stitute the components of the curvature tensor, this becomes a finite sum since Tr(Ω^{j}) = 0 if 2j > dim M. More generally, we can define P (Ω) if P (A) is an invariant formal power series by truncating the power series appropriately. As a differential form, P (∇) depends on the connection chosen. We show P (∇) is always closed. As an element of de Rham cohomology, P (∇) is independent of the connection and defines a cohomology class we shall denote by P (V). Lemma 2.1.2. Let P be an invariant polynomial. (a) dP (∇) = 0 so P (∇) is a closed differential form. (b) Given two connections ∇_{0} and ∇_{1}, we can define a differential form T P (∇_{0}, ∇_{1}) so that P (∇_{1}) − P (∇_{0}) = d{T P (∇_{1}, ∇_{0})}. Proof: By decomposing P as a sum of homogeneous polynomials, we may assume without loss of generality that P is homogeneous of order k. Let P (A_{1},…, A_{k}) denote the complete polarization of P. We expand P (t_{1}A_{1}+ ⋯+t_{k}A_{k}) as a polynomial in the variables {t_{j}}. 1/k! times the coefficient of t_{1} … t_{k} is the polarization of P. P is a multi-linear symmetric function of its arguments. For example, if P (A) = Tr(A^{3}), then the polarization is given by _{1 2} Tr(A_{1}A_{2}A_{3} + A_{2}A_{1}A_{3}) and P (A) = P (A, A, A). Fix a point x_{0} of M and choose a frame so ω(x_{0}) = dΩ(x_{0}) = 0. Then dP (Ω)(x_{0}) = dP (Ω,…, Ω)(x_{0}) = kP (dΩ, Ω,…, Ω)(x_{0}) = 0. Since x_{0} is arbitrary and since dP (Ω) is independent of the frame chosen this proves dP (Ω) = 0 which proves (a).

Let ∇_{t} = t∇_{1} + (1 − t)∇_{0} have connection 1-form ω_{t} = ω_{0} + tθ where θ = ω_{1} − ω_{0}. The transformation law for ω implies relative to a new frame: θ′ = ω_{′1} − ω′0 = (dh · h^{−1} + hω_{1}h^{−1}) − (dh · h^{−1} + hω_{0}h^{−1}) = h(ω^{1} − ω^{0})h^{−1} = hθh−1 so θ transforms like a tensor. This is of course nothing but the fact that the difference between two first order operators with the same leading symbol is a 0^{th} order operator. Let Ω_{t} be the curvature of the connection ∇_{t}. This is a matrix valued 2-form. Since θ is a 1-form, it commutes with 2-forms and we can define P (θ, Ω_{t},…, Ω_{t}) ∈ Λ^{2k−1}(T^{∗}M) by substitution. Since P is invariant, its complete polarization is also in- variant so P (h^{−1}θh, h^{−1}Ω_{t}h,…, h^{−1}Ω_{t}h) = P (θ, Ω_{t},…, Ω_{t}) is invariantly defined independent of the choice of frame. We compute that: P (∇_{1}) − P (∇_{0}) = ∫_{1 0} d dtP (Ω_{t},…, Ω_{t}) dt = k ∫_{1 0} P (Ω_{′t}, Ω_{t},…, Ω_{t}) dt. We define:  T P (∇_{1}, ∇_{0}) = k ∫_{1 0} P (θ, Ω_{t},…, Ω_{t}) dt. To complete the proof of the Lemma, it suffices to check dP (θ, Ω_{t},…, Ω_{t}) = P (Ω_{′t}, Ω_{t},…, Ω_{t}). Since both sides of the equation are invariantly defined, we can choose a suitable local frame to simplify the computation. Let x_{0} ∈ M and fix t_{0}. We choose a frame so ω_{t}(x_{0}, t_{0}) = 0 and dΩ_{t}(x_{0}, t_{0}) = 0. We compute: Ω′t = {dω^{0} + tdθ − ω^{t} ∧ ω^{t}}′ = dθ − ω_{′t} ∧ ω_{t} − ω_{t} ∧ ω′t Ω_{′t}(x_{0}, t_{0}) = dθ and  dP (θ, Ω_{t},…, Ω_{t})(x_{0}, t_{0}) = P (dθ, Ω_{t},…, Ω_{t})(x_{0}, t_{0}) = P (Ω_{′t}, Ω_{t},…, Ω_{t})(x_{0}, t_{0}) which completes the proof. T P is called the transgression of P and will play an important role in our discussion of secondary characteristic classes in Chapter 4 when we discuss the eta invariant with coefficients in a locally flat bundle.

We suppose that the matrix A = diag(λ_{1},…, λ_{k}) is diagonal. Then modulo suitable normalizing constants (_{i 2π} )^{j}, it is immediate that c_{j}(A) is the j^{th} elementary symmetric function of the λ_{i}’s since det(I + A) = ∏(1 + λ_{j}) = 1 + s_{1}(λ) + ⋯ + s_{k}(λ). If P (·) is any invariant polynomial, then P (A) is a symmetric function of the λ_{i}’s. The elementary symmetric functions form an algebra basis for the symmetric polynomials so there is a unique polynomial Q so that P (A) = Q(c_{1},…, c_{k})(A) i.e., we can decompose P as a polynomial in the c_{i}’s for diagonal matrices. Since P is invariant, this is true for diagonalizable matrices. Since the diagonalizable matrices are dense and P is continuous, this is true for all A. This proves: Lemma 2.1.3. Let P (A) be a invariant polynomial. There exists a unique polynomial Q so that P = Q(c_{1}(A),…, c_{k}(A)). It is clear that any polynomial in the c_{k}’s is invariant, so this completely characterizes the ring of invariant polynomials; it is a free algebra in k variables {c_{1},…, c_{k}}. If P (A) is homogeneous of degree k and is defined on k × k matrices, we shall see that if P (A) ≠ 0 as a polynomial, then there exists a holomor- phic manifold M so that if T_{c}(M) is the holomorphic tangent space, then ∫_{M} P (T_{c}(M)) ≠ 0 where dim M = 2k. This fact can be used to show that in general if P (A) ≠ 0 as a polynomial, then there exists a manifold M and a bundle V over M so P (V) ≠ 0 in cohomology. We can apply functorial constructions on connections. Define: ∇_{1} ⊕ ∇2 on C^{∞}(V_{1} ⊕ V_{2}) ; ∇_{1} ⊗ ∇2 on C^{∞}(V_{1} ⊗ V_{2}) ; ∇^{∗} 1  on C^{∞}(V_{∗ 1}) by:  (∇_{1} ⊕ ∇_{2})(s_{1} ⊕ s_{2}) = (∇_{1}s_{1}) ⊕ (∇_{2}s_{2}) with ω = ω_{1} ⊕ ω_{2} and Ω = Ω_{1} ⊕ Ω2 (∇_{1} ⊗ ∇_{2})(s_{1} ⊗ s_{2}) = (∇_{1}s_{1}) ⊗ (∇_{2}s_{2}) with ω = ω_{1} ⊗ 1 + 1 ⊗ ω_{2} and Ω = Ω_{1} ⊗ 1 + 1 ⊗ Ω2 (∇_{1}s_{1}, s_{∗ 1}) + (s_{1}, ∇_{∗ 1}s_{∗ 1}) = d(s_{1}, s_{∗ 1}) relative to the dual frame ω = −ω_{t1} and Ω = −Ω_{t1}.

94 2.1. Characteristic Classes In a similar fashion we can define an induced connection on Λ^{p}(V) (the bundle of p-forms) and S^{p}(V) (the bundle of symmetric p-tensors). If V has a given Hermitian fiber metric, the connection ∇ is said to be unitary or Riemannian if (∇s_{1}, s_{2}) + (s_{1}, ∇s_{2}) = d(s_{1}, s_{2}). If we identify V with V^{∗} using the metric, this simply means ∇ = ∇^{∗}. This is equivalent to the condition that ω is a skew adjoint matrix of 1-forms relative to a local orthonormal frame. We can always construct Riemannian connections lo- cally by taking ω = 0 relative to a local orthonormal frame and then using a partition of unity to construct a global Riemannian connection. If we restrict to Riemannian connections, then it is natural to consider polynomials P (A) which are defined for skew-Hermitian matrices A+A^{∗} = 0 and which are invariant under the action of the unitary group. Exactly the same argument as that given in the proof of Lemma 2.1.3 shows that we can express such a P in the form P (A) = Q(c_{1}(A),…, c_{k}(A)) so that the GL(·, C) and the U(·) characteristic classes coincide. This is not true in the real category as we shall see; the Euler form is a characteristic form of the special orthogonal group which can not be defined as a characteristic form using the general linear group GL(·, R). The Chern form and the Chern character satisfy certain identities with respect to functorial constructions. Lemma 2.1.4. (a) c(V_{1} ⊕ V_{2}) = c(V_{1})c(V_{2}) c(V^{∗}) = 1 − c_{1}(V) + c_{2}(V) − ⋯ + (−1)^{k}c_{k}(V). (b) ch(V_{1} ⊕ V_{2}) = ch(V_{1}) + ch(V_{2}) ch(V_{1}) ⊕ V_{2}) = ch(V_{1})ch(V_{2}). Proof: All the identities except the one involving c(V^{∗}) are immediate from the definition if we use the direct sum and product connections. If we fix a Hermitian structure on V, the identification of V with V^{∗} is conjugate linear. The curvature of ∇^{∗} is −Ω^{t} so we compute: c(V^{∗}) = det (I − i 2π Ω^{t}) = det (I − i 2π Ω) which gives the desired identity. If we choose a Riemannian connection on V, then Ω + Ω̄t = 0. This immediately yields the identities: ch(V) = ch(V) and c(V) = c(V)

so these are real cohomology classes. In fact the normalizing constants were chosen so c_{k}(V) is an integral cohomology class—i.e., if N_{2k} is any oriented submanifold of dimension 2k, then ∫_{N_{2}k} c_{k}(V) ∈ Z. The ch_{k}(V) are not integral for k > 1, but they are rational cohomology classes. As we shall not need this fact, we omit the proof. The characteristic classes give cohomological invariants of a vector bun- dle. We illustrate this by constructing certain examples over even dimen- sional spheres; these examples will play an important role in our later development of the Atiyah-Singer index theorem. Definition. A set of matrices \{e_{0},…, e_{m}\} are Clifford matrices if the e_{i} are self-adjoint and satisfy the commutation relations e_{i}e_{j} + e_{j}e_{i} = 2δij where δ_{ij} is the Kronecker symbol. For example, if m = 2 we can take: e^{0} = ( {1 0 \atop 0 −1} ) ,  e^{1} = ( {0 1 \atop 1 0} ) ,  e^{2} = ( {0 i \atop −i 0} ) to be the Dirac matrices. More generally, we can take the e_{i}’s to be given by the spin representations. If x ∈ R^{m+1}, we define: e(x) = _{∑ i} x_{i}e_{i}  so e(x)^{2} = |x|^{2}I. Conversely let e(x) be a linear map from R^{m} to the set of self-adjoint matrices with e(x) = |x|^{2}I. If \{v_{0},…, v_{m}\} is any orthonormal basis for R^{m}, \{e(v_{0}),…, e(v_{m})\} forms a set of Clifford matrices. If x ∈ S^{m}, we let Π_{±}(x) be the range of _{1 2} (1 + e(x)) = π_{±}(x). This is the span of the ±1 eigenvectors of e(x). If e(x) is a 2k × 2k matrix, then dim Π_{±}(x) = k. We have a decomposition S^{m} × C^{2k} = Π_{+} ⊕ Π_{−}. We project the flat connection on S^{m} × C^{2k} to the two sub-bundles to define connections ∇_{±} on Π_{±}. If e_{0 ±} is a local frame for Π_{±}(x_{0}), we define e_{±}(x) = π_{±}e_{0 ±} as a frame in a neighborhood of x_{0}. We compute ∇_{±}e_{±} = π_{±} dπ_{±}e_{0 ±},  Ω_{±}e_{±} = π_{±} dπ_{±} dπ_{±}e_{0 ±}. Since e_{0 ±} = e_{±}(x_{0}), this yields the identity: Ω_{±}(x_{0}) = π_{±} dπ_{±} dπ_{±}(x_{0}). Since Ω is tensorial, this holds for all x.

Let m = 2j be even. We wish to compute ch_{j}. Suppose first x_{0} = (1, 0,…, 0) is the north pole of the sphere. Then: π_{+}(x_{0}) = 1 2(1 + e_{0}) dπ_{+}(x_{0}) = 1 2 _{∑ i≥1} dx_{i}ei Ω_{+}(x_{0}) = 1 2(1 + e^{0})(1 2 ^{∑ i≥1} dx^{i}e^{i})2 Ω^{+}(x^{0})j = 1 2(1 + e^{0})(1 2 ^{∑ i≥1} dx^{i}e^{i})2j = 2^{−m−1} m!(1 + e_{0})(e_{1} … e_{m})(dx_{1} ∧ ⋯ ∧ dx_{m}). The volume form at x_{0} is dx_{1} ∧ ⋯ ∧ dx_{m}. Since e_{1} anti-commutes with the matrix e_{1} … e_{m}, this matrix has trace 0 so we compute: ch_{j}(Ω_{+})(x_{0}) = ( i 2π )^{j} 2^{−m−1} m! Tr(e_{0} … e_{m}) dvol(x_{0})/j! . A similar computation shows this is true at any point x_{0} of S^{m} so that: ∫_{S m} ch_{j}(Π_{+}) = ( i 2π )^{j} 2^{−m−1} m! Tr(e_{0} … e_{m}) vol(s^{m})/j! . Since the volume of S^{m} is j!2^{m+1}π^{j}/m! we conclude: Lemma 2.1.5. Let e(x) be a linear map from R^{m+1} to the set of self- adjoint matrices. We suppose e(x)^{2} = |x|^{2}I and define bundles Π_{±}(x) over S^{m} corresponding to the ±1 eigenvalues of e. Let m = 2j be even, then: ∫_{S m} ch_{j}(Π_{+}) = i^{j}2^{−j} Tr(e_{0} … e_{m}). In particular, this is always an integer as we shall see later when we consider the spin complex. If e^{0} = ( {1 0 \atop 0 −1} ) ,  e^{1} = ( {0 1 \atop 1 0} ) ,  e^{2} = ( {0 i \atop −i 0} ) then Tr(e_{0}e_{1}e_{2}) = −2i so ∫_{S 2} ch_{1}(Π_{+}) = 1 which shows in particular Π+ is a non-trivial line bundle over S^{2}.

Of a Complex Vector Bundle 97

There are other characteristic classes arising in the index theorem. These are most conveniently discussed using generating functions. Let x_{j} = i 2π λj be the normalized eigenvalues of the matrix A. We define: c(A) = _{k ∏ j=1}(1 + x_{j}) and ch(A) = _{k ∑ j=1} e^{xj} . If P (x) is a symmetric polynomial, we express P (x) = Q(c_{1}(x),…, c_{k}(x)) to show P (A) is a polynomial in the components of A. More generally if P is analytic, we first express P in a formal power series and then collect homogeneous terms to define P (A). We define: Todd class: Td(A) = ^{k ∏ j=1} xj 1 − e−xj = 1 + 1 2c_{1}(A) + 1 12(c_{2 1} + c_{2})(A) + 1 24c_{1}(A)c_{2}(A) + ⋯ The Todd class appears in the Riemann-Roch theorem. It is clear that it is multiplicative with respect to direct sum—Td(V_{1} ⊕ V_{2}) = Td(V_{1}) Td(V_{2}). The Hirzebruch L-polynomial and the Â polynomial will be discussed in the next subsection. These are real characteristic classes which also are defined by generating functions. If V is a bundle of dimension k, then V ⊕1 will be a bundle of dimension k+1. It is clear c(V ⊕1) = c(V) and Td(V ⊕1) = Td(V) so these are stable characteristic classes. ch(V) on the other hand is not a stable characteristic class since ch_{0}(V) = dim V depends explicitly on the dimension of V and changes if we alter the dimension of V by adding a trivial bundle.

2.2. Characteristic Classes of a Real Vector Bundle. Pontrjagin and Euler Classes. Let V be a real vector bundle of dimension k and let V_{c} = V ⊗ C denote the complexification of V. We place a real fiber metric on V to reduce the structure group from GL(k, R) to the orthogonal group O(k). We re- strict henceforth to Riemannian connections on V, and to local orthonor- mal frames. Under these assumptions, the curvature is a skew-symmetric matrix of 2-forms: Ω + Ω^{t} = 0. Since V_{c} arises from a real vector bundle, the natural isomorphism of V with V^{∗} defined by the metric induces a complex linear isomorphism of Vc with V_{∗ c} so c_{j}(V_{c}) = 0 for j odd by Lemma 2.1.4. Expressed locally, det (I + i 2π A) = det (I + i 2π A^{t}) = det (I − i 2π A) if A + A^{t} = 0 so c(A) is an even polynomial in A. To avoid factors of i we define the Pontrjagin form: p(A) = det (I + 1 2π A) = 1 + p_{1}(A) + p_{2}(A) + ⋯ where p_{j}(A) is homogeneous of order 2j in the components of A; the cor- responding characteristic class p_{j}(V) ∈ H^{4j}(M; R). It is immediate that: p_{j}(V) = (−1)^{j}c_{2j}(V_{c}) where the factor of (−1)^{j} comes form the missing factors of i. The set of skew-symmetric matrices is the Lie algebra of O(k). Let P (A) be an invariant polynomial under the action of O(k). We define P (Ω) for ∇ a Riemannian connection exactly as in the previous subsections. Then the analogue of Lemma 2.1.3 is: Lemma 2.2.1. Let P (A) be a polynomial in the components of a ma- trix A. Suppose P (A) = P (gAg^{−1}) for every skew-symmetric A and for every g ∈ O(k). Then there exists a polynomial Q(p_{1},…) so P (A) = Q(p_{1}(A),…) for every skew-symmetric A. Proof: It is important to note that we are not asserting that we have P (A) = Q(p_{1}(A),…) for every matrix A, but only for skew-symmetric A. For example, P (A) = Tr(A) vanishes for skew symmetric A but does not vanish in general.

