Quaternionic Structures in Mathematics and Physics SISSA - Trieste, 5-9 September 1994

Stokes’ Formula and the Dirac Operator on Embedded Manifolds

J. Cnops^{∗}

ABSTRACT

In this paper we give an alternative approach to the definition of the Dirac operator on manifolds, taking as a starting point Stokes’ formula. This definition can be used on a quite general class of manifolds, and has the advantage of being very simple when the structure of the manifold is. In order to be able to use Stokes’ formula on manifolds with a non-definite metric the formula is slightly changed so as to avoid the use of the notion of outward normal vector. Conformal transformation properties of the operator are given, as well as some examples.

∗ Post-doc researcher NFWO, Belgium. Address: RUG, Galglaan 2, B-9000 Gent, Belgium.

1 Introduction

The definition of the Dirac operator as we construct it here is inspired both by the idea of the areolar derivative as introduced by Theodoresco in [7] and the derivative of a differential form, which can be found e.g. in [4].

Theodoresco defined for Euclidean space the areolar derivative as follows. Given a function f defined and, say, C^{1} in a neighbourhood of some point x⃗. Then the areolar derivative of f in x⃗ is defined as the limit for r → 0 of the integral over the sphere, centred in x⃗ with radius r, of f multiplied with the outer normal of the sphere. In order to perform this multiplication with a vector, Theodoresco used the Clifford algebra IR_{0n}.

We shall only define the Dirac operator for (pseudo-)Riemannian, orientable manifolds which can be conformally embedded in a (pseudo-)Euclidean space.

This is not a restriction, really, since (up to a smoothness condition) any metric manifold can be isometrically (and hence conformally) embedded in some (pseudo-)Euclidean space of sufficiently big dimension. This was proved for the Riemannian case by Nash [6], where the embedding is in Euclidean space, and in general by Clarke, [2].

Quite often however a manifold is embedded in a natural way in a pseudo-Riemannian space, and the structure of the embedding space is readily at hand. The general notion of embedding provides then a link between several possible embeddings. Due to the embedding in a pseudo-Euclidean space we could do without the notion of tangent space in the abstract sense, and only use the definition of tangent space in its classical geometrical meaning. Moreover, since the Clifford algebra IR_{pq} is available, the notion of forms could be disposed with (even if the notion of course is present in the background), and we use Clifford algebra valued functions instead. We have defined integrals in terms of classical integrals on manifolds, where also the notion of k-forms is not explicitly introduced, even if, again, this notion is implicitly present in the background.

Regarding terms and notations, recall that a pseudo-Riemannian manifold M is a manifold such that, at any point x⃗ of the manifold there is a nondegenerate symmetric bilinear form ⟨·, ·⟩x⃗ on the tangent space Tx⃗, where the mapping x⃗ → ⟨·, ·⟩x⃗ is C^{∞}. The manifold is called Riemannian if this bilinear form is positive definite.

Clifford algebras

Definitions. Basic properties. Let IR^{pq} be the real n-dimensional space, where n = p+q, endowed with the nondegenerate inner product B(·, ·) of signature (p, q). We assume that an orthonormal basis e_{1}, …, e_{n} is given, that is a basis such that

\begin{matrix}B(e_{i}, e_{i}) = & −1, & i = 1, …, p \\ B(e_{i}, e_{i}) = & 1, & i = p + 1, …, n \\ B(e_{i}, e_{j}) = & 0, & i ≠ j. \end{matrix}

It should be noticed that the notation is taken such that Euclidean space with positive definite inner product is written as IR^{0n} (which will sometimes be abbreviated to IR^{n}).

The Clifford algebra over IR^{pq} is denoted by IR_{pq}. It is the free algebra generated by the e_{i} modulo the relations

e_{i}e_{j} + e_{j}e_{i} = −2B(e_{i}, e_{j}).  (1)

Elements of IR_{pq} are called Clifford numbers or Clifford scalars. As a vector space the Clifford algebra has a basis of elements of the form e_{t_{1}} … e_{t_{k}} where t_{1} < … < t_{k}. We shall use two shorthand notations for such a product:

e_{t_{1}} … e_{t_{k}} = e_{t_{1}...t_{k}} = e_{{t_{1},...,t_{k}}}, t_{1} < … < t_{k}.

The dimension of IR_{pq} over IR is equal to 2^{n}.

If A has k elements, then e_{A} is called a k-vector. Likewise any linear combination of k-vectors is called a k-vector, and the vector space of k-vectors is written as IR_{pq}^{k}. Obviously IR_{pq} is the direct sum of all IR_{pq}^{k} for k ≤ n, and the projection of a Clifford scalar a on IR_{pq}^{k} will be written as [a]_{k}. Instead of 1-vectors the term vectors is used, and vectors are identified with elements of IR^{pq}. They will be denoted by x⃗, y⃗ etc. Also the term bivectors is used for 2-vectors. The n-vector e_{{1,...,n}} is called the pseudoscalar. It is often written as e_{N}.

The following (anti)automorphisms will be used:

- The main antiautomorphism defined by

¯x⃗ = −x⃗ \overline{(ab)} = b̄ā.

- The main automorphism is defined by

x′ = −x⃗ (ab)′ = a′b′.

These (anti)morphisms will mainly be applied to products of vectors. Explicitly one obtains for a product of k vectors:

\overline{(x⃗_{1} … x⃗_{k})} = (−x⃗_{k}) … (−x⃗_{1})
(x⃗_{1} … x⃗_{k})′ = (−x⃗_{1}) … (−x⃗_{k}).

It is not uncommon to define a so-called dot product on the Clifford algebra. We shall only need it when both arguments are vector valued, and in this case the dot product is given by

x⃗ · y⃗ = \frac{1}{2}(x⃗y⃗ + y⃗x⃗) = −B(x⃗, y⃗).

The wedge product between two vectors is given by x⃗ ∧ y⃗ = \frac{1}{2}(x⃗y⃗ − y⃗x⃗). If b is a k-vector, then

x⃗ ∧ b = \frac{1}{2}(x⃗b + (−1)^{k}bx⃗),

giving a k+1-vector. This can be extended in a unique way to the general case a∧b, using linearity and demanding associativity, to the wedge product of the so-called Grassman algebra, but it will be needed here only when the first argument is a vector.

Notice that, if x⃗ is parallel (orthogonal) to y⃗, then x⃗ · y⃗ (x⃗ ∧ y⃗) is equal to x⃗y⃗: parallel vectors commute, while orthogonal vectors anticommute. As a consequence x^{2} is real for any vector x⃗, and x⃗ is invertible as an element of the Clifford algebra if and only if x^{2} ≠ 0. Indeed, if x^{2} = 0, then x⃗ is a zero divisor and hence not invertible, while in the opposite case we can put x^{−1} = x⃗(x^{2})^{−1}.

Let V be a right module over IR_{pq}. An inner product on V is a real-bilinear mapping V × V → IR_{pq} such that it is IR_{pq}-linear in the second argument and moreover either (f, g) = \overline{(g, f)} (symmetric) or (f, g) = −\overline{(g, f)} (antisymmetric inner product), for all f and g. From this a real valued inner product can be derived putting

(f, g)_{IR} = [(f, g)]_{0}.

Analoguous definitions can be given for left modules. Notice that here the inner product is linear in the first argument.

Subalgebra and subgroups. The even subalgebra IR_{pq}^{+} is the set of Clifford numbers a such that a = a′. Alternatively it can be defined as the subalgebra generated by products of even numbers of vectors. Several groups in the Clifford algebra will be of interest. We have

- The Clifford group Γ(p, q), sometimes called the Lipschitz group. It is generated, as a group, by the invertible vectors. Notice that for an invertible vector x⃗, x⃗¯x⃗ = B(x⃗, x⃗) is real and non-zero. By induction, for any a in Γ(p, q), aā is real and non-zero.

- The Pin group Pin(p, q) is the subgroup of Γ(p, q) of elements a for which aā = ±1. It can be considered as the group generated by the unit vectors, i.e. vectors with x⃗¯x⃗ = ±1.