It is not possible in general to diagonalize a skew-symmetric real matrix. We can, however, put it in block diagonal form: A = 0 −λ1 0 0 . . . . . . ; λ1 0 0 0 . . . . . . ; 0 0 0 −λ2 . . . . . . ; 0 0 λ2 0 . . . . . . ; . . . . . . . . . . . . . . . . . . . . . . . . . . ; . . . . . . . . . . . . . . . . . . . . . . . . . . If k is odd, then the last block will be a 1 × 1 block with a zero in it. We let x_{j} = −λ_{j}/2π; the sign convention is chosen to make the Euler form have the right sign. Then: p(A) = _{∏ j}(1 + x_{2 j}) where the product ranges from 1 through [_{k 2} ]. If P (A) is any invariant polynomial, then P is a symmetric function in the {x_{j}}. By conjugating A by an element of O(k), we can replace any xj by −x_{j} so P is a symmetric function of the {x_{2 j}}. The remainder of the proof of Lemma 2.2.1 is the same; we simply express P (A) in terms of the elementary symmetric functions of the {x_{2 j}}. Just as in the complex case, it is convenient to define additional charac- teristic classes using generating functions. The functions: z/ tanh z  and z{2 sinh(z/2)} are both even functions of the parameter z. We define: The Hirzebruch L-polynomial: L(x) = ^{∏ j} xj tanh xj = 1 + 1 3p_{1} + 1 45(7p_{2} − p_{2 1}) ⋯ The Â (A-roof ) genus: A(x) = ^{∏ j}  xj 2 sinh(x^{j}/2) = 1 − 1 24p_{1} + 1 5760(7p_{2 1} − 4p_{2}) + · · · . These characteristic classes appear in the formula for the signature and spin complexes.

For both the real and complex case, the characteristic ring is a pure polynomial algebra without relations. Increasing the dimension k just adds generators to the ring. In the complex case, the generators are the Chern classes {1, c_{1},…, c_{k}}. In the real case, the generators are the Pontrjagin classes {1, p_{1},…, p_{[k/2]}} where [k/2] is the greatest integer in k/2. There is one final structure group which will be of interest to us. Let V be a vector bundle of real dimension k. V is orientable if we can choose a fiber orientation consistently. This is equivalent to assuming that the real line bundle Λ^{k}(V) is trivial. We choose a fiber metric for V and an orientation for V and restrict attention to oriented orthonormal frames. This restricts the structure group from O(k) to the special orthogonal group SO(k). If k is odd, no new characteristic classes result from the restriction of the fiber group. We use the final 1 × 1 block of 0 in the representation of A to replace any λ_{i} by −λ_{i} by conjugation by an element of SO(k). If n is even, however, we cannot do this as we would be conjugating by a orientation reversing matrix. Let P (A) be a polymonial in the components of A. Suppose P (A) = P (gAg^{−1}) for every skew-symmetric A and every g ∈ SO(k). Let k = 2k̄ be even and let P (A) = P (x_{1},…, x_{k̄}). Fix g_{0} ∈ O(k) − SO(k) and define: P_{0}(A) = 1 2(P (A) + P (g_{0}Ag_{−1 0})) and P_{1}(A) = 1 2(P (A) − P (g_{0}Ag_{−1 0})). Both P_{0} and P_{1} are SO(k) invariant. P_{0} is O(k) invariant while P_{1} changes sign under the action of an element of O(k) − SO(k). We can replace x_{1} by −x_{1} by conjugating by a suitable orientation re- versing map. This shows: P_{1}(x_{1}, x_{2},…) = −P (−x_{1}, x_{2},…) so that x_{1} must divide every monomial of P_{0}. By symmetry, x_{j} divides every monomial of P_{1} for 1 ≤ j ≤ k̄ so we can express: P_{1}(A) = x_{1} … x_{k̄}P_{′ 1}(A) where P_{′ 1}(A) is now invariant under the action of O(k). Since P_{′ 1} is poly- nomial in the {x_{2 j}}, we conclude that both P_{0} and P_{′ 1} can be represented as polynomials in the Pontrjagin classes. We define: e(A) = x_{1} … x_{k}  so e(A)^{2} = det(A) = p_{k̄}(A) = _{∏ j} x^{2} j and decompose:  P = P_{0} + e(A)P_{′ 1}.

e(A) is a square root of the determinant of A. It is not, of course, immediate that e(A) is a polynomial in the com- ponents of A. Let {v_{i}} be an oriented orthonormal basis for R^{k}. We let Av_{i} = ∑_{j} A_{ij}v_{j} and define ω(A) = 1 2π _{∑ i<j} A_{ij}v_{i} ∧ v_{j} ∈ Λ^{2}(R^{k}). We let v_{1} ∧ ⋯ ∧ v_{k} be the orientation class of R^{k} and define: e(A) = (ω(A)^{k̄}, v_{1} ∧ ⋯ ∧ v_{k})/k! where ( , ) denotes the natural inner product on Λ^{k}R^{k} ≃ R. It is clear from the definition that e(A) is invariant under the action of SO(k) since ω(A) is invariantly defined for skew-symmetric matrices A. It is clear that e(A) is polynomial in the components of A. If we choose a block basis so that: Av_{1} = λ_{1}v_{2}, Av_{2} = −λ_{1}v_{1}, Av_{3} = λ_{2}v_{4}, Av_{4} = −λ_{2}v_{3},… then we compute: ω(A) = {−λ_{1}v_{1} ∧ v_{2} − λ_{2}v_{3} ∧ v_{4} … }/2π = x_{1}v_{1} ∧ v_{2} + x_{2}v_{3} ∧ v_{4} ⋯ e(A) = x_{1}x_{2} … This new characteristic class is called the Euler class. While the Pontrja- gin classes can be computed from the curvature of an arbitrary connection, the Euler class can only be computed from the curvature of a Riemannian connection. If Ω_{ij} are the matrix components of the curvature of V relative to some oriented orthomormal basis, then: e(Ω) = (−4π)^{−k̄}/k!_{∑ ρ} sign(ρ)Ω_{ρ(1)ρ(2)} … Ω_{ρ(k−1)ρ(k)} ∈ Λ^{k}(M) for 2k̄ = k = dim(V). The sum ranges over all possible permutations ρ. We define e(V) = 0 if dim(V) is odd. It is immediate that e(V_{1} ⊕ V_{2}) = e(V_{1})e(V_{2}) if we give the natural orientation and fiber metric to the direct sum and if we use the direct sum connection. We illustrate this formula (and check that we have the correct normal- izing constants) by studying the following simple example. Let m = 2 and let M = S^{2} be the unit sphere. We calculate e(T S^{2}). Parametrize M using spherical coordinates f(u, v) = (cos u sin v, sin u sin v, cos v) for 0 ≤ u ≤ 2π and 0 ≤ v ≤ π. Define a local orthonormal frame for T (R^{3}) over S^{2} by: e1 = (sin v)^{−1}∂/∂u = (− sin u, cos u, 0) e2 =  ∂/∂v = (cos u cos v, sin u cos v, − sin v) e3 =  N = (cos u sin v, sin u sin v, cos v).

The Euclidean connection ∇ is easily computed to be: ∇e1e1 = (sin v)^{−1}(− cos u, − sin u, 0) = −(cos v/ sin v)e_{2} − e3 ∇e1e2 = (cos v/ sin v)(− sin u, cos u, 0) = (cos v/ sin v)e1 ∇e2e1 = 0 ∇e2e2 = (− cos u sin v, − sin u sin v, − cos v) = −e_{3}. Covariant differentiation ∇ on the sphere is given by projecting back to T (S^{2}) so that: ∇_{e1}e1 = (− cos v/ sin v)e2  ∇_{e1}e2 = (cos v/ sin v)e1 ∇_{e2}e1 = 0  ∇_{e2}e2 = 0 and the connection 1-form is given by: ∇e1 = (− cos v/ sin v)e^{1} ⊗ e2 so ω^{11} = 0 and ω^{12} = −(cos v/ sin v)e1 ∇e2 = (cos v/ sin v)e^{1} ⊗ e1 so ω_{22} = 0 and ω_{21} = (cos v/ sin v)e^{1}. As du = e^{1}/ sin v we compute ω_{12} = − cos v du so Ω_{12} = sin v dv ∧ du = −e^{1} ∧ e^{2} and Ω_{21} = e^{1} ∧ e^{2}. From this we calculate that: e(Ω) = − 1 4π (Ω^{12} − Ω^{21}) = e^{1} ∧ e2 2π and consequently ∫_{S 2} E_{2} = vol(S^{2})/2π = 2 = χ(S^{2}). There is a natural relation between the Euler form and c_{k̄}. Let V be a complex vector space of dimension k̄ and let V_{r} be the underlying real vector space of dimention 2k̄ = k. If V has a Hermitian inner product, then V_{r} inherits a natural real inner product. V_{r} also inherits a natural orientation from the complex structure on V. If {e_{j}} is a unitary basis for V, then {e_{1}, ie_{1}, e_{2}, ie_{2},… } is an oriented orthonormal basis for V_{r}. Let A be a skew-Hermitian matrix on V. The restriction of A to V_{r} defines a skew- symmetric matrix A_{r} on V_{r}. We choose a basis {e_{j}} for V so Ae_{j} = iλ_{j}e_{j}. Then:  x_{j} = −λ_{j}/2π  and c_{k̄}(A) = x_{1} … x_{}k̄ defines the k̄th Chern class. If v_{1} = e_{1}, v_{2} = ie_{1}, v_{3} = e_{2}, v_{4} = ie_{2},… then: A_{r}v_{1} = λ_{1}v_{2}, A_{r}v_{2} = −λ_{1}v_{1}, A_{r}v_{3} = λ_{2}v_{4}, A_{r}v_{4} = −λ_{2}v_{3},… so that  e(A_{r}) = c_{k̄}(A). We summarize these results as follows:

Pontrjagin and Euler Classes 103

Lemma 2.2.2. Let P (A) be a polynomial with P (A) = P (gAg^{−1}) for every skew symmetric A and every g ∈ SO(k). (a) If k is odd, then P (A) is invariant under O(k) and is expressible in terms of Pontrjagin classes. (b) If k = 2k̄ is even, then we can decompose P (A) = P_{0}(A) + e(A)P_{1}(A) where P_{i}(A) are O(k) invariant and are expressible in terms of Pontrjagin classes. (c) e(A) is defined by: e(A) = (−4π)^{−k̄} _{∑ ρ} sign(ρ)A_{ρ(1)ρ(2)} … A_{ρ(k−1)ρ(k)} /k! This satisfies the identity e(A)^{2} = p_{k̄}(A). (We define e(A) = 0 if k is odd). This completely describes the characteristic ring. We emphasize the conclusions are only applicable to skew-symmetric real matrices A. We have proved that e(A) has the following functorial properties: Lemma 2.2.3. (a) If we take the direct sum connection and metric, then e(V_{1} ⊕ V_{2}) = e(V_{1})e(V_{2}). (b) If we take the metric and connection of V_{r} obtained by forgetting the complex structure on a complex bundle V then e(V_{r}) = c_{k̄}(V). This lemma establishes that the top dimensional Chern class c_{k̄} does not really depend on having a complex structure but only depends on having an orientation on the underlying real vector bundle. The choice of the sign in computing det(A)^{1/2} is, of course, motivated by this normalization.

2.3. Characteristic Classes of Complex Projective Space. So far, we have only discussed covariant differentiation from the point of view of the total covariant derivative. At this stage, it is convenient to introduce covariant differentiation along a direction. Let X ∈ T (M) be a tangent vector and let s ∈ C^{∞}(V) be a smooth section. We define ∇_{X}s ∈ C^{∞}(V) by:  ∇_{X}s = X · ∇s where “·” denotes the natural pairing from T (M) ⊗ T^{∗}(M) ⊗ V → V. Let [X, Y] = XY − Y X denote the Lie-bracket of vector fields. We define: Ω(X, Y) = ∇_{X}∇_{Y} − ∇_{Y} ∇_{X} − ∇[X,Y ] and compute that: Ω(fX, Y)s = Ω(X, fY)s = Ω(X, Y)fs = fΩ(X, Y)s for f ∈ C^{∞}(M), s ∈ C^{∞}(V), and X, Y ∈ C^{∞}(T M). If {e_{i}} is a local frame for T (M), let {e^{i}} be the dual frame for T^{∗}(M). It is immediate that:  ∇s = _{∑ i} e^{i} ⊗ ∇_{ei}(s) so we can recover the total covariant derivative from the directional deriva- tives. We can also recover the curvature: Ωs = _{∑ i<j} e^{i} ∧ e^{j} ⊗ Ω(e_{i}, e_{j})s. If V = T (M), there is a special connection called the Levi-Cività con- nection on T (M). It is the unique Riemannian connection which is torsion free—i.e., (∇_{X}Y, Z) + (Y, ∇_{X}Z) = X(Y, Z) ∇_{X}Y − ∇_{Y} X − [X, Y] = 0  (Riemannian) (Torsion free). Let x = (x_{1},…, x_{m}) be a system of local coordinates on M and let {∂/∂x_{i}} be the coordinate frame for T (M). The metric tensor is given by: ds^{2} = _{∑ i,j} g_{ij} dx^{i} ⊗ dx^{j}  for g_{ij} = (∂/∂x_{i}, ∂/∂x_{j}). We introduce the Christoffel symbols Γ_{k ij} and Γ_{ijk} by: ∇_{∂/∂xi} ∂/∂x_{j} = _{∑ k} Γ_{k ij}∂/∂xk (∇_{∂/∂xi} ∂/∂x_{j}, ∂/∂x_{k}) = Γ_{ijk}.

Characteristic Classes 105

They are related by the formula: Γijk = ∑ l Γlijglk,  Γk ij = ∑ l Γijlglk where g^{lk} = (dx_{l}, dx_{k}) is the inverse of the matrix g_{ij}. It is not difficult to compute that:  Γ_{ijk} = _{1 2} {g_{jk/i} + g_{ik/j} − g_{ij/k}} where we use the notation “/” to denote (multiple) partial differentiation. The complete curvature tensor is defined by: Ω(∂/∂x_{i}, ∂/∂x_{j})∂/∂x_{k} = _{∑ l} R_{ijk}^{l}∂/∂xl or equivalently if we lower indices: (Ω(∂/∂x_{i}, ∂/∂x_{j})∂/∂x_{k}, ∂/∂x_{l}) = R_{ijkl}. The expression of the curvature tensor in terms of the derivatives of the metric is very complicated in general. By making a linear change of coordi- nates we can always normalize the metric so g_{ij}(x_{0}) = δ_{ij} is the Kronecker symbol. Similarly, by making a quadratic change of coordinates, we can further normalize the metric so g_{ij/k}(x_{0}) = 0. Relative to such a choice of coordinates:  Γ_{k ij}(x_{0}) = Γ_{ijk}(x_{0}) = 0 and R_{ijkl}(x_{0}) = Rijk^{l}(x_{0}) = _{1 2} {g_{jl/ik} + g_{ik/lj} − g_{jk/il} − g_{il/jk}}(x_{0}). At any other point, of course, the curvature tensor is not as simply ex- pressed. In general it is linear in the second derivatives of the metric, quadratic in the first derivatives of the metric with coefficients which de- pend smoothly on the g_{ij}’s. We choose a local orthonormal frame {e_{i}} for T^{∗}(M). Let m = 2m̄ be even and let orn = ∗ · 1 = dvol be the oriented volume. Let e = E_{m}(g) orn be the Euler form. If we change the choice of the local orientation, then e and orn both change signs so E_{m}(g) is scalar invariant of the metric. In terms of the curvature, if 2m̄ = m, then: c(m) = (−1)m̄(8π)−m̄ 1 (m̄)! Em = c(m) ∑ ρ,τ sign(ρ) sign(τ)Rρ(1)ρ(2)τ(1)τ(2) … Rρ(m−1)ρ(m)τ(m−1)τ(m)

106 2.3. Characteristic Classes where the sum ranges over all permutations ρ, τ of the integers 1 through m. For example: E2 = −(2π)^{−1}R1212 E4 = (2π)^{−2} _{∑ i,j,k,l}{R_{ijij}R_{klkl} + R_{ijkl}R_{ijkl} − 4R_{ijik}R_{ljlk}}/8. Let dvol be the Riemannian measure on M. If M is oriented, then ∫_{M} E_{m} dvol = ∫_{M} e is independent of the orientation of M and of the metric. If M is not orientable, we pass to the double cover to see ∫_{M} E_{m} dvol is a topological invariant of the manifold M. We shall prove later this integral is the Euler characteristic χ(M) but for the moment simply note it is not dependent upon a choice of orientation of M and is in fact defined even if M is not orientable. It is worth computing an example. We let S^{2} be the unit sphere in R^{3}. Since this is homogeneous, E_{2} is constant on S^{2}. We compute E_{2} at the north pole (0, 0, 1) and parametrize S^{2} by (u, v, (1 − u^{2} − v^{2})^{1/2}). Then ∂/∂u = (1, 0, −u/(1 − u^{2} − v^{2})^{1/2}), ∂/∂v = (0, 1, −v/(1 − u^{2} − v^{2})^{1/2}) g11 = 1 + u^{2}/(1 − u^{2} − v^{2}) g22 = 1 + v^{2}/(1 − u^{2} − v^{2}) g12 = uv/(1 − u^{2} − v^{2}). It is clear g_{ij}(0) = δ_{ij} and g_{ij/k}(0) = 0. Therefore at u = v = 0, E2 = −(2π)^{−1}R_{1212} = −(2π)^{−1}{g_{11/22} + g_{22/11} − 2g_{12/12} }/2 = −(2π)^{−1}{0 + 0 − 2}/2 = (2π)−1 so that ∫_{S 2} E_{2} dvol = (4π)(2π)^{−1} = 2 = χ(S^{2}). More generally, we can let M = S^{2} × ⋯ × S^{2} where we have m̄ factors and 2m̄ = m. Since e(V_{1} ⊕ V_{2}) = e(V_{1})e(V_{2}), when we take the product metric we have E_{m} = (E_{2})^{m̄} = (2π)^{−m̄} for this example. Therefore: ∫_{S^{2}×···×S 2} E_{m} dvol = 2^{m̄} = χ(S^{2} × ⋯ × S^{2}). We will use this example later in this chapter to prove ∫_{M} E_{m} = χ(M) in general. The natural examples for studying the Euler class are products

of two dimensional spheres. Unfortunately, the Pontrjagin classes vanish identically on products of spheres so we must find other examples. It is convenient at this point to discuss the holomorphic category. A manifold M of real dimension m = 2m̄ is said to be holomorphic if we have local coordinate charts z = (z_{1},…, z_{m̄}): M → C^{m̄} such that two charts are related by holomorphic changes of coordinates. We expand z_{j} = x_{j} + iyj and define: ∂/∂zj = _{1 2} (∂/∂x_{j} − i∂/∂y_{j}), dzj = dx_{j} + i dy_{j},  ∂/∂z̄j = _{1 2} (∂/∂x_{j} + i∂/∂y_{j}) dz̄j = dx_{j} + i dy_{j}. We complexify T (M) and T^{∗}(M) and define: T_{c}(M) = span{∂/∂z_{j}}, Λ^{1,0}(M) = span{dz_{j}}, Λ^{0,1}(M) = span{dz̄j}. Then  Λ^{1}(M) = Λ^{1,0}(M) ⊕ Λ^{0,1}(M) T_{c}(M)∗ = Λ^{1,0}(M). As complex bundles, T_{c}(M) ≃ Λ^{0,1}(M). Define: ∂(f) = _{∑ j} ∂f/∂z_{j} dz^{j}: C^{∞}(M) → C^{∞}(Λ^{1,0}(M)) ¯∂(f) = _{∑ j} ∂f/∂z̄j dz̄j: C^{∞}(M) → C^{∞}(Λ^{0,1}(M)) then the Cauchy-Riemann equations show f is holomorphic if and only if̄ ∂f = 0. This decomposes d = ∂ + ¯∂ on functions. More generally, we define: Λp,q = span{dz_{i1} ∧ ⋯ ∧ dz_{ip} ∧ dz̄j1 ∧ ⋯ ∧ dz̄jq} Λn = _{⨁ p+q=n} Λ^{p,q}. This spanning set of Λ^{p,q} is closed. We decompose d = ∂ + ¯∂ where ∂: C^{∞}(Λ^{p,q}) → C^{∞}(Λ^{p+1,q})  and  ¯∂: C^{∞}(Λ^{p,q}) → C^{∞}(Λ^{p,q+1} ) so that ∂∂ = ¯∂ ¯∂ = ¯∂∂ + ∂ ¯∂ = 0. These operators and bundles are all invariantly defined independent of the particular holomorphic coordinate system chosen.