- The Pin plus group Pin^{+}(p, q) is the group of those elements a of Pin(p, q) such that aā = +1. For p = 0, Pin^{+}(0, n) = Pin(0, n). Notice that Pin^{+}(p, q) usually is not the group generated by the vectors such that x⃗¯x⃗ = +1. So, e.g., Pin^{+}(n, 0) is not trivial for n > 1, while there is no vector in IR^{n0} with x⃗¯x⃗ = 1.

- The Spin group Spin(p, q) is the intersection of Pin(p, q) with IR_{pq}^{+}. It is the group consisting of the products of even numbers of unit vectors.

If p = 0 often the notations Γ(n) etc. instead of Γ(0, n) etc. will be used, in analogy with the short notation IR^{n} for IR^{0n} It should be noticed that for any element of the Clifford group the equality

a′ = εa

holds, where ε = +1 when a is in the even subalgebra and ε = −1 if this is not the case. For an element s of the Pin group a transformation of IR_{pq} is defined by

χ(s)a = sas^{′−1}.

This clearly defines a group action. χ(s) leaves the space of k-vectors invariant for every k, and is usually only applied to IR^{pq}. As an example we calculate χ(x⃗) for a vector x⃗ as it acts on IR^{pq}. A vector y⃗ can be split into a part λx⃗ parallel to x⃗ (λ real), and a part y⃗_{⊥} orthogonal to x⃗, and hence anticommuting with x⃗. Since x′ = −x⃗ one obtains

χ(x⃗)y⃗ = x⃗y⃗x^{′−1} = −x⃗(λx⃗ + y⃗_{⊥})x^{−1}
= −λx⃗(x⃗x^{−1}) + y⃗_{⊥}(x⃗x^{−1})
= −λx⃗ + y⃗_{⊥}.

So χ(x⃗) is the orthogonal reflection into the hyperplane orthogonal to x⃗, and χ(a) is an orthogonal transformation for any a ∈ Pin(p, q). It is easily seen that χ(1) and χ(−1) both are the identity, and it is possible to prove that no other a results in the identity. This way Pin(p, q) gives a double covering of the orthogonal group O(p, q), while Spin(p, q) gives a double covering of SO(p, q).

When dealing with conformal mappings use will be made of mappings of the form

x⃗ → ax⃗a^{∗},

where a is an element of the Clifford group. For such a the element b = a/\sqrt{|aā|} is in the Pin group, and b^{′−1} = εb^{−1} = εā\sqrt{aā}/(aā), where ε = ±1. Hence ax⃗ā is up to a constant χ(b), and the mapping x⃗ → ax⃗ā is the product of a dilatation and an orthogonal transformation.

Monogenic functions

Definitions. Let f be a C^{1}, IR_{0n}-valued function on a domain in IR^{0n}. The Dirac operator D for Euclidean space is given by

Df(x⃗) = ∑_{i=1}^{n} e_{i}∂_{i}f(x⃗).

It is a linearisation of the Laplacian, D^{2} = −Δ.

Let spherical coordinates of Euclidean space be given by the equation x⃗ = rξ, where r = |x⃗| and ξ is a unit vector. The Dirac operator can be split into a radial and a spherical part:

D = ξ (∂_{r} + \frac{Γ}{r})

where ∂_{r} = ξ · D = ∑_{i}(ξ · e_{i})∂_{i} and Γ = ξ ∧ D = ∑_{i}(ξ ∧ e_{i})∂_{i}. Writing out both sides of the equation −D^{2} = Δ in spherical coordinates gives

∂_{r}^{2} + \frac{Γ − ξΓξ}{r} ∂_{r} − \frac{Γ + ξΓξΓ}{r^{2}} = ∂_{r}^{2} + \frac{m − 1}{r}∂_{r} + \frac{Δ_{LB}}{r^{2}} ,

where Δ_{LB} is the Laplace-Beltrami operator on the sphere of IR^{0n}. This results in

Γ − ξΓξ = n − 1
−ξΓξΓ − Γ = (2 − n + Γ)Γ = Δ_{LB}, (2)

2 Dirac Operators

It is possible, for the definition of the Dirac operator, to allow milder assumptions on the continuity of the functions involved (such as the embedding mapping) than C^{∞}, as only derivatives of some small order are needed. However this would need ad hoc adaptation of the continuity demands, and this would be at the expense of clarity. Definition. A metric manifold M is said to be conformally embedded in IR_{pq} if there exists a one to one C^{∞} mapping ϕ of M to a submanifold ϕ(M) of IR^{pq} such that there exists a strictly positive C^{∞} function α on ϕ(M) describing the metric of M.