108 2.3. Characteristic Classes A bundle V over M is said to be holomorphic if we can cover M by holomorphic charts U_{α} and find frames s_{α} over each U_{α} so that on U_{α} ∩Uβ the transition functions s_{α} = f_{αβ}s_{β} define holomorphic maps to GL(n, C). For such a bundle, we say a local section s is holomorphic if s = ∑_{ν} a_{ν α}s^{ν} α for holomorphic functions a_{ν α}. For example, T_{c}(M) and Λ^{p,0}(M) are holo- morphic bundles over M; Λ^{0,1} is anti-holomorphic. If V is holomorphic, we use a partition of unity to construct a Hermitian fiber metric h on V. Let h_{α} = (s_{α}, s_{α}) define the metric locally; this is a positive definite symmetric matrix which satisfies the transition rule h_{α} = f_{αβ}h_{β}f̄αβ. We define a connection 1-form locally by: ω_{α} = ∂h_{α}h^{−1} α and compute the transition rule: ω_{α} = ∂{f_{αβ}h_{β}f̄αβ}f̄ −1 αβ h_{−1 β}f_{−1 αβ}. Since f_{αβ} is holomorphic, ¯∂f_{αβ} = 0. This implies df_{αβ} = ∂f_{αβ} and ∂f̄αβ = 0 so that: ω_{α} = ∂f_{αβ}f_{−1 αβ} + f_{αβ}∂h_{β}h_{−1 β}f_{−1 αβ} = df_{αβ}f_{−1 αβ} + f_{αβ}ω_{β}f_{−1 αβ}. Since this is the transition rule for a connection, the {ω_{α}} patch together to define a connection ∇_{h}. It is immediate from the definition that: (∇_{h}s_{α}, s_{α}) + (s_{α}, ∇_{h}s_{α}) = ω_{α}h_{α} + h_{α}ω_{∗ α} = ∂h_{α} + ¯∂h_{α} = dhα so ∇_{h} is a unitary connection on V. Since ∇_{h}s_{α} ∈ C^{∞}(Λ^{1,0} ⊗ V) we conclude ∇_{h} vanishes on holomorphic sections when differentiated in anti- holomorphic directions (i.e., ∇_{h} is a holomorphic connection). It is easily verified that these two properties determine ∇_{h}. We shall be particularly interested in holomorphic line bundles. If L is a line bundle, then h_{α} is a positive function on U_{α} with h_{α} = |f_{αβ}|^{2}h_{β}. The curvature in this case is a 2-form defined by: Ω_{α} = dω_{α} − ω_{α} ∧ ω_{α} = d(∂h_{α}h_{−1 α}) = ¯∂∂ log(h_{α}) = −∂ ¯∂ log(h_{α}). Therefore: c_{1}(L) = 1 2πi∂ ¯∂ log(h) is independent of the holomorphic frame chosen for evaluation. c_{1}(L) ∈ C^{∞}(Λ^{1,1}(M)) and dc_{1} = ∂c_{1} = ¯∂c_{1} = 0 so c_{1}(L) is closed in all possible senses.

Let CP_{n} be the set of all lines through 0 in C^{n+1}. Let C^{∗} = C − 0 act on C^{n+1} − 0 by complex multiplication, then CP_{n} = (C^{n+1} − 0)/C^{∗} and we give CP_{n} the quotient topology. Let L be the tautological line bundle over CP_{n}:  L = { (x, z) ∈ CP_{n} × C^{n+1} : z ∈ x }. Let L^{∗} be the dual bundle; this is called the hyperplane bundle. Let (z_{0},…, z_{n}) be the usual coordinates on C^{n+1}. Let U_{j} = { z : z_{j} ≠ 0 }. Since U_{j} is C^{∗} invariant, it projects to define an open set on CP_{n} we again shall denote by U_{j}. We define z_{j k} = z_{k}/zj over U_{j}. Since these functions are invariant under the action of C^{∗}, they extend to continuous functions on U_{j}. We let z^{j} = (z_{j 0},…, z^{j} j ,…, z_{j n}) where we have deleted z_{j j} = 1. This gives local coordinates on U_{j} in CP_{n}. The transition relations are: z_{j ν} = (z_{k j})^{−1}z^{k} ν which are holomorphic so CP_{n} is a holomorphic manifold. We let s^{j} = (z^{j 0},…, z^{j n}) be a section to L over U^{j}. Then s^{j} = (z^{k j})^{−1}sk transform holomorphically so L is a holomorphic line bundle over CP_{n}. The coordinates z_{j} on C^{n+1} give linear functions on L and represent global holomorphic sections to the dual bundle L^{∗}. There is a natural inner product on the trivial bundle CP_{n} × C^{n+1} which defines a fiber metric on L. We define:  x = −c_{1}(L) = i 2π ∂ ¯∂ log(1 + |z^{j}|^{2}) then this is a closed 2-form over CP_{n}. U(n + 1) acts on C^{n+1}; this action induces a natural action on CP_{n} and on L. Since the metric on L arises from the invariant metric on C^{n+1}, the 2-form x is invariant under the action of U(n + 1). Lemma 2.3.1. Let x = −c_{1}(L) over CP_{n}. Then: (a) ∫_{CPn} x^{n} = 1. (b) H^{∗}(CP_{n}; C) is a polynomial ring with additive generators {1, x,…, x^{n}}. (c) If i: CP_{n−1} → CP_{n} is the natural inclusion map, then i^{∗}(x) = x. Proof: (c) is immediate from the naturality of the constructions involved. Standard methods of algebraic topology give the additive structure of H^{∗}(CP_{n}; C) = C ⊕ 0 ⊕ C ⊕ 0 ⋯ ⊕ C. Since x ∈ H^{2}(CP_{n}; C) satisfies x^{n} ≠ 0, (a) will complete the proof of (b). We fix a coordinate chart U_{n}. Since CP_{n} − U_{n} = CP_{n−1}, it has measure zero. It suffices to check that: ∫_{Un} x^{n} = 1.

110 2.3. Characteristic Classes We identify z ∈ U_{n} = C^{n} with (z, 1) ∈ C^{n+1}. We imbed U(n) in U(n + 1) as the isotropy group of the vector (0,…, 0, 1). Then U(n) acts on (z, 1) in C^{n+1} exactly the same way as U(n) acts on z in C^{n}. Let dvol = dx^{1} ∧ dy^{1} ∧ ⋯ ∧ dx^{n} ∧ dy^{n} be ordinary Lebesgue measure on Cn without any normalizing constants of 2π. We parametrize C^{n} in the form (r, θ) for 0 ≤ r < θ and θ ∈ S^{2n−1} . We express x^{n} = f(r, θ) dvol. Since x is invariant under the action of U(n), f(r, θ) = f(r) is spherically symmetric and does not depend on the parameter θ. We compute that: x = − 1 2πi∂ ¯∂ log (1 + ∑ z^{j}z̄j) = − 1 2πi∂(^{∑ j} z^{j} dz̄j/(1 + |z|^{2})) = − 1 2πi{_{∑ j} dz_{j} ∧ dz̄j/(1 + r^{2}) −_{∑ j,k} z_{j}z̄k dz_{k} ∧ dz̄j/(1 + r^{2})^{2}}. We evaluate at the point z = (r, 0,…, 0) to compute: x = − 1 2πi{^{∑ j} dz^{j} dz̄j/(1 + r^{2}) − r^{2} dz^{1} ∧ dz̄1/(1 + r^{2})^{2}} = − 1 2πi{dz_{1} ∧ dz̄1/(1 + r^{2})^{2} +_{∑ j>1} dz_{j} ∧ dz̄j/(1 + r^{2})}. Consequently at this point, xn = (− 1 2πi)^{n} n!(1 + r^{2})^{−n−1} dz_{1} ∧ dz̄1 ∧ ⋯ ∧ dz_{n} ∧ dz̄n = π^{−n}n!(1 + r^{2})^{−n−1} dx_{1} ∧ dy_{1} ∧ ⋯ ∧ dx_{n} ∧ dyn = π^{−n}n!(1 + r^{2})^{−n−1} dvol . We use the spherical symmetry to conclude this identitiy holds for all z. We integrate over U_{n} = C^{n} and use spherical coordinates: ∫ xn = n! π^{−n} ∫ (1 + r^{2})^{−n−1} dvol = n! π^{−n} ∫ (1 + r^{2})^{−n−1}r^{2n−1} dr dθ = n! π^{−n} vol(S^{2n−1} ) ∫_{∞ 0}(1 + r^{2})^{−n−1}r^{2n−1} dr = 1 2n! π^{−n} vol(S^{2n−1} ) ∫_{∞ 0}(1 + t)^{−n−1}t^{n−1} dt. We compute the volume of S^{2n−1} using the identity √π = ∫_{∞ −∞} e^{−t2} dt. Thus: πn = ∫ e^{−r2} dvol = vol(S^{2n−1} ) ∫_{∞ 0} r^{2n−1} e^{−r2} dr = 1 2 vol(S^{2n−1} ) ∫_{∞ 0} t^{n−1}e^{−t} dt = (n − 1)! 2 vol(S^{2n−1} ).

We solve this for vol(S^{2n−1} ) and substitute to compute: ∫ xn = n ∫_{∞ 0}(1 + t)^{−n−1}t^{n−1} dt = (n − 1) ∫_{∞ 0}(1 + t)^{−n}t^{n−2} dt = ⋯ = ∫_{∞ 0}(1 + t)^{−2} dt = 1 which completes the proof. x can also be used to define a U(n + 1) invariant metric on CP_{n} called the Fubini-Study metric. We shall discuss this in more detail in Chapter 3 as this gives a Kaehler metric for CP_{n}. There is a relation between L and Λ^{1,0}(CP_{n}) which it will be convenient to exploit: Lemma 2.3.2. Let M = CP_{n}. (a) There is a short exact sequence of holomorphic bundles: 0 → Λ^{1,0}(M) → L ⊗ 1^{n+1} → 1 → 0. (b) There is a natural isomorphism of complex bundles: T_{c}(M) ⊕ 1 ≃ L^{∗} ⊗ 1^{n+1} = _{L^{∗} ⊕ ⋯ ⊕ L∗ n+1 times}. Proof: Rather than attempting to give a geometric proof of this fact, we give a combinatorial argument. Over U_{k} we have functions z_{k i} which give coordinates 0 ≤ i ≤ n for i ≠ k. Furthermore, we have a section s^{k} to L. We let {s_{k i}}_{n i=0} give a frame for L ⊗ 1^{n+1} = L ⊕ ⋯ ⊕ L. We define F: Λ^{1,0}(M) → L ⊗ 1^{n+1} on U_{k} by: F (dz_{k i}) = s_{k i} − z_{k i}s_{k k}. We note that z_{k k} = 1 and F (dz_{k k}) = 0 so this is well defined on U_{k}. On the overlap, we have the relations z_{k i} = (z_{j k})^{−1}z_{j i} and: s_{k i} = (z_{j k})^{−1}s_{ji} and dz_{k i} = (z_{j k})^{−1} dz_{j i} − (z_{j k})^{−2}z_{j i} dz_{j k}. Thus if we compute in the coordinate system U_{j} we have: F (dz_{k i}) = (z_{j k})^{−1}(s_{ji} − z_{j i}s_{jj}) − (z_{j k})^{−2}z_{j i}(s_{jk} − z_{j k}s_{jj}) = s_{k i} − z_{j i}s_{k j} − z_{k i}s_{k k} + z_{j i}s_{k j} = s_{k i} − z_{k i}s^{k} k which agrees with the definition of F on the coordinate system U_{k}. Thus F is invariantly defined. It is clear F is holomorphic and injective. We let

ν = L ⊗ 1^{n+1}/ image(F), and let π be the natural projection. It is clear that s_{k k} is never in the image of F so πs_{k k} ≠ 0. Since F (dz_{k j}) = s_{k j} − z_{k j}s_{k k} = z_{k j}(s_{jj} − s_{k k}) we conclude that πs_{jj} = πs_{k k} so s = πs_{k k} is a globally defined non-zero section to ν. This completes the proof of (a). We dualize to get a short exact sequence: 0 → 1 → L^{∗} ⊗ 1^{n+1} → T_{c}(M) → 0. These three bundles have natural fiber metrics. Any short exact sequence of vector bundles splits (although the splitting is not holomorphic) and this proves (b). From assertion (b) it follows that the Chern class of T_{c}(CP_{n}) is given by: c(T_{c}) = c(T_{c} ⊕ 1) = c(L^{∗} ⊕ ⋯ ⊕ L^{∗}) = c(L^{∗})^{n+1} = (1 + x)^{n+1}. For example, if m = 1, then c(T_{c}) = (1 + x)^{2} = 1 + 2x. In this case, c_{1}(T_{c}) = e(T (S^{2})) and we computed ∫_{S 2} e(T (S)) = 2 so ∫_{S 2} x = 1 which checks with Lemma 2.3.1. If we forget the complex structure on T_{c}(M) when M is holomorphic, then we obtain the real tangent space T (M). Consequently: T (M) ⊗ C = T_{c} ⊕ T^{∗} c and  c(T (CP_{n}) ⊗ C) = c(T_{c}(CP_{n}) ⊕ T_{∗ c}(CP_{n})) = (1 + x)^{n+1}(1 − x)^{n+1} = (1 − x^{2})^{n+1}. When we take into account the sign changes involved in defining the total Pontrjagin form, we conclude: Lemma 2.3.3. (a) If M = S^{2}× ⋯×S^{2} has dimension 2n, then ∫_{M} e(T (M)) = χ(M) = 2^{n}. (b) If M = CP_{n} has dimension 2n, then c(T_{c}(M)) = (1 + x)^{n+1}  and p(T (M)) = (1 + x^{2})^{n+1}. x ∈ H^{2}(CP_{n}; C) is the generator given by x = c_{1}(L^{∗}) = −c_{1}(L), L is the tautological line bundle over CP_{n}, and L^{∗} is the dual, the hyperplane bundle. The projective spaces form a dual basis to both the real and complex characteristic classes. Let ρ be a partition of the positive integer k in the

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form k = i_{1} + ⋯ + i_{j} where we choose the notation so i_{1} ≥ i_{2} ≥ · · ·. We let π(k) denote the number of such partitions. For example, if k = 4 then π(4) = 5 and the possible partitions are: 4 = 4, 4 = 3 + 1, 4 = 2 + 2, 4 = 2 + 1 + 1, 4 = 1 + 1 + 1 + 1. We define classifying manifolds: M^{c ρ} = CP^{i1} × ⋯ × CP^{ij} and M^{r ρ} = CP^{2i1} × ⋯ × CP2ij to be real manifolds of dimension 2k and 4k. Lemma 2.3.4. Let k be a positive integer. Then: (a) Let constants c(ρ) be given. There exists a unique polynomial P (A) of degree k in the components of a k × k complex matrix which is GL(k, C) invariant such that the characteristic class defined by P satisfies: ∫_{M^{c} ρ} P (T_{c}(M_{c ρ})) = c(ρ) for every such partition ρ of k. (b) Let constants c(ρ) be given. There exists a polynomial P (A) of degree 2k in the components of a 2k ×2k real matrix which is GL(2k, R) invariant such that the characteristic class defined by P satisfies: ∫_{M^{r} ρ} P (T (M_{r ρ})) = c(ρ) for every such partition ρ of k. If P′ is another such polynomial, then P (A) = P′(A) for every skew-symmetric matrix A. In other words, the real and complex characteristic classes are completely determined by their values on the appropriate classifying manifolds. Proof: We prove (a) first. Let P_{k} denote the set of all such polynomials P (A). We define c_{ρ} = c_{i1} … c_{ij} ∈ P_{k}, then by Lemma 2.1.3 the {c_{ρ}} form a basis for P_{k} so dim(P_{k}) = π(k). The {c_{ρ}} are not a very convenient basis to work with. We will define instead: H_{ρ} = ch_{i1} … ch_{ij} ∈ Pk and show that the matrix: a(ρ, τ) = ∫_{M^{c} τ} H_{ρ}(T_{c}(M_{c τ})) is a non-singular matrix. This will prove the H_{ρ} also form a basis for Pk and that the M_{c τ} are a dual basis. This will complete the proof.

The advantage of working with the Chern character rather than with the Chern class is that: ch_{i}(T_{c}(M_{1} × M_{2})) = ch_{i}(T_{c}M_{1} ⊕ T_{c}M_{2}) = ch_{i}(T_{c}M_{1}) + ch_{i}(T_{c}M_{2}). Furthermore, ch_{i}(T_{c}M) = 0 if 2i > dim(M). We define the length ℓ(ρ) = j to be the number of elements in the partition ρ. Then the above remarks imply:  a(ρ, τ) = 0  if ℓ(τ) > ℓ(ρ). Furthermore, if ℓ(τ) = ℓ(ρ), then a(ρ, τ) = 0 unless τ = ρ. We define the partial order τ > ρ if ℓ(τ) > ℓ(ρ) and extend this to a total order. Then a(ρ, τ) is a triangular matrix. To show it is invertible, it suffices to show the diagonal elements are non-zero. We first consider the case in which ρ = τ = k. Using the identity T_{c}(CP_{k})⊕1 = (L^{∗} ⊕ ⋯⊕L^{∗}) (k +1 times), it is clear that ch_{k}(T_{c}CP_{k}) = (k + 1)ch_{k}(L^{∗}). For a line bundle, ch_{k}(L^{∗}) = c_{1}(L^{∗})^{k}/k! . If x = c_{1}(L^{∗}) is the generator of H^{2}(CP_{k}; C), then ch_{k}(T_{c}CP_{k}) = (k + 1)x^{k}/k! which does not integrate to zero. If ρ = τ = {i_{1},…, i_{j}} then: a(ρ, ρ) = c_{ℓ(ρ) ∏ ν=1} a(i_{ν}, i_{ν}) where c is a positive constant related to the multiplicity with which the iν appear. This completes the proof of (a). To prove (b) we replace M_{c ρ} by M_{c 2ρ} = M_{r ρ} and H_{ρ} by H_{2ρ}. We compute ch_{2i}(T (M)) ^{def} = ch_{2i}(T (M) ⊗ C) = ch_{2i}(T_{c}M ⊕ T_{∗ c}M) = ch_{2i}(T_{c}M) + ch_{2i}(T_{∗ c}M) = 2ch_{2i}(T_{c}M). Using this fact, the remainder of the proof of (b) is immediate from the calculations performed in (a) and this completes the proof. The Todd class and the Hirzebruch L-polynomial were defined using generating functions. The generating functions were chosen so that they would be particularly simple on the classifying examples: Lemma 2.3.5. (a) Let x_{j} = −λ_{j}/2πi be the normalized eigenvalues of a complex matrix A. We define Td(A) = Td(x) = ∏_{j} x_{j}/(1−e^{−xj}) as the Todd class. Then: ∫_{M^{c} ρ} Td(T^{c}(M^{c ρ})) = 1  for all ρ.

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(b) Let x_{j} = λ_{j}/2π where the eigenvalues of the skew-symmetric real ma- trix A are {±λ_{1},…}. We define L(A) = L(x) = ∏_{j} x_{j}/ tanh x_{j} as the Hirzebruch L-polynomial. Then ∫_{M^{r} ρ} L(T (M^{r})) = 1 for all ρ. We will use this calculation to prove the integral of the Todd class gives the arithmetic genus of a complex manifold and that the integral of the Hirzebruch L-polynomial gives the signature of an oriented real manifold. In each case, we only integrate the part of the total class which is of the same degree as the dimension of the manifold. Proof: Td is a multiplicative class: Td(T_{c}(M_{1} × M_{2})) = Td(T_{c}(M_{1}) ⊕ T_{c}(M_{2})) = Td(T_{c}(M_{1})) Td(T_{c}(M_{2})). Similarly the Hirzebruch polynomial is multiplicative. This shows it suffices to prove Lemma 2.3.5 in the case ρ = k so M = CP_{k} or CP_{2k}. We use the decomposition T_{c}(CP_{k}) ⊕ 1 = L^{∗} ⊕ ⋯ ⊕ L^{∗} (k + 1 times) to compute Td(T_{c}(CP_{k})) = [Td(x)]^{k+1} where x = c_{1}(L^{∗}) is the generator of H^{2}(CP_{k}; R). Since x^{k} integrates to 1, it suffices to show the coefficient of x^{k} in Td(x)^{k+1} is 1 or equivalently to show: Res_{x=0} x^{−k−1}[Td(x)^{k+1}] = Res_{x=0}(1 − e^{−x})^{−k−1} = 1. If k = 0, then: (1 − e^{−x})^{−1} = (x − 1 2x^{2} + ⋯)^{−1} = x^{−1} (1 + 1 2x + ⋯) and the result follows. Similarly, if k = 1 (1 − e^{−x})^{−2} = x^{−2}(1 + x + ⋯) and the result follows. For larger values of k, proving this directly would be a combinatorial nightmare so we use instead a standard trick from complex variables. If g(x) is any meromorphic function, then Res_{x=0} g′(x) = 0. We apply this to the function g(x) = (1 − e^{−x})^{−k} for k ≥ 1 to conclude: Res_{x=0}(1 − e^{−x})^{−k−1}e^{−x} = 0. This implies immediately that: Res_{x=0}(1 − e^{−x})−k−1 = Res_{x=0}(1 − e^{−x})^{−k−1}(1 − e^{−x}) = Res_{x=0}(1 − e^{−x})^{−k} = 1 by induction which completes the proof of assertion (a).

We now assume k is even and study CP_{k}. Again, using the decomposition T_{c}(CP_{k}) ⊕ 1 = L^{∗} ⊕ ⋯ ⊕ L^{∗} it follows that L(T (CP_{k})) = [L(x)]^{k+1} = ^{xk+1} tanh^{k+1} x. Since we are interested in the coefficient of x^{k}, we must show: Res_{x=0} tanh^{−k−1} x = 1  if k is even or equivalently Res_{x=0} tanh^{−k} x = 1  if k is odd. We recall that  tanh x = e^{x} − e−x e^{x} + e−x tanh^{−1} x = 2 + x^{2} + ⋯ 2x + ⋯ so the result is clear if k = 1. We now proceed by induction. We differen- tiate tanh^{−k} x to compute: (tanh^{−k} x)′ = −k(tanh^{−k−1} x)(1 − tanh^{2} x). This implies Res_{x=0} tanh^{−k+1} x = Res_{x=0} tanh^{−k−1} x for any integer k. Consequently Res_{x=0} tanh^{−k} x = 1 for any odd integer k since these residues are periodic modulo 2. (The residue at k even is, of course, zero). This completes the proof.

2.4. The Gauss-Bonnet Theorem. Let P be a self-adjoint elliptic partial differential operator of order d > 0. If the leading symbol of P is positive definite, we derived an asymptotic expansion for Tr{e^{−tP} } in section 1.7. This is too general a setting in which to work so we shall restrict attention henceforth to operators with leading symbol given by the metric tensor, as this is the natural category in which to work. Let P: C^{∞}(V) → C^{∞}(V) be a second order operator. We choose a local frame for V. Let x = (x_{1},…, x_{m}) be a system of local coordinates. Let ds^{2} = g_{ij} dx^{i} dx^{j} be the metric tensor and let g^{ij} denote the inverse matrix; dx^{i} · dx^{j} = g^{ij} is the metric on the dual space T^{∗}(M). We assume P has the form:  P = −g^{ij} ∂2 ∂x_{i}∂xj I + a_{j} ∂ ∂xj + b where we sum over repeated indices. The a_{j} and b are sections to END(V). We introduce formal variables for the derivatives of the symbol of P. Let g_{ij/α} = d_{α x}g_{ij},  a_{j/α} = d_{α x}a_{j},  b_{/α} = d_{α x}b. We will also use the notation g_{ij/kl...}, a_{j/kl...}, and b_{/kl...} for multiple partial derivatives. We emphasize that these variables are not tensorial but depend upon the choice of a coordinate system (and local frame for V). There is a natural grading on these variables. We define: ord(g_{ij/α}) = |α|,  ord(a_{j/α}) = 1 + |α|,  ord(b_{/α}) = 2 + |α|. Let P be the non-commutative polynomial algebra in the variables of pos- itive order. We always normalize the coordinate system so x_{0} corresponds to the point (0,…, 0) and so that: g_{ij}(x_{0}) = δ_{ij}  and g_{ij/k}(x_{0}) = 0. Let K(t, x, x) be the kernel of e^{−tP}. Expand K(t, x, x) ∼ _{∑ n≥0} t^{^{n−m} 2} e_{n}(x, P) where e_{n}(x, P) is given by Lemma 1.7.4. Lemma 1.7.5(c) implies: Lemma 2.4.1. e_{n}(x, P) ∈ P is a non-commutative polynomial in the jets of the symbol of P which is homogeneous of order n. If a_{n}(x, P) = Tr e_{n}(x, P) is the fiber trace, then a_{n}(x, P) is homogeneous of order n in

118 2.4. The Gauss-Bonnet Theorem the components (relative to some local frame) of the jets of the total symbol of P. We could also have proved Lemma 2.4.1 using dimensional analysis and the local nature of the invariants e_{n}(x, P) instead of using the combinatorial argument given in Chapter 1. We specialize to the case where P = Δ_{p} is the Laplacian on p-forms. If p = 0, then Δ_{0} = −g^{−1}∂/∂x_{i}{gg^{ij}∂/∂x_{j}}; g = det(g_{ij})^{1/2} defines the Riemannian volume dvol = g dx. The leading symbol is given by the metric tensor; the first order symbol is linear in the 1-jets of the metric with coefficients which depend smoothly on the metric tensor. The 0th order symbol is zero in this case. More generally: Lemma 2.4.2. Let x = (x_{1},…, x_{m}) be a system of local coordinates on M. We use the dx^{I} to provide a local frame for Λ(T^{∗}M). Relative to this frame, we expand σ(Δ_{p}) = p_{2} + p_{1} + p_{0}. p_{2} = |ξ|^{2}I. p_{1} is linear in the 1-jets of the metric with coefficients which depend smoothly on the g_{ij}’s. p_{0} is the sum of a term which is linear in the 2-jets of the metric and a term which is quadratic in the 1-jets of the metric with coefficients which depend smoothly on the g_{ij}’s. Proof: We computed the leading symbol in the first chapter. The re- mainder of the lemma follows from the decomposition Δ = dδ + δd = ±d ∗ d ∗ ± ∗ d ∗ d. In flat space, Δ = − ∑_{i} ∂^{2}/∂x_{2 i} as computed earlier. If the metric is curved, we must also differentiate the matrix representing the Hodge ∗ operator. Each derivative applied to “∗” reduces the order of differentiation by one and increases the order in the jets of the metric by one. Let a_{n}(x, P) = Tr e_{n}(x, P). Then: Tr e^{−tP} ∼ _{∑ n≥0} t^{^{n−m} 2} ∫_{M} a_{n}(x, P) dvol(x). For purposes of illustration, we give without proof the first few terms in the asymptotic expansion of the Laplacian. We shall discuss such formulas in more detail in the fourth chapter. Lemma 2.4.3. (a) a_{0}(x, Δ_{p}) = (4π)^{−1} dim(Λ^{p}) = (4π)^{−1}(_{m p}). (b) a_{2}(x, Δ_{0}) = (4π)^{−1}(−R_{ijij})/6. (c) a_{4}(x, Δ_{0}) = 1 4π · −12R_{ijij;kk} + 5R_{ijij}R_{klkl} − 2R_{ijik}R_{ljlk} + 2R_{ijkl}Rijkl 360 . In this lemma, “ ; ” denotes multiple covariant differentiation. a_{0}, a_{2}, a_{4}, and a_{6} have been computed for Δ_{p} for all p, but as the formulas are extremely complicated, we shall not reproduce them here.

The Gauss-Bonnet Theorem 119

We consider the algebra generated by the variables {g_{ij/α}}_{|α|≥2}. If X is a coordinate system and G a metric and if P is a polynomial in these variables, we define P (X, G)(x_{0}) by evaluation. We always normal- ize the choice of X so g_{ij}(X, G)(x_{0}) = δ_{ij} and g_{ij/k}(X, G)(x_{0}) = 0, and we omit these variables from consideration. We say that P is invariant if P (X, G)(x_{0}) = P (Y, G)(x_{0}) for any two normalized coordinate systems X and Y. We denote the common value by P (G)(x_{0}). For example, the scalar curvature K = −_{1 2} R_{ijij} is invariant as are the a_{n}(x, Δ_{p}). We let P_{m} denote the ring of all invariant polynomials in the derivatives of the metric for a manifold of dimension m. We defined ord(g_{ij/α}) = |α|; let P_{m,n} be the subspace of invariant polynomials which are homogeneous of order n. This is an algebraic characterization; it is also useful to have the following coordinate free characterization: Lemma 2.4.4. Let P ∈ P_{m}, then P ∈ P_{m,n} if and only if P (c^{2}G)(x_{0}) = c^{−n}P (G)(x_{0}) for every c ≠ 0. Proof: Fix c = 0 and let X be a normalized coordinate system for the metric G at the point x_{0}. We assume x_{0} = (0,…, 0) is the center of the coordinate system X. Let Y = cX be a new coordinate system, then: ∂/∂yi = c^{−1}∂/∂xi d^{α} y = c^{−|α|}d^{α} x c^{2}G(∂/∂y_{i}, ∂/∂y_{j}) = G(∂/∂x_{i}, ∂/∂x_{j}) g_{ij/α}(Y, c^{2}G) = c^{−|α|}g_{ij/α}(X, G). This implies that if A is any monomial of P that: A(Y, c^{2}G)(x_{0}) = c^{− ord(A)}A(X, G)(x_{0}). Since Y is normalized coordinate system for the metric c^{2}G, P (c^{2}G)(x_{0}) = P (Y, c^{2}G)(x_{0}) and P (G)(x_{0}) = P (X, G)(x_{0}). This proves the Lemma. If P ∈ P_{m} we can always decompose P = P_{0} + ⋯ + P_{n} into homoge- neous polynomials. Lemma 2.4.4 implies the P_{j} are all invariant separately. Therefore P_{m} has a direct sum decomposition P_{m} = P_{m,0} ⊕ P_{m,1} ⊕ ⋯ ⊕ P_{m,n} ⊕ ⋯ and has the structure of a graded algebra. Using Taylor’s theo- rem, we can always find a metric with the g_{ij/α}(X, G)(x_{0}) = c_{ij,α} arbitrary constants for |α| ≥ 2 and so that g_{ij}(X, G)(x_{0}) = δ_{ij}, g_{ij/k}(X, G)(x_{0}) = 0. Consequently, if P ∈ P_{m} is non-zero as a polynomial, then we can always find G so P (G)(x_{0}) ≠ 0 so P is non-zero as a formula. It is for this reason we work with the algebra of jets. This is a pure polynomial algebra. If we work instead with the algebra of covariant derivatives of the curvature tensor, we must introduce additional relations which correspond to the Bianchi identities as this algebra is not a pure polynomial algebra. We note finally that P_{m,n} is zero if n is odd since we may take c = −1. Later in this chapter, we will let P_{m,n,p} be the space of p-form valued invariants which are homogeneous of order n. A similar argument will show P_{m,n,p} is zero if n − p is odd. Lemmas 2.4.1 and 2.4.2 imply:

120 2.4. The Gauss-Bonnet Theorem Lemma 2.4.5. a_{n}(x, Δ_{p}) defines an element of P_{m,n}. This is such an important fact that we give another proof based on Lemma 2.4.4 to illustrate the power of dimensional analysis embodied in this lemma. Fix c > 0 and let Δ_{p}(c^{2}G) = c^{−2}Δ_{p}(G) be the Laplacian cor- responding to the new metric. Since dvol(c^{2}G) = c^{m} dvol(G), we conclude:^{e−tΔ^{p}(c^{2}G) = e−tc^{−2}Δ^{p}(G)} K(t, x, x, Δ_{p}(c^{2}G)) dvol(c^{2}G) = K(c^{−2}t, x, x, Δ_{p}(G)) dvol(G) K(t, x, x, Δ_{p}(c^{2}G)) = c^{−m}K(c^{−2}t, x, x, Δ_{p}(G))_{∑ n} t^{^{n−m} 2} a_{n}(x, Δ_{p}(c^{2}G)) ∼ _{∑ n} c^{−m}c^{m}c^{−n}t^{^{n−m} 2} a_{n}(x, Δ_{p}(G)) a_{n}(x, Δ_{p}(c^{2}G)) = c^{−n}a_{n}(x, Δ_{p}(G)). We expand a_{n}(x, Δ_{p}(G)) = ∑_{ν} a_{n,ν,p} + r in a finite Taylor series about g_{ij} = δ_{ij} in the g_{ij/α} variables. Then if a_{n,ν,p} is the portion which is homogeneous of order ν, we use this identity to show a_{n,ν,p} = 0 for n ≠ ν and to show the remainder in the Tayor series is zero. This shows a_{n} is a homogeneous polynomial of order n and completes the proof. Since a_{n}(x, Δ_{p}) = 0 if n is odd, in many references the authors replace the asymptotic series by t^{−^{m} 2} ∑_{n} t^{n}a_{n}(x, Δ_{p}), They renumber this sequence a_{0}, a_{1},… rather than a_{0}, 0, a_{2}, 0, a_{4},…. We shall not adopt this notational convention as it makes dealing with boundary problems more cumbersome. H. Weyl’s theorem on the invariants of the orthogonal group gives a spanning set for the spaces P_{m,n}: Lemma 2.4.6. We introduce formal variables R_{i_{1}i_{2}i_{3}i_{4};i_{5}...ik} for the mul- tiple covariant derivatives of the curvature tensor. The order of such a variable is k + 2. We consider the polynomial algebra in these variables and contract on pairs of indices. Then all possible such expressions generate P_{m}. In particular: {1} spans P_{m,0}, {R_{ijij}} spans Pm,2 {R_{ijij;kk}, R_{ijij}R_{klkl}, R_{ijik}R_{ljlk}, R_{ijkl}R_{ijkl}} spans P_{m,4}. This particular spanning set for P_{m,4} is linearly independent and forms a basis if m ≥ 4. If m = 3, dim(P_{3,4}) = 3 while if m = 2, dim(P_{2,4}) = 2 so there are relations imposed if the dimension is low. The study of these additional relations is closely related to the Gauss-Bonnet theorem. There is a natural restriction map r: P_{m,n} → Pm−1,n