This description of the metric is to be understood as follows: as a submanifold of IR^{pq} the tangent space T_{ϕ(x⃗)} is described in a natural way in IR^{pq}, leading to the metric ⟨·, ·⟩_{ϕ(x⃗)} on the tangent space T_{ϕ(x⃗)}, while ϕ induces a mapping Tx⃗ → T_{ϕ(x⃗)}, which will be denoted by ϕx⃗. We say that α describes the metric of M on ϕ(M) whenever

α(ϕ(x⃗))^{2}⟨t, s⟩x⃗ = ⟨ϕx⃗(t), ϕx⃗(s)⟩_{ϕ(x⃗)}

for any x⃗ in M and t and s in Tx⃗. The natural metric on ϕ(M) will in the sequel be denoted by N if there is no confusion about the conformal embedding used, and the manifold M will then be identified with ϕ(M). That the metric is described by α will then be abbreviated by saying that the metric on M is given by N/α.

It is of course true that the embedding is not unique. Indeed, a conformal mapping of IR_{pq} into itself (we assume for simplicity that no point of M is mapped to infinity) gives a new conformal embedding. As an example we can take the punctured sphere Sn−1 ; s⃗, which is the sphere S^{n−1} with the south pole s⃗ excluded, with its natural metric as a subset of IR^{n}. Using a Möbius transformation (the so-called Cayley transformation) this sphere can be mapped conformally to the hyperplane IR^{n−1}, and the metric in terms of the natural metric N_{n−1} of IR^{n−1} becomes N_{n−1}(x⃗)/(1 + |x⃗|^{2}). As will be shown however, the relation between the relevant terms for different conformal embeddings, such as integration over cycles with codimension 1, satisfy simple relations which make it easy to relate the different models. It should be remarked however that the notion of constant function cannot be transferred this way and is peculiar to the conformal embedding used.

2.1 Integration over cycles

Functions on M or on a cycle C will always be assumed to take values in an algebra A containing IR_{pq} as a subalgebra, and having an antiautomorphism a → ā which coincides on IR_{pq} with the classical main antiautomorphism. If one has one conformal embedding of M in IR_{pq}, then of course A can be taken to be IR_{pq} itself. However, if we want to compare two embeddings of M, say in IR_{pq} and IR_{p′q′}, then it is useful to consider functions with values in a Clifford algebra containing both IR_{pq} and IR_{p′q′}.

Moreover, if any function space is mentioned, it is always implicitly supposed to be a space of functions with values in A, and to be regarded as a right module over IR_{pq}, where the action is defined by pointwise multiplication.

Let M be embedded in IR^{pq} with metric N/α and let C be a k-cycle in M. Then the notion of oriented elementary k-dimensional surface element can be described in terms of the Clifford algebra, and so the notion

∫_{C} F (x⃗)dM_{k}(x⃗)G(x⃗)

is clear.

It is possible to give the definition in terms of the classical integral over C. Assume that t = (t_{1}, t_{2}, …, t_{k}) is a set of coordinates on C with orientation compatible with that of C (global coordinates are assumed here; patching together local coordinates is left to the reader), such that ψ maps a domain Ω in IR^{k} to C. For t inside Ω fixed the vectors ∂_{i}ψ(t) are well defined and

∫_{C} F (x⃗)dM_{k}(x⃗)G(x⃗) = ∫_{Ω} F (ψ(t)) ( ⋀_{i=1}^{k} ∂_{i}ψ(t)) G(ψ(t))\frac{d_{k}(t)}{α(ψ(t))^{k}}

where d_{k} is k-dimensional Lebesgue measure. With the integral over C it is possible to define an A-valued inner product on the space of A-valued functions on C (which is not defined for all functions; we refrain from specifying a domain for this inner product as it may be used for duality relations) by the formula

(F, G)_{C} = ∫_{C} \overline{F (x⃗)}dM_{k}(x⃗)G(x⃗).

Whether this inner product is symmetric or antisymetric depends on k.

2.2 The Dirac operator

The Dirac operator of a manifold M, D_{M} is defined using Stokes’ theorem as a basis. Notice that, for an (m − 1) cycle C we define the orientation by a vector w⃗ not parallel to the cycle in a point x⃗ as follows: let a set of local coordinates (t_{2}, …, t_{m}) of C be given, and t_{1} an extra coordinate such that (t_{1}, …, t_{m}) is a set of local coordinates on M such that t_{1} increases in the direction of w⃗. Then (t_{2}, …, t_{m}) agrees with the orientation of C if and only if (t_{1}, …, t_{m}) agrees with the orientation of M. For a cycle C its boundary has the inherited orientation which is defined in any point by a vector pointing outwards. Definition. Let Ω be a domain in the m-dimensional manifold M. A real-linear operator P : C^{1}(Ω) → C^{0}(Ω) such that, for any f and g in C^{1}(Ω), Stokes’ theorem holds is called a pre-Dirac operator on M in Ω.

Stokes’ theorem says that, for any m-dimensional bounded cycle C in Ω having boundary ∂C with the inherited orientation,

∫_{∂C} \overline{f(y⃗)}dM_{m−1}(y⃗)g(y⃗)  (3)
= ∫_{C} \overline{P f(x⃗)}dM_{m}(x⃗)g(x⃗) + (−1)^{m} ∫_{C} \overline{f(x⃗)}dM_{m}(x⃗)P g(x⃗),  (4)

or, in inner product notation (f, g)_{∂C} = (P f, g)_{C} + (−1)^{m}(f, P g)_{C}.

It is clear that P is IR_{pq}-linear. Indeed, if one takes a function g with invertible values everywhere in Ω and replaces in Stokes’ formula f by fλ, then necessarily P (fλ) = (P f)λ. It is not necessarily true that there is only one pre-Dirac operator in a domain Ω. Later on we shall describe all possible pre-Dirac operators of a manifold in terms of the Dirac operator. In order to distinguish a special operator, which will be the Dirac operator of the manifold, we have to introduce an extra condition. Definition. A pre-Dirac operator D_{M} which is such that for any real valued g and each x⃗ in Ω, D_{M}g(x⃗) is a vector tangent to M in x⃗, is called a Dirac operator. Remark This definition of a Dirac operator differs rather considerably from other definitions, since a curvature term here is missing. As a result the notion of Dirac operator is not completely independent of the embedding of the manifold. However this rather wilful dependence of the Dirac operator on the embedding will serve to clarify the inherent differences of Spin representations and of possible inner products to make these unitary in the case where different embeddings are available.

Theorem 2.1 The Dirac operator, if it exists, is unique.

Proof. Let L and T be two Dirac operators. It is sufficient to prove that T f = Lf for real valued functions. Take such f and let Z_{f} the interior of the set where f is zero. If Z_{f} is not empty, then for any cycle C within Z_{f} and any function g both ∫_{∂C} \overline{f(y⃗)}dM_{m−1}(y⃗)g(y⃗) and ∫_{C} \overline{f(x⃗)}dM_{m}(x⃗)P g(x⃗) are zero, and so, by continuity P f(x⃗) is zero in Z_{f} for any pre-Dirac operator. Hence Lf(x⃗) = T f(x⃗) in Z_{f}. Take now a cycle C outside Z_{f}, and subtract Stokes’ formula for T and L. This gives

∫_{C} \overline{(T − L)f(x⃗)}dM_{m}(x⃗)g(x⃗) + (−1)^{m} ∫_{C} \overline{f(x⃗)}dM_{m}(x⃗)(T − L)g(x⃗) = 0.