The Gauss-Bonnet Theorem 121

which is defined algebraically as follows. We let deg_{k}(g_{ij/α}) = δ_{i,k} + δ_{j,k} + α(k) be the number of times an index k appears in the variable g_{ij/α}. Let r(g^{ij/α}) = { g_{ij/α} ∈ Pm−1 if deg_{m}(g_{ij/α}) = 0 0 if deg_{m}(g_{ij/α}) > 0. We extend r: P_{m} → P_{m−1} to be an algebra morphism; r(P) is a polynomial in the derivatives of a metric on a manifold of dimension m − 1. It is clear r preserves the degree of homogeneity. r is the dual of a natural extension map. Let G′ be a metric on a manifold M′ of dimension m − 1. We define i(G′) = G + dθ^{2} on the manifold M = M′ × S^{1} where S^{1} is the unit circle with natural parameter θ. If X′ is a local coordinate system on M′, then i(X′) = (x, θ) is a local coordinate system on M. It is clear that: (rP)(X′, G′)(x_{′0}) = P (i(X′), i(G′))(x_{′0}, θ_{0}) for any θ_{0} ∈ S^{1}; what we have done by restricting to product manifolds M′ × S^{1} with product metrics G′ + dθ^{2} is to introduce the relation which says the metric is flat in the last coordinate. Restiction is simply the dual of this natural extension; rP is invariant if P is invariant. Therefore r : P_{m,n} → P_{m,n−1} . There is one final description of the restriction map which will be useful. In discussing a H. Weyl spanning set, the indices range from 1 through m. We define the restriction by letting the indices range from 1 through m − 1. Thus R_{ijij} ∈ P_{m,2} is its own restriction in a formal sense; of course r(R_{ijij}) = 0 if m = 2 since there are no non-trivial local invariants over a circle. Theorem 2.4.7. (a) r: P_{m,n} → P_{m−1,n} is always surjective. (b) r: P_{m,n} → P_{m−1,n} is bijective if n < m. (c) r: P_{m,m} → P_{m−1,m} has 1-dimensional kernel spanned by the Euler class E_{m} if m is even. If m is odd, P_{m,m} = P_{m−1,m} = 0. (c) is an axiomatic characterization of the Euler form. It is an expression of the fact that the Euler form is a unstable characteristic class as opposed to the Pontrjagin forms which are stable characteristic classes. Proof: (a) is consequence of H. Weyl’s theorem. If we choose a H. Weyl spanning set, we let the indices range from 1 to m instead of from 1 to m − 1 to construct an element in the inverse image of r. The proof of (b) and of (c) is more complicated and will be postponed until the next subsection. Theorem 2.4.7 is properly a theorem in invariance theory, but we have stated it at this time to illustrate how invariance theory can be used to prove index theorems using heat equation methods:

Theorem 2.4.8. Let a_{n}(x, d + δ) = ∑_{p}(−1)^{p}a_{n}(x, Δ_{p}) ∈ P_{n,m} be the invariant of the de Rham complex. We showed in Lemma 1.7.6 that: ∫^{M} a^{n}(x, d + δ) dvol(x) = { χ(M) if n = m 0  if n ≠ m. Then: (a) a_{n}(x, d + δ) = 0 if either m is odd or if n < m. (b) a_{m}(x, d + δ) = E_{m} is m is even so χ(M) = ∫_{M} E_{m} dvol(x) (Gauss- Bonnet theorem). Proof: We suppose first that m is odd. Locally we can always choose an orientation for T (M). We let ∗ be the Hodge operator then ∗Δ_{p} = ±Δ_{m−p}∗ locally. Since these operators are locally isomorphic, their local invariants are equal so a_{n}(x, Δ_{p}) = a_{n}(x, Δ_{m−p}). If m is odd, these terms cancel in the alternating sum to give a_{n}(x, d + δ) = 0. Next we suppose m is even. Let M = M′ × S^{1} with the product metric. We decompose any ω ∈ Λ(T^{∗}M) uniquely in the form ω = ω_{1} + ω_{2} ∧ dθ  for ω_{i} ∈ Λ(T^{∗}M′). We define:  F (ω) = ω_{1} ∧ dθ + ω2 and compute easily that F Δ = ΔF since the metric is flat in the S 1 direction. If we decompose Λ(M) = Λ^{e}(M) ⊕ Λ^{o}(M) into the forms of even and odd degree, then F interchanges these two factors. Therefore, a_{n}(x, Δ_{e}) = a_{n}(x, Δ_{0}) so a_{n}(x, d + δ) = 0 for such a product metric. This implies r(a_{n}) = 0. Therefore a_{n} = 0 for n < m by Theorem 2.4.7. Furthermore:  a_{m} = c_{m}Em for some universal constant c_{m}. We show c_{m} = 1 by integrating over the classifying manifold M = S^{2} × ⋯ × S^{2}. Let 2m̄ = m, then 2^{m̄} = χ(M) = ∫_{M} a_{m}(x, d + δ) dvol(x) = ∫_{M} E_{m} dvol(x) by Lemma 2.3.4. This completes the proof of the Gauss-Bonnet theorem.

2.5. Invariance Theory and the Pontrjagin Classes of the Tangent Bundle. In the previous subsection, we gave in Theorem 2.4.7 an axiomatic char- acterization of the Euler class in terms of functorial properties. In this subsection we will complete the proof of Theorem 2.4.7. We will also give a similar axiomatic characterization of the Pontrjagin classes which we will use in our discussion of the signature complex in Chapter 3. Let T: R^{m} → R^{m} be the germ of a diffeomorphism. We assume that: T (0) = 0  and dT (x) = dT (0) + O(x^{2})  for dT (0) ∈ O(k). If X is any normalized coordinate system for a metric G, then T X is another normalized coordinate system for G. We define an action of the group of germs of diffeomorphisms on the polynomial algebra in the {g_{ij/α}}, |α| ≥ 2 variables by defining the evaluation: (T^{∗}P)(X, G)(x_{0}) = P (T X, G)(x_{0}). Clearly P is invariant if and only if T^{∗}P = P for every such diffeomorphism T.Let P be invariant and let A be a monomial. We let c(A, P) be the coefficient of A in P; c(A, P) defines a linear functional on P for any monomial A. We say A is a monomial of P if c(A, P) ≠ 0. Let T_{j} be the linear transformation: T^{j}(x^{k}) = { −xj  if k = j xk  if k ≠ j. This is reflection in the hyperplane defined by x_{j} = 0. Then T_{∗ j}(A) = (−1)^{deg_{j}(A)}A for any monomial A. Since T_{∗ j}P = ∑(−1)^{deg_{j}(A)}c(A, P)A = P = ∑ c(A, P)A, we conclude deg_{j}(A) must be even for any monomial A of P. If A has the form:  A = g^{i_{1}j_{1}/α1} … gi^{r}j^{r}/αr we define the length of A to be: ℓ(A) = r. It is clear 2ℓ(A) + ord(A) = ∑_{j} deg_{j}(A) so ord(A) is necessarily even if A is a monomial of P. This provides another proof P_{m,n} = 0 if n is odd.

In addition to the hyperplane reflections, it is convenient to consider coordinate permutations. If ρ is a permutation, then T_{∗ ρ}(A) = A^{ρ} is defined by replacing each index i by ρ(i) in the variables g_{ij/k...}. Since T_{∗ ρ}(P) = P, we conclude the form of P is invariant under coordinate permutations or equivalently c(A, P) = c(A^{ρ}, P) for every monomial A of P. We can use these two remarks to begin the proof of Theorem 2.4.7. Fix P ≠ 0 with r(P) = 0 and P ∈ P_{m,n}. Let A be a monomial of P. Then r(P) = 0 implies deg_{m}(A) > 0 is even. Since P is invariant under coordinate permutations, deg_{k}(A) ≥ 2 for 1 ≤ k ≤ m. We construct the chain of inequalities: 2m ≤ _{∑ k} deg_{k}(A) = 2ℓ(A) + ord(A) = _{∑ ν}(2 + |α_{ν}|) ≤ _{∑ ν} 2|α_{ν}| = 2 ord(A) = 2n. We have used the fact |α_{ν}| ≥ 2 for 1 ≤ ν ≤ ℓ(A). This implies m ≤ n so in particular P = 0 if n < m which proves assertion (b) of Theorem 2.4.7. If n = m, then all of the inequalities in this string must be equalities. This implies 2ℓ(A) = m, deg_{k}(A) = 2 for all k, and |α_{ν}| = 2 for all ν. Thus the monomial A must have the form: A = g_{i_{1}j_{1}/k_{1}l1} … g_{i_{r}j_{r}/k_{r}lr}  for 2r = m and in particular, P only involves the second derivatives of the metric. In order to complete the proof of Theorem 2.4.7, we must use some results involving invariance under circle actions: Lemma 2.5.1. Parametrize the circle SO(2) by z = (a, b) for a^{2} +b^{2} = 1. Let T_{z} be the coordinate transformation: y_{1} = ax_{1} + bx_{2},  y_{2} = −bx_{1} + ax_{2},  y_{k} = x_{k}  for k > 2. Let P be a polynomial and assume T_{∗ z}P = P for all z ∈ S^{1}. Then: (a) If g_{12/α} divides a monomial A of P for some α, then g_{11/β} divides a monomial B of P for some β. (b) If g_{ij/α} divides a monomial of A of P for some (i, j), then g_{kl/β} divides a monomial B of P for some β and some (k, l) where β(1) = α(1) + α(2) and β(2) = 0. Of course, the use of the indices 1 and 2 is for convenience only. This lemma holds true for any pair of indices under the appropriate invariance assumption. We postpone the proof of this lemma for the moment and use it to complete the proof of Theorem 2.4.7. Let P ≠ 0 ∈ P_{m,m} with r(P) = 0.

Let A be a monomial of P. We noted deg_{k}(A) = 2 for 1 ≤ k ≤ m and A is a polynomial in the 2-jets of the metric. We decompose A = g_{i_{1}j_{1}/k_{1}l1} … g_{i_{r}j_{r}/k_{r}lr}  for 2r = m. By making a coordinate permutation we can choose A so i_{1} = 1. If j_{1} ≠ 1, we can make a coordinate permutation to assume j_{1} = 2 and then apply Lemma 2.5.1(a) to assume i_{1} = j_{1} = 1. We let P_{1} = ∑ c(A, P)A where the sum ranges over A of the form:^{A = g11/k_{1}l1 … gi_{r}j_{r}/k_{r}lr} which are monomials of P. Then P_{1} ≠ 0 and P_{1} is invariant under coor- dinate transformations which fix the first coordinate. Since deg_{1}(A) = 2, the index 1 appears nowhere else in A. Thus k_{1} is not 1 so we may make a coordinate permutation to choose A so k_{1} = 2. If l_{2} ≠ 2, then l_{1} ≥ 3 so we may make a coordinate permutation to assume l_{1} = 3. We then apply Lemma 2.5.1(b) to choose A a monomial of P_{1} of the form: A = g_{11/22} A′. We have deg_{1}(A) = deg_{2}(A) = 2 so deg_{1}(A′) = deg_{2}(A′) = 0 so these indices do not appear in A′. We define P_{2} = ∑ c(A, P)A where the sum ranges over those monomial A of P which are divisible by g_{11/22} , then P_{2} ≠ 0. We proceed inductively in this fashion to show finally that^{A0 = g11/22 g33/44 … gm−1,m−1/mm} is a monomial of P so c(A_{0}, P) ≠ 0. The function c(A_{0}, P) is a separating linear functional on the kernel of r in P_{m,m} and therefore dim({ P ∈ P_{m,m} : r(P) = 0 }) ≤ 1. We complete the proof of Theorem 2.4.7(c) by showing that r(E_{m}) = 0. If E_{m} is the Euler form and M = M′ × S^{1}, then: E_{m}(M) = E_{m}(M′ × S^{1}) = E_{m−1}(M′)E_{1}(S^{1}) = 0 which completes the proof; dim N(r) = 1 and E_{m} ∈ N(r) spans. Before we begin the proof of Lemma 2.5.1, it is helpful to consider a few examples. If we take A = g_{11/11} then it is immediate: ∂/∂y1 = a∂/∂x_{1} + b∂/∂x_{2} ,  ∂/∂y_{2} = −b∂/∂x_{1} + a∂/∂x2 ∂/∂yk = ∂/∂x_{k}  for k > 2.

126 2.5. Invariance Theory We compute T_{∗ z}(A) by formally replacing each index 1 by a1 + b2 and each index 2 by −b1 + a2 and expanding out the resulting expression. Thus, for example:^{T_{∗ z}(g_{11/11} ) = ga1+b2,a1+b2/a1+b2,a1+b2} = a^{4}g_{11/11} + 2a^{3}bg_{11/12} + 2a^{3}bg12/11 + a^{2}b^{2}g_{11/22} + a^{2}b^{2}g_{22/11} + 4a^{2}b^{2}g12/12 + 2ab^{3}g_{12/22} + 2ab^{3}g_{22/12} + b^{4}g_{22/22} . We note that those terms involving a^{3}b arose from changing exactly one index to another index (1 → 2 or 2 → 1); the coefficient reflects the multi- plicity. Thus in particular this polynomial is not invariant. In computing invariance under the circle action, all other indices remain fixed so T_{∗ z}(g_{34/11} + g_{34/22} ) = (a^{2}g_{34/11} + b^{2}g_{34/22} + 2abg34/12 + a^{2}g_{34/22} + b^{2}g_{34/11} − 2abg_{34/12} ) = (a^{2} + b^{2})(g_{34/11} + g_{34/22} ) is invariant. Similarly, it is easy to compute: T_{∗ z}(g_{11/22} + g_{22/11} − 2g_{12/12} ) = (a^{2} + b^{2})^{2}(g_{11/22} + g_{22/11} − 2g_{12/12} ) so this is invariant. We note this second example is homogeneous of degree 4 in the (a, b) variables since deg_{1}(A) + deg_{2}(A) = 4. With these examples in mind, we begin the proof of Lemma 2.5.1. Let P be invariant under the action of the circle acting on the first two coor- dinates. We decompose P = P_{0} + P_{1} + ⋯ where each monomial A of Pj satisfies deg_{1}(A) + deg_{2}(A) = j. If A is such a monomial, then T_{∗ z}(A) is a sum of similar monomials. Therefore each of the P_{j} is invariant separately so we may assume P = P_{n} for some n. By setting a = b = −1, we see n must be even. Decompose: T^{∗ z}(P) = a^{n}P^{(0)} + ba^{n−1}P^{(1)} + ⋯ + b^{n}P (n) where P = P^{(0)}. We use the assumption T_{∗ z}(P) = P and replace b by −b to see:0 = T_{∗ (a,b)}(P) − T_{∗ (a,−b)} (P) = 2ba^{n−1}P^{(1)} + 2b^{3}a^{n−3}P^{(3)} + ⋯ We divide this equation by b and take the limit as b → 0 to show P^{(1)} = 0. (In fact, it is easy to show P^{(2j+1)} = 0 and P^{(2j)} = (_{n/2 j})P but as we shall not need this fact, we omit the proof).

We let A_{0} denote a variable monomial which will play the same role as the generic constant C of Chapter 1. We introduce additional notation we shall find useful. If A is a monomial, we decompose T_{∗ (a,b)}A = a^{n}A + a^{n−1}bA^{(1)} + · · ·. If B is a monomial of A^{(1)} then deg_{1}(B) = deg_{1}(A) ± 1 and B can be constructed from the monomial A by changing exactly one index 1 → 2 or 2 → 1. If deg_{1}(B) = deg_{1}(A) + 1, then B is obtained from A by changing exactly one index 2 → 1; c(B, A^{(1)}) is a negative integer which reflects the multiplicity with which this change can be made. If deg_{1}(B) = deg_{1}(A) − 1, then B is obtained from A by changing exactly one index 1 → 2; c(B, A^{(1)}) is a positive integer. We define: A(1 → 2) = ∑ c(B, A^{(1)})B  summed over deg_{1}(B) = deg_{1}(A) − 1 A(2 → 1) = ∑ −c(B, A^{(1)})B  summed over deg_{1}(B) = deg_{1}(A) + 1 A(1) = A(1 → 2) − A(2 → 1). For example, if A = (g_{12/33} )^{2}g_{11/44} then n = 6 and: A(1 → 2) = 2g_{12/33} g_{22/33} g_{11/44} + 2(g_{12/33} )^{2}g12/44 A(2 → 1) = 2g_{12/33} g_{11/33} g_{11/44} . It is immediate from the definition that:_{∑ B} c(B, A(1 → 2)) = deg_{1}(A)  and  _{∑ B} c(B, A(2 → 1)) = deg_{2}(A). Finally, it is clear that c(B, A^{(1)}) ≠ 0 if and only if c(A, B^{(1)}) ≠ 0, and that these two coefficients will be opposite in sign, and not necessarily equal in magnitude. Lemma 2.5.2. Let P be invariant under the action of SO(2) on the first two coordinates and let A be a monomial of P. Let B be a monomial of A^{(1)}. Then there exists a monomial of A_{1} different from A so that c(B, A^{(1)})c(A, P)c(B, A_{(1) 1})c(A_{1}, P) < 0. Proof: We know P^{(1)} = 0. We decompose P^{(1)} = _{∑ A} c(A, P)A^{(1)} = _{∑ B} c(A, P)c(B, A^{(1)})B. Therefore c(B, P^{(1)}) = 0 implies_{∑ A} c(A, P)c(B, A^{(1)}) = 0

128 2.5. Invariance Theory for all monomials B. If we choose B so c(B, A^{(1)}) ≠ 0 then there must be some other monomial A_{1} of P which helps to cancel this contribution. The signs must be opposite which proves the lemma. This lemma is somewhat technical and formidable looking, but it is ex- actly what we need about orthogonal invariance. We can now complete the proof of Lemma 2.5.1. Suppose first that A = (g_{12/α})^{k}A_{0} for k > 0 where g_{12/α} does not divide A_{0}. Assume c(A_{,}P) ≠ 0. Let B = g_{11/α}(g_{12/α})^{k−1}A0 then c(B, A^{(1)}) = −k ≠ 0. Choose A_{1} ≠ A so c(A_{1}, P) ≠ 0. Then c(A_{1}, B^{(1)}) ≠ 0. If c(A_{1}, B(2 → 1)) ≠ 0 then A_{1} is constructed from B by changing a 2 → 1 index so A_{1} = g_{11/β} … has the desired form. If c(A_{1}, B(1 → 2)) ≠ 0, we expand B(1 → 2) = A+ terms divisible by g11/β for some β. Since A_{1} ≠ A, again A_{1} has the desired form which proves (a). To prove (b), we choose g_{ij/α} dividing some monomial of P. Let β be cho- sen with β(1)+β(2) = α(1)+α(2) and β(k) = α(k) for k > 2 with α(1) max- imal so g_{uv/β} divides some monomial of P for some (u, v). Suppose β(2) ≠ 0, we argue for a contradiction. Set γ = (β(1)+1, β(2)−1, β(3),…, β(m)). Expand A = (g_{uv/β})^{k}A_{0} and define B = g_{uv/γ}(g_{uv/β})^{k−1}A_{0} where guv/β does not divide A_{0}. Then c(B, A^{(1)}) = −β(2)k ≠ 0 so we may choose A_{1} ≠ A so c(A_{1}, B^{(1)}) ≠ 0. If c(A_{1}, B(2 → 1)) ≠ 0 then either A_{1} is divisible by g_{u′v′/γ} or by g_{uv/γ′} where γ′(1) = γ(1) + 1, γ′(2) = γ(2) − 1, and γ′(j) = α(j) for j > 2. Either possibility contradicts the choice of β as maximal so c(A_{1}, B(1 → 2)) ≠ 0. However, B(1 → 2) = β(1)A+ terms divisible by g_{u′v′/γ} for some (u′, v′). This again contradicts the maximal- ity as A ≠ A_{1} and completes the proof of Lemma 2.5.1 and thereby of Theorems 2.4.7 and 2.4.8. If I = {1 ≤ i_{1} ≤ ⋯ ≤ i_{p} ≤ m}, let |I| = p and dx^{I} = dx_{i1} ∧ ⋯ ∧ dx_{ip}. A p-form valued polynomial is a collection {P_{I}} = P for |I| = p of polynomials P_{I} in the derivatives of the metric. We will also sometimes write P = ∑_{|I|=p} P_{I} dx^{I} as a formal sum to represent P. If all the {P_{I}} are homogeneous of order n, we say P is homogeneous of order n. We define: P (X, G)(x_{0}) = _{∑ I} P_{I}(X, G) dx^{I} ∈ Λ^{p}(T^{∗}M) to be the evaluation of such a polynomial. We say P is invariant if P (X, G)(x_{0}) = P (Y, G)(x_{0}) for every normalized coordinate systems X and Y; as before we denote the common value by P (G)(x_{0}). In analogy with Lemma 2.4.4 we have: Lemma 2.5.3. Let P be p-form valued and invariant. Then P is homo- geneous of order n if and only if P (c^{2}G)(x_{0}) = c^{p−n}P (G)(x_{0}) for every c ≠ 0. The proof is exactly the same as that given for Lemma 2.4.4. The only new feature is that dy^{I} = c^{p} dx^{I} which contributes the extra feature of cp in this equation.