It is true that \overline{(T − L)f(x⃗)}dM_{m}(x⃗) = (−1)^{m}dM_{m}(x⃗)(T − L)f(x⃗), because dM_{m}(x⃗), like the pseudoscalar of the tangent space in x⃗, commutes with any tangent vector if m is odd, and anticommutes with any such vector if m is even. Hence the equation can be written, if we take g = f, as 2 ∫_{C} dM_{m}(x⃗)f(x⃗)(T − L)f(x⃗) = 0. Since this holds for any C outside Z_{f}, and f ≠ 0 almost everywhere there, this means that (L − T)f = 0.

Remark The definition of the Dirac operator is locally independent of the domain Ω chosen. Indeed, let Ω_{1} and Ω_{2} be two domains in M, with the two Dirac operators D_{1} and D_{2}. If the intersection is non-empty, and f is a function in C^{1}(Ω_{1} ∪ Ω_{2}), then by the uniqueness theorem, D_{1}f = D_{2}f in Ω_{1}∩Ω_{2}, while C^{1}(Ω_{1}∪Ω_{2}) = C^{1}(Ω_{1})⊕C^{1}(Ω_{2}) if the intersection is empty. In both cases the extension of the Dirac operator to C^{1}(Ω_{1}∪Ω_{2}) is straightforward.

Using the Dirac operator it is now possible to describe all pre-Dirac operators.

Theorem 2.2 Let M be a manifold with Dirac operator D_{M}, and let T be a pre-Dirac operator. Then there exists a C^{0} function λ satisfying \overline{λ(x⃗)}dM_{m}(x⃗) = (−1)^{m−1}dM_{m}(x⃗)λ(x⃗) such that

T f(x⃗) = (D_{M} − λ(x⃗))f(x⃗).

Proof. Take the constant function g(x⃗) = 1, and define λ(x⃗) = (D_{M} − T)g(x⃗). If for any C and any f, we subtract Stokes’ formula for T from that for D_{M} we obtain

0 = ∫_{C} \overline{(D_{M} − T)f(x⃗)}dM_{m}(x⃗) + (−1)^{m} ∫_{C} \overline{f(x⃗)}dM_{m}(x⃗)λ(x⃗).

For f = 1 this gives \overline{λ(x⃗)}dM_{m}(x⃗) = (−1)^{m−1}dM_{m}(x⃗)λ(x⃗); then for arbitrary f it gives \overline{(D_{M} − T)f(x⃗)} = \overline{f(x⃗)} \overline{λ(x⃗)}.

The following theorem is a generalisation of the chain rule.

Theorem 2.3 Let g be real valued and let T be a pre-Dirac operator on M. Then, for any f,

T (gf) = (T g − g(T 1))f + g(T f)

Proof. We first prove the theorem for the Dirac operator. It is obvious that (fg, 1)_{∂C} = (f, g)_{∂C}. Applying Stokes’ theorem to both sides of the equation shows that

(D_{M}(fg), 1)_{C} + (−1)^{m}(fg, D_{M}1)_{C} = (D_{M}f, g)_{C} + (−1)^{m}(f, D_{M}g)_{C}

Since D_{M}g is vector valued, we get \overline{D_{M}g}dM_{n} = (−1)^{m}dM_{n}D_{M}g, and the same holds for D_{M}1. The equation can thus be rewritten as ([D_{M}(fg) + (D_{M}1)fg − g(D_{M}f) − (D_{M}g)f], 1)_{C} = 0. Since this holds for all C, the term between square brackets is zero everywhere. From the expression T f(x⃗) = D_{M}f(x⃗) − λ(x⃗)f(x⃗) the general case easily follows.

We finally prove that, even if the conformal embedding chosen influences the explicit form of the Dirac operator, different embeddings essentially lead to the same operator. Assume that for a given manifold we have two conformal embeddings, ϕ in IR^{uv} and ψ in IR^{sr}. To be able to make comparisons we define the Dirac operator for functions with values in IR_{pq}, where p = max{u, s} and q = max{v, r}. Of course we can then consider both ϕ and ψ as conformal embeddings in IR^{pq}, so in the following theorem we can immediately assume that both embeddings are in the same space, and moreover that ψ is the identical mapping. For any point x⃗ of M we again consider the mapping ϕ between the tangent spaces Tx⃗ and T_{ϕx⃗}. This mapping can be extended to a mapping Rx⃗ on IR^{pq} which is up to a constant an orthogonal transformation (even for fixed x⃗ this is not uniquely defined, unless m = p + q). Since however ϕ is an isometry, the constant in consideration must be 1, and since the orientations of M and ϕ(M) are the same, one can take the orthogonal transformation to be direct. Hence such a transformation can be described by an element of the Spin(p, q) group, Sx⃗. Now, while the mapping x⃗ → Rx⃗ can be chosen in such a way that it is continuous, this is not necessarily true for the mapping x⃗ → Sx⃗ (locally it can be defined to be continuous, but after travelling around it might be that we end up with the sign changed), and we must make an extra assumption to relate Stokes’ formula on M and on ϕ(M):