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We define P_{m,n,p} to be the vector space of all p-form valued invariants which are homogeneous of order n on a manifold of dimension m. If Pm,∗,p denotes the vector space of all p-form valued invariant polynomials, then we have a direct sum decomposition P_{m,∗,p} = ⨁_{n} P_{m,n,p} exactly as in the scalar case p = 0. We define deg_{k}(I) to be 1 if k appears in I and 0 if k does not appear in I. Lemma 2.5.4. Let P ∈ P_{m,n,p} with P ≠ 0. Then n − p is even. If A is a monomial then A is a monomial of at most one P_{I}. If c(A, P_{I}) ≠ 0 then: deg_{k}(A) + deg_{k}(I) is always even. Proof: Let T be the coordinate transformation defined by T (x_{k}) = −xk and T (x_{j}) = x_{j} for j ≠ k. Then P = T^{∗}(P) = ^{∑ I,A}(−1)^{deg_{k}(A)+deg_{k}(I)} c(A, P^{I}) dxI which implies deg_{k}(A) + deg_{k}(I) is even if c(A, P_{I}) ≠ 0. Therefore_{∑ k} deg_{k}(A) + deg_{k}(I) = 2ℓ(A) + ord(A) + p = 2ℓ(A) + n + p must be even. This shows n+p is even if P ≠ 0. Furthermore, if c(A, P_{I}) ≠ 0 then I is simply the ordered collection of indices k so deg_{k}(A) is odd which shows A is a monomial of at most one P_{I} and completes the proof. We extend the Riemannian metric to a fiber metric on Λ(T^{∗}M) ⊗ C. It is clear that P · P = ∑_{I} P_{I}P̄I since the {dx^{I}} form an orthonormal basis at x_{0}. P · P is a scalar invariant. We use Lemma 2.5.1 to prove: Lemma 2.5.5. Let P be p-form valued and invariant under the action of O(m). Let A be a monomial of P. Then there is a monomial A_{1} of P with deg_{k}(A_{1}) = 0 for k > 2ℓ(A). Proof: Let r = ℓ(A) and let P_{′ r} = ∑_{ℓ(B)=r} c(B, P_{I}) dx^{I} ≠ 0. Since this is invariant under the action of O(m), we may assume without loss of generality P = P_{r}. We construct a scalar invariant by taking the inner product Q = (P, P). By applying Lemma 2.5.1(a) and making a coordinate permutation if necessary, we can assume g_{11/α1} divides some monomial of Q. We apply 2.5.1(b) to the indices > 1 to assume α_{1}(k) = 0 for k > 2. g_{11/α1} must divide some monomial of P. Let P^{1} = _{∑ A_{0},I} c(g^{11/α1} A^{0}, P^{I})g^{11/α1} A^{0} dx^{I} ≠ 0.

130 2.5. Invariance Theory This is invariant under the action of O(m−2) on the last m−2 coordinates. We let Q_{1} = (P_{1}, P_{1}) and let g_{i_{2}j_{2}/α2} divide some monomial of Q_{1}. If both i_{2} and j_{2} are ≤ 2 we leave this alone. If i_{2} ≤ 2 and j_{2} ≥ 3 we perform a coordinate permutation to assume j_{2} = 3. In a similar fashion if i_{2} ≥ 3 and j_{2} ≤ 2, we choose this variable so i_{2} = 3. Finally, if both indices ≥ 3, we apply Lemma 2.5.1(a) to choose this variable so i_{2} = j_{2} = 3. We apply Lemma 2.5.1(b) to the variables k ≥ 4 to choose this variable so α_{2}(k) = 0 for k > 4. If A_{2} = g_{11/α1} g_{i_{2}j_{2}/α2} then: deg_{k}(A_{2}) = 0 for k > 4 and A_{2} divides some monomial of P. We continue inductively to construct A_{r} = g_{11/α1} … g_{i_{r}j_{r}/αr} so that deg_{k}(A_{r}) = 0 for k > 2r and A_{r} divides some monomial of P. Since every monomial of P has length r, this implies A_{r} itself is a monomial of P and completes the proof. Let P_{j}(G) = p_{j}(T M) be the j^{th} Pontrjagin form computed relative to the curvature tensor of the Levi-Civita connection. If we expand p_{j} in terms of the curvature tensor, then p_{j} is homogeneous of order 2j in the {R_{ijkl}} tensor so p_{j} is homogeneous of order 4j in the jets of the metric. It is clear p_{j} is an invariantly defined 4j-form so p_{j} ∈ P_{m,4j,4j}. The algebra generated by the p_{j} is called the algebra of the Pontrjagin forms. By Lemma 2.2.2, this is also the algebra of real characteristic forms of T (M). If ρ is a partition of k = i_{1} + ⋯ + i_{j} we define p_{ρ} = p_{i1} … p_{ij} ∈ P_{m,4k,4k}. The {p_{ρ}} form a basis of the Pontrjagin 4k forms. By Lemma 2.3.4, these are linearly independent if m = 4k since the matrix ∫_{M^{r} τ} p_{ρ} is non-singular. By considering products of these manifolds with flat tori T^{m−4k} we can easily show that the {p_{ρ}} are linearly independent in P_{m,4k,4k} if 4k ≤ m. We let π(k) be the number of partitions of k; this is the dimension of the Pontrjagin forms. The axiomatic characterization of the real characteristic forms of the tangent space which is the analogue of the axiomatic characterization of the Euler class given in Theorem 2.4.7 is the following: Lemma 2.5.6. (a) P_{m,n,p} = 0 if n < p. (b) P_{m,n,n} is spanned by the Pontrjagin forms—i.e., P_{m,n,n} = 0 if n is not divisible by 4k, P_{m,4k,4k} = span{p_{ρ}} for 4k ≤ m has dimension π(k). Proof: By decomposing P into its real and imaginary parts, it suffices to prove this lemma for polynomials with real coefficients. Let 0 ≠ P ∈ Pm,n,p

Classes of the Tangent Bundle 131

and let A be a monomial of P. Use Lemma 2.5.5 to find a monomial A_{1} of some P_{I} where I = {1 ≤ i_{1} < ⋯ < i_{p} ≤ m} so deg_{k}(A) = 0 for k > 2ℓ(A). Since deg_{ip}(A) is odd (and hence non-zero) and since 2ℓ(A) ≤ ∑ |α_{ν}| = n as A is a polynomial in the jets of order 2 and higher, we estimate: p ≤ i_{p} ≤ 2ℓ(A) ≤ n. This proves P = 0 if n < p, which proves (a). If n = p, then all of these inequalities must have been equalities. This shows that the higher order jets of the metric do not appear in P so P is a polynomial in the {g_{ij/kl}} variables. Furthermore, i_{p} = p and there is some monomial A so that deg_{k}(A) = 0 for k > p and A dx^{1} ∧ ⋯ ∧ dx^{p} appears in P. There is a natural restriction map r: P_{m,n,p} → P_{m−1,n,p} defined in the same way as the restriction map r: P_{m,n,0} → P_{m−1,n,0} discussed earlier. This argument shows r: P_{m,n,n} → P_{m−1,n,n} is injective for n < m since r(A dx^{1} ∧ ⋯ ∧ dx^{p}) = A dx^{1} ∧ ⋯ ∧ dx^{p} appears in r(P). The Pontrjagin forms have dimension π(k) for n = 4k. By induction, r^{m−n}: P_{m,n,n} → P_{n,n,n} is injective so dim P_{m,n,n} ≤ dim P_{n,n,n}. We shall prove Lemma 2.5.6(b) for the special case n = m. If n is not divisible by 4k, then dim P_{n,n,n} = 0 which implies dim P_{m,n,n} = 0. If n = 4k, then dim P_{n,n,n} = π(k) implies that π(k) ≤ dim P_{m,n,n} ≤ dim P_{n,n,n} ≤ π(k) so dim P_{m,n,n} = π(k). Since the Pontrjagin forms span a subspace of exactly dimension π(k) in P_{m,n,n} this will complete the proof of (b). This lemma is at the heart of our discussion of the index theorem. We shall give two proofs for the case n = m = p. The first is based on H. Weyl’s theorem for the orthogonal group and follows the basic lines of the proof given in Atiyah-Bott-Patodi. The second proof is purely combinatorial and follows the basic lines of the original proof first given in our thesis. The H. Weyl based proof has the advantage of being somewhat shorter but relies upon a deep theorem we have not proved here while the second proof although longer is entirely self-contained and has some additional features which are useful in other applications. We review H. Weyl’s theorem briefly. Let V be a real vector space with a fixed inner product. Let O(V) denote the group of linear maps of V → V which preserve this inner product. Let ⨂^{k}(V) = V ⊗ ⋯ ⊗ V denote the k^{th} tensor prodect of V. If g ∈ O(V), we extend g to act orthogonally on ⨂^{k}(V). We let z ↦ g(z) denote this action. Let f: ⨂^{k}(V) → R be a multi-linear map, then we say f is O(V) invariant if f(g(z)) = f(z) for every g ∈ O(V). By letting g = −1, it is easy to see there are no O(V) invariant maps if k is odd. We let k = 2j and construct a map

f_{0} : ⨂^{k}(V) = (V ⊗ V) ⊗ (V ⊗ V) ⊗ ⋯ ⊗ (V ⊗ V) → R using the metric to map (V ⊗ V) → R. More generally, if ρ is any permutation of the integers 1 through k, we define z ↦ z_{ρ} as a map from ⨂^{k}(V) → ⨂^{k}(V) and let f_{ρ}(z) = f_{0}(z_{ρ}). This will be O(V) invariant for any permutation ρ. H. Weyl’s theorem states that the maps \{f_{ρ}\} define a spanning set for the collection of O(V) invariant maps. For example, let k = 4. Let \{v_{i}\} be an orthonormal basis for V and express any z ∈ ⨂^{4}(V) in the form a_{ijkl}v_{i} ⊗ v_{j} ⊗ v_{k} ⊗ v_{l} summed over repeated indices. Then after weeding out duplications, the spanning set is given by: f_{0}(z) = a_{iijj},  f_{1}(z) = a_{ijij},  f_{2}(z) = aijji where we sum over repeated indices. f_{0} corresponds to the identity permu- tation; f_{1} corresponds to the permutation which interchanges the second and third factors; f_{2} corresponds to the permutation which interchanges the second and fourth factors. We note that these need not be linearly in- dependent; if dim V = 1 then dim(⨂^{4} V) = 1 and f_{1} = f_{2} = f_{3}. However, once dim V is large enough these become linearly independent. We are interested in p-form valued invariants. We take ⨂^{k}(V) where k − p is even. Again, there is a natural map we denote by f^{p}(z) = f_{0}(z_{1}) ∧ Λ(z_{2}) where we decompose ⨂^{k}(V) = ⨂^{k−p}(V) ⊗ ⨂^{p}(V). We let f_{0} act on the first k − p factors and then use the natural map ⨂^{p}(V) ^{Λ} → Λ^{p}(V) on the last p factors. If ρ is a permutation, we set f_{p ρ}(z) = f^{p}(z_{ρ}). These maps are equivariant in the sense that f_{p ρ}(gz) = gf_{p ρ}(z) where we extend g to act on Λ^{p}(V) as well. Again, these are a spanning set for the space of equivariant multi-linear maps from ⨂^{k}(V) to Λ^{p}(V). If k = 4 and p = 2, then after eliminating duplications this spanning set becomes: {f_{1}(z) = a_{iijk}v_{j} ∧ v_{k} , f_{2}(z) = a_{ijik}v_{j} ∧ v_{k} , f_{3}(z) = a_{ijki}v_{j} ∧ vk \atop f_{4}(z) = a_{jiki}v_{j} ∧ v_{k} , f_{5}(z) = a_{jiik}v_{i} ∧ v_{k} , f_{6}(z) = a_{jkii}v_{j} ∧ v_{k} .} Again, these are linearly independent if dim V is large, but there are re- lations if dim V is small. Generally speaking, to construct a map from ⨂^{k}(V) → Λ^{p}(V) we must alternate p indices (the indices j, k in this ex- ample) and contract the remaining indices in pairs (there is only one pair i, i here). Theorem 2.5.7 (H. Weyl’s theorem on the invariants of the orthogonal group). The space of maps \{f_{p ρ}\} constructed above span the space of equivariant multi-linear maps from ⨂^{k} V → Λ^{p}V. The proof of this theorem is beyond the scope of the book and will be omitted. We shall use it to give a proof of Lemma 2.5.6 along the lines

of the proof given by Atiyah, Bott, and Patodi. We will then give an independent proof of Lemma 2.5.6 by other methods which does not rely on H. Weyl’s theorem. We apply H. Weyl’s theorem to our situation as follows. Let P ∈ Pn,n,n then P is a polynomial in the 2-jets of the metric. If we let X be a system of geodesic polar coordinates centered at x_{0}, then the 2-jets of the the metric are expressible in terms of the curvature tensor so we can express P as a polynomial in the {R_{ijkl}} variables which is homogeneous of order n/2. The curvature defines an element R ∈ ⨂^{4}(T (M)) since it has 4 indices. There are, however, relations among the curvature variables: R_{ijkl} = R_{klij},  R_{ijkl} = −R_{jikl},  and R_{ijkl} + R_{iklj} + R_{iljk} = 0. We let V be the sub-bundle of ⨂^{4}(T (M)) consisting of tensors satisfying these 3 relations. If n = 2, then P: V → Λ^{2}(T (M)) is equivariant while more generally, P: ⨂^{n/2}(V) → Λ^{n}(T (M)) is equivariant under the action of O(T (M)). (We use the metric tensor to raise and lower indices and identify T (M) = T^{∗}(M)). Since these relations define an O(T (M)) invariant subspace of ⨂^{2n}(T (M)), we extend P to be zero on the orthogonal complement of ⨂^{n/2}(V) in ⨂^{2n}(T (M)) to extend P to an equivariant action on the whole tensor algebra. Consequently, we can use H. Weyl’s theorem to span Pn,n,n by expressions in which we alternate n indices and contract in pairs the remaining n indices. For example, we compute: p^{1} = C · R^{ijab}R^{ijcd} dx^{a} ∧ dx^{b} ∧ dx^{c} ∧ dxd for a suitable normalizing constant C represents the first Pontrjagin form. In general, we will use letters a, b, c,… for indices to alternate on and indices i, j, k,… for indices to contract on. We let P be such an element given by H. Weyl’s theorem. There are some possibilities we can eliminate on a priori grounds. The Bianchi identity states: R_{iabc} dx^{a} ∧ dx^{b} ∧ dx^{c} = 1 3(R_{iabc} + R_{ibca} + R_{icab}) dx^{a} ∧ dx^{b} ∧ dx^{c} = 0 so that three indices of alternation never appear in any R_{...} variable. Since there are n/2 R variables and n indices of alternation, this implies each R variable contains exactly two indices of alternation. We use the Bianchi identity again to express: R^{iajb} dx^{a} ∧ dxb = 1 2(R^{iajb} − R^{ibja}) dx^{a} ∧ dxb = 1 2(R^{iajb} + R^{ibaj}) dx^{a} ∧ dxb = −1 2R^{ijba} dx^{a} ∧ dx^{b} = 1 2R^{ijab} dx^{a} ∧ dx^{b}.

134 2.5. Invariance Theory This together with the other curvature identities means we can express P in terms of R_{ijab} dx^{a} ∧ dx^{b} = Ω_{ij} variables. Thus P = P (Ω_{ij}) is a polynomial in the components of the curvature matrix where we regard Ω_{ij} ∈ Λ^{2}(T^{∗}M). (This differs by various factors of 2 from our previous definitions, but this is irrelevant to the present argument). P (Ω) is an O(n) invariant polynomial and thus is a real characteristic form. This completes the proof. The remainder of this subsection is devoted to giving a combinatorial proof of this lemma in the case n = m = p independent of H. Weyl’s theorem. Since the Pontrjagin forms span a subspace of dimension π(k) if n = 4k we must show P_{n,n,n} = 0 if n is not divisible by 4 and that dim P_{4k,4k,4k} ≤ π(k) since then equality must hold. We showed n = 2j is even and any polynomial depends on the 2-jets of the metric. We improve Lemma 2.5.1 as follows: Lemma 2.5.8. Let P satisfy the hypothesis of Lemma 2.5.1 and be a polynomial in the 2-jets of the metric. Then: (a) Let A = g_{12/α}A_{0} be a monomial of P. Either by interchanging 1 and 2 indices or by changing two 2 indices to 1 indices we can construct a monomial A_{1} of the form A_{1} = g_{11/β} A_{′0} which is a monomial of P. (b) Let A = g_{ij/12}A_{0} be a monomial of P. Either by interchanging 1 and 2 indices or by changing two 2 indices to 1 indices we can construct a monomial of A_{1} of the form A_{1} = g_{i′j′/11}A_{′0} which is a monomial of P. (c) The monomial A_{1} ≠ A. If deg_{1}(A_{1}) = deg_{1}(A)+2 then c(A, P)c(A_{1}, P) > 0. Otherwise deg_{1}(A_{1}) ≠ deg_{1}(A) and c(A, P)c(A_{1}, P) < 0. Proof: We shall prove (a) as (b) is the same; we will also verify (c). Let B = g_{11/α}A_{0} so c(B, A(2 → 1)) ≠ 0. Then deg_{1}(B) = deg_{1}(A) + 1. Apply Lemma 2.5.2 to find A_{1} ≠ A so c(B, A_{(1) 1}) ≠ 0. We noted earlier in the proof of 2.5.1 that A_{1} must have the desired form. If c(A_{1}, B(2 → 1)) ≠ 0 then deg_{1}(A_{1}) = deg_{1}(B)+1 = deg_{1}(A)+2 so A_{1} is constructed from A by changing two 2 to 1 indices. Furthermore, c(B, A_{(1) 1}) > 0 and c(B, A^{(1)}) < 0 implies c(A, P)c(A_{1}, P) > 0. If, on the other hand, c(A_{1}, B(1 → 2)) ≠ 0 then deg_{1}(A_{1}) = deg_{1}(B) − 1 = deg_{1}(A) and A changes to A_{1} by inter- changing a 2 and a 1 index. Furthermore, c(B, A_{(1) 1}) < 0 and c(B, A^{(1)}) < 0 implies c(A, P)c(A_{1}, P) < 0 which completes the proof. Let P ∈ P_{n,n,n} and express P = P′ dx^{1} ∧ ⋯ ∧ dx^{n}. P′ = ∗P is a scalar invariant which changes sign if the orientation of the local coordinate system is reversed. We identify P with P′ for notational convenience and henceforth regard P as a skew-invariant scalar polynomial. Thus deg_{k}(A) is odd for every k and every monomial A of P. The indices with deg_{k}(A) = 1 play a particularly important role in our discussion. We say that an index i touches an index j in the monomial A

Classes of the Tangent Bundle 135

if A is divisible by a variable g_{ij/..} or g_{../ij} where “..” indicate two indices which are not of interest. Lemma 2.5.9. Let P ∈ P_{n,n,n} with P ≠ 0. Then there exists a monomial A of P so (a) deg_{k}(A) ≤ 3 all k. (b) If deg_{k}(A) = 1 there exists an index j(k) which touches k in A. The index j(k) also touches itself in A. (c) Let deg_{j} A + deg_{k} A = 4. Suppose the index j touches itself and the index k in A. There is a unique monomial A_{1} different from A which can be formed from A by interchanging a j and k index. deg_{j} A_{1} +deg_{k} A_{1} = 4 and the index j touches itself and the index k in A_{1}. c(A, P)+c(A_{1}, P) = 0. Proof: Choose A so the number of indices with deg_{k}(A) = 1 is minimal. Among all such choices, we choose A so the number of indices which touch themselves is maximal. Let deg_{k}(A) = 1 then k touches some index j = j(k) ≠ k in A. Suppose A has the form A = g_{jk/..}A_{0} as the other case is similar. Suppose first that deg_{j}(A) ≥ 5. Use lemma 2.5.8 to find A_{1} = g_{kk/..}A_{0}. Then deg_{k}(A_{1}) ≠ deg_{k}(A) implies deg_{k}(A_{1}) = deg_{k}(A) + 2 = 3. Also deg_{j}(A_{1}) = deg_{j}(A) − 2 ≥ 3 so A_{1} has one less index with degree 1 which contradicts the minimality of the choice of A. We suppose next deg_{j}(A) = 1. If T is the coordinate transformation interchanging x_{j} and x_{k}, then T reverses the orientation so T^{∗}P = −P. However deg_{j}(A) = deg_{k}(A) = 1 implies T^{∗}A = A which contradicts the assumption that A is a monomial of P. Thus deg_{j}(A) = 3 which proves (a). Suppose j does not touch itself in A. We use Lemma 2.5.8 to con- struct A_{1} = g_{kk/..}A_{′0}. Then deg_{k}(A_{1}) = deg_{k}(A) + 2 = 3 and deg_{j}(A_{1}) = deg_{j}(A) − 2 = 1. This is a monomial with the same number of indices of degree 1 but which has one more index (namely k) which touches itself. This contradicts the maximality of A and completes the proof of (b). Finally, let A_{ν} = { k : deg_{k}(A) = ν } for ν = 1, 3. The map k ↦ j(k) defines an injective map from A_{1} → A_{3} since no index of degree 3 can touch two indices of degree 1 as well as touching itself. The equalities: n = card(A_{1})+card(A_{3}) and 2n = _{∑ k} deg_{k}(A) = card(A_{1})+3 card(A_{3}) imply 2 card(A_{3}) = n. Thus card(A_{1}) = card(A_{3}) = n/2 and the map k ↦ j(k) is bijective in this situation. (c) follows from Lemma 2.5.8 where j = 1 and k = 2. Since deg_{k}(A) = 1, A_{1} cannot be formed by transforming two k indices to j indices so A_{1} must be the unique monomial different from A obtained by interchanging these indices. For example, if A = g_{jj/ab}g_{jk/cd}A_{0}, then A_{1} = g_{jk/ab}g_{jj/cd}A_{0}. The multiplicities involved are all 1 so we can conclude c(A, P)+c(A_{1}, P) = 0 and not just c(A, P)c(A_{1}, P) < 0.

Before further normalizing the choice of the monomial A, we must prove a lemma which relies on cubic changes of coordinates: Lemma 2.5.10. Let P be a polynomial in the 2-jets of the metric in- variant under changes of coordinates of the form y_{i} = x_{i} + c_{jkl}x_{j}x_{k}x_{l}. Then: (a) g_{12/11} and g_{11/12} divide no monomial A of P. (b) Let A = g_{23/11} A_{0} and B = g_{11/23} A_{0} then A is a monomial of P if and only if B is a monomial of P. Furthermore, c(A, P) and c(B, P) have the same sign. (c) Let A = g_{11/22} A_{0} and B = g_{22/11} A_{0}, then A is a monomial of P if and only if B is a monomial of P. Furthermore, c(A, P) and c(B, P) have the same sign. Proof: We remark the use of the indices 1, 2, 3 is for notational con- venience only and this lemma holds true for any triple of distinct indices. Since g_{ij}(X, G) = δ_{ij} + O(x^{2}), g^{ij}_{/kl}(X, G)(x_{0}) = −g_{ij/kl}(X, G)(x_{0}). Fur- thermore, under changes of this sort, d_{α y}(x_{0}) = d_{α x}(x_{0}) if |α| = 1. We consider the change of coordinates: y2 = x_{2} + cx_{3 1}, dy2 = dx_{2} + 3cx_{2 1} dx_{1},  yk = kx otherwise dyk = dxk otherwise with  g^{12}(Y, G) = g^{12}(X, G) + 3cx_{2 1} + O(x^{4}), g^{ij}(Y, G) = g^{ij}(X, G) + O(x^{4}) otherwise g_{12/11} (Y, G)(x_{0}) = g_{12/11} (X, G)(x_{0}) − 6c g_{ij/kl}(Y, G)(x_{0}) = g_{ij/kl}(X, G)(x_{0}) otherwise. We decompose A = (g_{12/11} )^{ν}A_{0}. If ν > 0, T^{∗}(A) = A−6νc(g_{12/11} )^{ν−1}A_{0}+ O(c^{2}). Since T^{∗}(P) = P, and since there is no way to cancel this additional contribution, A cannot be a monomial of P so g_{12/11} divides no monomial of P. Next we consider the change of coordinates: y1 = x_{1} + cx_{2 1}x_{2}, dy1 = dx_{1} + 2cx_{1}x_{2} dx_{1} + cx_{2 1} dx_{2},  yk = xk otherwise dyk = dxk otherwise with  g_{11/12} (Y, G)(x_{0}) = g_{11/12} (X, G)(x_{0}) − 4c g_{12/11} (Y, G)(x_{0}) = g_{12/11} (X, G)(x_{0}) − 2c g_{ij/kl}(Y, G)(x_{0}) = g_{ij/kl}(X, G)(x_{0})  otherwise. We noted g_{12/11} divides no monomial of P. If A = (g_{11/12} )^{ν}A_{0}, then ν > 0 implies T^{∗}(A) = A − 4cv(g_{11/12} )^{ν−1}A_{0} + O(c^{2}). Since there would be no

way to cancel such a contribution, A cannot be a monomial of P which completes the proof of (a). Let A_{′0} be a monomial not divisible by any of the variables g_{23/11} , g_{12/13} , g_{13/12} , g_{11/23} and let A(p, q, r, s) = (g_{23/11} )^{p}(g_{12/13} )^{q}(g_{13/12} )^{r}(g_{11/23} )^{s}A_{′0}. We set c(p, q, r, s) = c(A(p, q, r, s), P) and prove (b) by establishing some relations among these coefficients. We first consider the change of coordi- nates, y2 = x_{2} + cx_{2 1}x_{3}, dy2 = dx_{2} + 2cx_{1}x_{3} dx_{1} + cx_{2 1} dx_{3},  yk = xk otherwise dyk = dxk otherwise with  g_{12/13} (Y, G)(x_{0}) = g_{12/13} (X, G)(x_{0}) − 2c g_{23/11} (Y, G)(x_{0}) = g_{23/11} (X, G)(x_{0}) − 2c g_{ij/kl}(Y, G)(x_{0}) = g_{ij/kl}(X, G)(x_{0})  otherwise. We compute that: T^{∗}(A(p, q, r, s)) = A(p, q, r, s) + c{−2qA(p, q − 1, r, s) − 2pA(p − 1, q, r, s)} + O(c^{2}). Since T^{∗}(P) = P is invariant, we conclude pc(p, q, r, s) + (q + 1)c(p − 1, q + 1, r, s) = 0. By interchanging the roles of 2 and 3 in the argument we also conclude: pc(p, q, r, s) + (r + 1)c(p − 1, q, r + 1, s) = 0. (We set c(p, q, r, s) = 0 if any of these integers is negative.) Next we consider the change of coordinates: y1 = x_{1} + cx_{1}x_{2}x_{3}, dy1 = dx_{1} + cx_{1}x_{2} dx_{3} + cx_{1}x_{3} dx_{2} + cx_{2}x_{3} dx_{1}, yk = xk otherwise dyk = dxk otherwise with  g_{11/23} (Y, G)(x_{0}) = g_{11/23} (X, G)(x_{0}) − 2c g_{12/13} (Y, G)(x_{0}) = g_{12/13} (X, G)(x_{0}) − c g_{13/12} (Y, G)(x_{0}) = g_{13/12} (Y, G)(x_{0}) − c so that T^{∗}(A(p, q, r, s)) = A(p, q, r, s) + c{−2sA(p, q, r, s − 1) − rA(p, q, r − 1, s) − qA(p, q − 1, r, s)} + O(c^{2}).

This yields the identities (q + 1)c(p, q + 1, r, s − 1) + (r + 1)c(p, q, r + 1, s − 1) + 2sc(p, q, r, s) = 0. Let p ≠ 0 and let A = g_{23/11} A_{′0} be a monomial of P. Then c(p−1, q+1, r, s) and c(p−1, q, r+1, s) are non-zero and have the opposite sign as c(p, q, r, s). Therefore (q + 1)c(p − 1, q + 1, r, s) + (r + 1)c(p − 1, q, r + 1, s) is non-zero which implies c(p − 1, q, r, s + 1)(s + 1) is non-zero and has the same sign as c(p, q, r, s). This shows g_{11/23} A_{′0} is a monomial of P. Conversely, if A is not a monomial of P, the same argument shows g_{11/23} A_{′0} is not a monomial of P. This completes the proof of (b). The proof of (c) is essentially the same. Let A_{′0} be a monomial not divisible by the variables {g_{11/22} , g_{12/12} , g_{22/11} } and let A(p, q, r) = (g_{11/22} )^{p}(g_{12/12} )^{q}(g_{22/11} )^{r}. Let c(p, q, r) = c(A(p, q, r), P). Consider the change of coordinates: y1 = x_{1} + cx_{1}x_{2 2}, dy1 = dx_{1} + cx_{2 2} dx_{1} + 2cx_{1}x_{2} dx_{2}, yk = xk otherwise dyk = dxk otherwise with  g_{11/22} (Y, g)(x_{0}) = g_{11/22} (X, G)(x_{0}) − 4c g_{12/12} (Y, G)(x_{0}) = g_{12/12} (X, G)(x_{0}) − 2c g_{ij/kl}(Y, G)(x_{0}) = g_{ij/kl}(X, G)(x_{0})  otherwise. This yields the relation 2pc(p, q, r) + (q + 1)c(p − 1, q + 1, r) = 0. By interchanging the roles of 1 and 2 we obtain the relation 2rc(p, q, r) + (q + 1)c(p, q + 1, r − 1) = 0 from which (c) follows. This step in the argument is functionally equivalent to the use made of the {R_{ijkl}} variables in the argument given previously which used H. Weyl’s formula. It makes use in an essential way of the invariance of P under a wider group than just first and second order transformations. For the Euler form, by contrast, we only needed first and second order coordinate transformations. We can now construct classifying monomials using these lemmas. Fix n = 2n_{1} and let A be the monomial of P given by Lemma 2.5.9. By making a coordinate permutation, we may assume deg_{k}(A) = 3 for k ≤ n1 and deg_{k}(A) = 1 for k > n_{1}. Let x(i) = i + n_{1} for 1 ≤ i ≤ n_{0}; we may assume the index I touches itself and x(i) in A for i ≤ n_{0}. We further normalize the choice of A as follows. Either g_{11/ij} or gij/11 divides A. Since deg_{1}(A) = 3 this term is not g_{11/11} and by Lemma 2.5.10 it is not g_{11/1x(1)} nor g_{1x(1)/11} . By Lemma 2.5.10(b) or 2.5.10(c), we may assume A = g_{11/ij} … for i, j ≥ 2. Since not both i and j can have degree

1 in A, by making a coordinate permutation if necessary we may assume that i = 2. If j = 2, we apply Lemma 2.5.9(c) to the indices 2 and x(2) to perform an interchange and assume A = g_{11/2x(2)} A_{0}. The index 2 must touch itself elsewhere in A. We apply the same considerations to choose A in the form A = g_{11/2x(2)} g_{22/ij} A_{0}. If i or j is 1, the cycle closes and we express A = g_{11/2x(2)} g_{11/1x(1)} A_{0} where deg_{k}(A_{0}) = 0 for k = 1, 2, 1 + n_{1}, 2 + n_{1}. If i, j > 2 we continue this argument until the cycle closes. This permits us to choose A to be a monomial of P in the form: A = g_{11/2x(2)} g_{22/3x(3)} … g_{j−1,j−1/jx(j)}g_{jj/1x(1)}A0 where deg_{k}(A_{0}) = 0 for 1 ≤ k ≤ j and n_{1} + 1 ≤ k ≤ n_{1} + j. We wish to show that the length of the cycle involved is even. We apply Lemma 2.5.10 to show B = g_{2x(2)/11} g_{3x(3)/22} … g_{1x(1)/jj} A0 satisfies c(A, P)c(B, P) > 0. We apply Lemma 2.5.9 a total of j times to see  C = g_{22,1x(1)} g_{33/2x(2)} … g_{11/jx(j)}A0 satisfies c(B, P)c(C, P)(−1)^{j} > 0. We now consider the even permutation: ρ(1) = j ρ(k) = { k − 1, 2 ≤ k ≤ j k,  j < k ≤ n1 ρ(k(j)) = x(ρ(j))  for 1 ≤ j ≤ n1 to see that c(C_{ρ}, P)c(C, P) > 0. However A = C_{ρ} so c(A, P)^{2}(−1)^{j} > 0 which shows j is necessarily even. (This step is formally equivalent to using the skew symmetry of the curvature tensor to show that only polynomials of even degree can appear to give non-zero real characteristic forms). We decompose A_{0} into cycles to construct A inductively so that A has the form:A ={g_{11/2x(2)} … g_{i_{1}i_{1}/1x(1)} } {g_{i_{1}+1,i_{1}+1/i_{1}+2,x(i_{1}+2)} … g_{i_{1}+i_{2},i_{1}+i_{2}/i_{1},x(i_{1})} }… where we decompose A into cycles of length i_{1}, i_{2},… with n_{1} = i_{1}+ ⋯+i_{j}. Since all the cycles must have even length, ℓ(A) = n/2 is even so n is divisible by 4. We let n = 4k and let ρ be a partition of k = k_{1} + ⋯ + k_{j}. We let A_{ρ} be defined using the above equation where i_{1} = 2k_{1}, i_{2} = 2k_{2},…. By making a coordinate permutation we can assume i_{1} ≥ i_{2} ≥ · · · . We have shown

140 2.5. Invariance Theory that if P = 0, then c(A_{ρ}, P) = 0 for some ρ. Since there are exactly π(k) such partitions, we have constructed a family of π(k) linear functionals on P_{n,n,n} which form a seperating family. This implies dim P_{n,n,n} < π(k) which completes the proof. We conclude this subsection with a few remarks on the proofs we have given of Theorem 2.4.7 and Lemma 2.5.6. We know of no other proof of Theorem 2.4.7 other than the one we have given. H. Weyl’s theorem is only used to prove the surjectivity of the restriction map r and is inessential in the axiomatic characterization of the Euler form. This theorem gives an immediate proof of the Gauss-Bonnet theorem using heat equation meth- ods. It is also an essential step in settling Singer’s conjecture for the Euler form as we shall discuss in the fourth chapter. The fact that r(E_{m}) = 0 is, of course, just an invariant statement of the fact E_{m} is an unstable charac- teristic class; this makes it difficult to get hold of axiomatically in contrast to the Pontrjagin forms which are stable characteristic classes. We have discussed both of the proofs of Lemma 2.5.6 which exist in the literature. The proof based on H. Weyl’s theorem and on geodesic normal coordinates is more elegant, but relies heavily on fairly sophisticated theorems. The second is the original proof and is more combinatorial. It is entirely self-contained and this is the proof which generalizes to Kaehler geometry to yield an axiomatic characterization of the Chern forms of T_{c}(M) for a holomorphic Kaehler manifold. We shall discuss this case in more detail in section 3.7.

2.6. Invariance Theory and Mixed Characteristic Classes of the Tangent Space and of a Coefficient Bundle. In the previous subsection, we gave in Lemma 2.5.6 an axiomatic char- acterization of the Pontrjagin forms in terms of functorial properties. In discussing the Hirzebruch signature formula in the next chapter, it will be convenient to have a generalization of this result to include invariants which also depend on the derivatives of the connection form on an auxilary bundle. Let V be a complex vector bundle. We assume V is equipped with a Hermitian fiber metric and let ∇ be a Riemannian or unitary connection on V. We let s = (s_{1},…, s_{a},…, s_{v}) be a local orthonormal frame for V and introduce variables ω_{abi} for the connection 1-form; ∇(s_{a}) = ω_{abi} dx^{i} ⊗ s_{b},  i.e., ∇s = ω ⊗ s. We introduce variables ω_{abi/α} = d_{α x}(ω_{abi}) for the partial derivatives of the connection 1-form. We shall also use the notation ω_{abi/jk...}. We use indices 1 ≤ a, b, ⋯ ≤ v to index the frame for V and indices 1 ≤ i, j, k ≤ m for the tangent space variables. We define: ord(ω_{abi/α}) = 1 + |α|  and  deg_{k}(ω_{abi/α}) = δ_{i,k} + α(k). We let Q be the polynomial algebra in the {ω_{abi/α}} variables for |α| ≥ 1; if Q ∈ Q we define the evaluation Q(X, s, ∇)(x_{0}). We normalize the choice of frame s by requiring ∇(s)(x_{0}) = 0. We also normalize the coordinate sys- tem X as before so X(x_{0}) = 0, g_{ij}(X, G)(x_{0}) = δ_{ij}, and g_{ij/k}(X, G)(x_{0}) = 0. We say Q is invariant if Q(X, s, ∇)(x_{0}) = Q(Y, s′, ∇)(x_{0}) for any nor- malized frames s, s′ and normalized coordinate systems X, Y; we denote this common value by Q(∇) (although it also depends in principle on the metric tensor and the 1-jets of the metric tensor through our normalization of the coordinate system X). We let Q_{m,p,v} denote the space of all invari- ant p-form valued polynomials in the {ω_{abi/α}} variables for |α| ≥ 1 defined on a manifold of dimension m and for a vector bundle of complex fiber dimension v. We let Q_{m,n,p,v} denote the subspace of invariant polynomials homogeneous of order n in the jets of the connection form. Exactly as was done for the P_{∗} algebra in the jets of the metric, we can show there is a direct sum decomposition Q_{m,p,v} = _{⨁ n} Q_{m,n,p,v}  and Q_{m,n,p,v} = 0  for n − p odd. Let Q(gAg^{−1}) = Q(A) be an invariant polynomial of order q in the components of a v×v matrix. Then Q(Ω) defines an element of Q_{m,2q,q,v} for

142 2.6. Invariance Theory any m ≥ 2q. By taking Q · Q̄ we can define scalar valued invariants and by taking δ(Q) we can define other form valued invariants in Q_{m,2q+1,2q−1,v} . Thus there are a great many such form valued invariants. In addition to this algebra, we let R_{m,n,p,v} denote the space of p-form valued invariants which are homogeneous of order n in the {g_{ij/α}, ω_{abk/β}} variables for |α| ≥ 2 and |β| ≥ 1. The spaces P_{m,n,p} and Q_{m,n,p,v} are both subspaces of R_{m,n,p,v}. Furthermore wedge product gives a natural map P_{m,n,p} ⊗ Q_{m,n′,p′,v} → R_{m,n+n′,p+p′,v}. We say that R ∈ R_{m,p,p,v} is a char- acteristic form if it is in the linear span of wedge products of Pontrjagin forms of T (M) and Chern forms of V. The characteristic forms are char- acterized abstractly by the following Theorem. This is the generalization of Lemma 2.5.6 which we shall need in discussing the signature and spin complexes. Theorem 2.6.1. (a) R_{m,n,p,v} = 0 if n < p or if n = p and n is odd. (b) If R ∈ R_{m,n,n,v} then R is a characteristic form. Proof: The proof of this fact relies heavily on Lemma 2.5.6 but is much easier. We first need the following generalization of Lemma 2.5.1: Lemma 2.6.2. Using the notation of Lemma 2.5.1 we define T_{∗ z}(R) if R is a scalar invariant in the {g_{ij/α}, ω_{abj/β}} variables. (a) Let g_{12/α} divide some monomial of R, then g_{../β} divides some other monomial of R. (b) Let g_{../α} divide some monomial of R, then g_{../β} divides some other monomial of R where β(1) = α(1) + α(2) and β(2) = 0. (c) Let ω_{abi/α} divide some monomial of R for i > 2, then ω_{abi/β} divides some other monomial of R where β(1) = α(1) + α(2) and β(2) = 0. The proof of this is exactly the same as that given for Lemma 2.5.1 and is therefore omitted. The proof of Theorem 2.6.1 parallels the proof of Lemma 2.5.6 for a while so we summarize the argument briefly. Let 0 ≠ R ∈ R_{m,n,p,v}. The same argument given in Lemma 2.5.3 shows an invariant polynomial is homogeneous of order n if R(c^{2}G, ∇) = c^{p−n}R(G, ∇) which gives a invari- ant definition of the order of a polynomial. The same argument as given in Lemma 2.5.4 shows n − p must be even and that if A is a monomial of R, A is a monomial of exactly one of the R_{I}. If c(A, R_{I}) = 0 then deg_{k}(A) + deg_{k}(I) is always even. We decompose A in the form: A = g^{i_{1}j_{1}/α1} … g^{i_{q}j_{q}/αq}ω^{a_{1}b_{1}k_{1}/β1} … ω^{a_{r}b_{r}k_{r}/βr} = A^{g}Aω and define ℓ(A) = q + r to the length of A. We argue using Lemma 2.6.2 to choose A so deg_{k}(A^{g}) = 0 for k > 2q. By making a coordinate permutation we can assume that the k_{ν} ≤ 2q + r for 1 ≤ ν ≤ r. We

apply Lemma 2.6.2(c) a total of r times to choose the β_{i} so β_{1}(k) = 0 for k > 2q + r + 1, β_{2}(k) = 0 for k > 2q + r + 2,…, β_{r}(k) = 0 for k > 2q + 2r. This chooses A so deg_{k}(A) = 0 for k > 2ℓ(A). If A is a monomial of RI for I = {1 ≤ i_{1} < ⋯ < i_{p} ≤ m} then deg_{ip}(A) is odd. We estimate p ≤ i_{p} ≤ 2ℓ(A) ≤ ∑ |α_{ν}| + ∑(|β_{μ}| + 1) = n so that P_{m,n,p,v} = {0} if n < p or if n − p is odd which proves (a) of Theorem 2.6.1. In the limiting case, we must have equalities so |α_{ν}| = 2 and |β_{μ}| = 1. Furthermore, i_{p} = p so there is some monomial A so deg_{k}(A) = 0 for k > p = n and A dx^{1} ∧ ⋯∧dx^{p} appears in R. There is a natural restriction map  ^{r: Rm,n,p,v → Rm−1,n,p,v} and our argument shows r: R_{m,n,n,v} → R_{m−1,n,n,v} is injective for n < m. Since the restriction of a characteristic form is a characteristic form, it suffices to prove (b) of Theorem 2.6.1 for the case m = n = p. Let 0 ≠ R ∈ R_{n,n,n,v} then R is a polynomial in the {g_{ij/kl}, ω_{abi/j}} variables. The restriction map r was defined by considering products M_{1} × S^{1} but there are other functorial constructions which give rise to useful projections. Fix non-negative integers (s, t) so that n = s + t. Let M1 be a Riemannian manifold of dimension s. Let M_{2} be the flat torus of dimension t and let V_{2} be a vector bundle with connection ∇_{2} over M_{2}. Let M = M_{1} × M_{2} with the product metric and let V be the natural extension of V_{2} to M which is flat in the M_{1} variables. More exactly, if π_{2}: M → M_{2} is a projection on the second factor, then (V, ∇) = π_{∗ 2}(V_{2}, ∇_{2}) is the pull back bundle with the pull back connection. We define π_{(s,t)}(R)(G_{1}, ∇_{2}) = R(G_{1} × 1, ∇). Using the fact that P_{s,n_{1},p1} = 0 for s < p_{1} or n_{1} < p_{1} and the fact Q_{t,n_{2},p_{2},k} = 0 for t < p_{2} or n_{2} < p_{2} it follows that π_{(s,t)} defines a map π_{(s,t)}: R_{n,n,v} → P_{s,s,s} ⊗ Q_{t,t,t,v} . More algebraically, let A = A^{g}A^{ω} be a monomial, then we define: π^{(s,t)}(A) = { 0  if deg_{k}(A^{g}) > 0 for k > s or deg_{k}(A^{ω}) > 0 for k ≤ s A otherwise. The only additional relations imposed are to set g_{ij/kl} = 0 if any of these indices exceeds s and to set ω_{abi/j} = 0 if either i or j is less than or equal to s. We use these projections to reduce the proof of Theorem 2.6.1 to the case in which R ∈ Q_{t,t,t,v}. Let 0 ≠ R ∈ R_{n,n,n,v} and let A = A^{g}A^{ω} be a monomial of R. Let s = 2ℓ(A^{g}) = ord(A^{g}) and let t = n − s = 2ℓ(A^{ω}) =

ord(A^ω). We choose A so deg_k(A^g) = 0 for k > s. Since deg_k(A) ≥ 1 must be odd for each index k, we can estimate:

t ≤ ∑

k>s

deg_k(A) = ∑

k>s

deg_k(A^ω) ≤ ∑

k

deg_k(A^ω) = ord(A^ω) = t.

As all these inequalities must be equalities, we conclude deg_k(A^ω) = 0 for k ≤ s and deg_k(A^ω) = 1 for k > s. This shows in particular that π_{(s,t)}(R) ≠ 0 for some (s, t) so that ⨁

s+t=n

π_{(s,t)}: R_{n,n,n,v} → ⨁

s+t=n

P_{s,s,s} ⊗ Q_{t,t,t,v}

is injective.

We shall prove that Q_{t,t,t,v} consists of characteristic forms of V. We showed earlier that P_{s,s,s} consists of Pontrjagin forms of T (M). The characteristic forms generated by the Pontrjagin forms of T (M) and of V are elements of R_{n,n,n,v} and π_{(s,t)} just decomposes such products. Thererfore π is surjective when restricted to the subspace of characteristic forms. This proves π is bijective and also that R_{n,n,n,v} is the space of characteristic forms. This will complete the proof of Theorem 2.6.1.

We have reduced the proof of Theorem 2.6.1 to showing Q_{t,t,t,v} consists of the characteristic forms of V. We noted that 0 ≠ Q ∈ Q_{t,t,t,v} is a polynomial in the {ω_{abi/j}} variables and that if A is a monomial of Q, then deg_k(A) = 1 for 1 ≤ k ≤ t. Since ord(A) = t is even, we conclude Q_{t,t,t,v} = 0 if t is odd.

The components of the curvature tensor are given by:

Ω_{abij} = ω_{abi/j} − ω_{abi/j} and Ω_{ab} = ∑ Ω_{abij} dx_i ∧ dx_j

up to a possible sign convention and factor of 1/2 which play no role in this discussion. If A is a monomial of P, we decompose:

A = ω_{a1b1i1/i2} … ω_{aubuit−1/it} where 2u = t.

All the indices i_ν are distinct. If ρ is a permutation of these indices, then c(A, P) = sign(ρ)c(A^ρ, P). This implies we can express P in terms of the expressions:

Ā = (ω_{a1b1i1/i2} − ω_{a1b1i2/i1} ) … (ω_{aubuit−1/it} − ω_{aubuit/it−1} )

dx_{i1} ∧ ⋯ ∧ dx_{it}

= Ω_{a1b1i1i2} … Ω_{aubuit−1it} dx_{i1} ∧ ⋯ ∧ dx_{it}

Again, using the alternating nature of these expression, we can express P in terms of expressions of the form:

Ω_{a1b1} ∧ ⋯ ∧ Ω_{aubu}

so that Q = Q(Ω) is a polynomial in the components Ω_{ab} of the curvature. Since the value of Q is independent of the frame chosen, Q is the invariant under the action of U(v). Using the same argument as that given in the proof of Lemma 2.1.3 we see that in fact Q is a characteristic form which completes the proof. We conclude this subsection with some uniqueness theorems regarding the local formulas we have been considering. If M is a holomorphic man- ifold of real dimension m and if V is a complex vector bundle of fiber dimension v, we let R_{ch m,m,p,v} denote the space of p-form valued invariants generated by the Chern forms of V and of T_{c}(M). There is a suitable ax- iomatic characterization of these spaces using invariance theory for Kaehler manifolds which we shall discuss in section 3.7. The uniqueness result we shall need in proving the Hirzebruch signature theorem and the Riemann- Roch theorem is the following: Lemma 2.6.3. (a) Let 0 ≠ R ∈ R_{m,m,m,v} then there exists (M, V) so M is oriented and ∫_{M} R(G, ∇) ≠ 0. (b) Let 0 ≠ R ∈ R_{ch m,m,m,v} then there exists (M, V) so M is a holomorphic manifold and ∫_{M} R(G, ∇) ≠ 0. Proof: We prove (a) first. Let ρ = {1 ≤ i_{1} ≤ ⋯ ≤ i_{ν}} be a partition k(ρ) = i_{1}+ ⋯+i_{ν} for 4k = s ≤ m. Let {M_{r ρ}} be the collection of manifolds of dimension s discussed in Lemma 2.3.4. Let P_{ρ} be the corresponding real characteristic form so  ∫_{M^{r} ρ} P_{τ}(G) = δ_{ρ,τ}. The {P_{ρ}} forms a basis for P_{m,s,s} for any s ≤ m. We decompose R = _{∑ ρ} P_{ρ}Q_{ρ}  for Q_{ρ} ∈ Q_{m,t,t,k}, where t + s = m. This decomposition is, of course, nothing but the decomposition defined by the projections π_{(s,t)} discussed in the proof of the previous lemma. Since R ≠ 0, at least one of the Q_{ρ} ≠ 0. We choose ρ so k(ρ) is maximal with Q_{ρ} ≠ 0. We consider M = M_{r ρ} × M_{2} and (V, ∇) = π_{∗ 2}(V_{2}, ∇_{2}) where (V_{2}, ∇_{2}) is a bundle over M_{2} which will be specified later. Then we compute:  ∫_{M} P_{τ}(G)Q_{τ}(∇) = 0 unless ord(Q_{τ}) ≤ dim(M_{2}) since ∇ is flat along M_{r ρ}. This implies k(τ) ≥ k(ρ) so if this integral is non-zero k(τ) = k(ρ) by the maximality of ρ. This implies ∫_{M} P^{τ}(G)Q^{τ}(∇) = ∫_{M^{r} ρ} P^{τ}(G) · ∫_{M2} Q^{τ}(∇^{2}) = δ^{ρ,τ} ∫_{M2} Q^{τ}(∇^{2}).

146 2.6. Invariance Theory This shows that ∫_{M} R(G, ∇) = ∫_{M2} Q_{ρ}(∇_{2}) and reduces the proof of this lemma to the special case Q ∈ Q_{m,m,m,v}. Let A be a v × v complex matrix and let {x_{1},…, x_{v}} be the normalized eigenvalues of A. If 2k = m and if ρ is a partition of k we define x_{ρ} = x_{i1 1} … x_{iν ν} , then x_{ρ} is a monomial of Q(A) for some ρ. We let M = M_{1} × ⋯ × Mν with dim(M_{j}) = 2i_{j} and let V = L_{1} ⊕ ⋯ ⊕ L_{ν} ⊕ 1^{v−ν} where the L_{j} are line bundles over M_{j}. If c(x_{ρ}, Q) is the coefficient of x in Q, then: ∫_{M} Q(∇) = c(x_{ρ}, Q)_{ν ∏ j=1} ∫_{Mj} c_{1}(L_{j})^{ij} . This is, of course, nothing but an application of the splitting principle. This reduces the proof of this lemma to the special case Q ∈ Q_{m,m,m,1} . If Q = c_{k 1} we take M = S^{2} × ⋯ × S^{2} to be the k-fold product of two dimensional spheres. We let L^{j} be a line bundle over the j^{th} factor of S 2 and let L = L_{1} ⊗ ⋯ ⊗ L_{k}. Then c_{1}(L) = c_{1}(L_{j}) so ∫_{M} c_{1}(L)^{k} = k!_{k ∏ j=1} ∫_{S 2} c_{1}(L_{j}) which reduces the proof to the case m = 2 and k = 1. We gave an ex- ample in Lemma 2.1.5 of a line bundle over S^{2} so ∫_{S 2} c_{1}(L) = 1. Al- ternatively, if we use the Gauss-Bonnet theorem with L = T_{c}(S^{2}), then ∫_{S 2} c_{1}(T_{c}(S^{2})) = ∫_{S 2} e_{2}(T (S^{2})) = χ(S^{2}) = 2 ≠ 0 completes the proof of (a). The proof of (b) is the same where we replace the real manifolds M_{r ρ} by the corresponding manifolds M_{c ρ} and the real basis P_{ρ} by the correspond- ing basis P_{c ρ} of characteristic forms of T_{c}(M). The remainder of the proof is the same and relies on Lemma 2.3.4 exactly as for (a) and is therefore omitted in the interests of brevity.