Theorem 2.4 Assume that the Spin field Sx⃗ can be chosen such as to be continuous. Define the IR_{pq} linear mapping Φ : F (M) → F (ϕ(M)) (here F (M) means the space of all A-valued functions on M) by

Φf(y⃗) = Sx⃗f(x⃗)

where ϕ(x⃗) = y⃗. Then Let C be any cycle on M and g and f be two functions. Then if, for any of the two pairs of integrals

∫_{∂C} \overline{f(x⃗)}dM_{m−1}(x⃗)g(x⃗)  ∫_{∂ϕC} \overline{Φf(y⃗)}d(ϕ(M))_{m−1}(y⃗)Φg(y⃗)
∫_{C} \overline{f(x⃗)}dM_{m}(x⃗)g(x⃗)  ∫_{ϕC} \overline{Φf(y⃗)}d(ϕM)_{m}(y⃗)Φg(y⃗)

one of the integral makes sense, then so does the other. Moreover they are either equal, or differ in sign (this does not depend on C, only on the chosen field S).

Proof. The action of Sx⃗ can be extended from IR^{pq} to the Clifford algebra and obviously we have (with a bit of abuse of notation) that, if y⃗ = ϕ(x⃗), then

d(ϕ(M))_{m−1}(y⃗) = Sx⃗dM_{m−1}(x⃗)S′−1 ; x⃗
d(ϕ(M))_{m}(y⃗) = Sx⃗dM_{m}(x⃗)S′−1 ; x⃗

Now, since Sx⃗ is in the Spin group, the expression ε = \overline{Sx⃗}Sx⃗S′−1 ; x⃗Sx⃗ is either +1 or −1, and from the continuity of S it follows that this expression is independent of x⃗, so

\overline{Sx⃗}d(ϕ(M))_{m−1}(y⃗)Sx⃗ = εdM_{m−1}(x⃗)
\overline{Sx⃗}d(ϕ(M))_{m}(y⃗)Sx⃗ = εdM_{m}(x⃗)

Remark The condition that a continuous function S exists is of course closely linked to the condition of the existence of a Spin structure on the manifold. Moreover it should be noticed that the mapping Φ cannot be defined in classical theory, since it does not map sections of the Clifford bundle of M to sections of the Clifford bundle of ϕ(M).

Corollary 2.5 If M has a pre-Dirac operator T, then the operator T_{ϕ} defined on ϕ(M) by

T_{ϕ}f = ΦT Φ^{−1}f

is a pre-Dirac operator on ϕ(M).

Proof. ¿From the previous theorem it is clear that Stokes’ theorem holds for T_{ϕ}, and so T_{ϕ} is a pre-Dirac operator.

Let now D_{M} be the Dirac operator on M, and let D_{ϕ} be defined by

D_{ϕ}f = ΦD_{M}Φ^{−1}f.

For any real valued function g we have that

[D_{ϕ}(g(y⃗)Sx⃗)] (Sx⃗)^{′−1} = [(D_{ϕ}g)Sx⃗ + g(D_{ϕ}Sx⃗) − (D_{ϕ}1)gSx⃗] (Sx⃗)^{′−1} =
Sx⃗(D_{M}G(x⃗))(Sx⃗)^{′−1} = (D_{ϕ}g)ε + gSx⃗(D_{M}1)(Sx⃗)^{′−1} − (D_{ϕ}1)gε

where ε = Sx⃗(Sx⃗)^{′−1} = ±1, and G(x⃗) = g(y⃗). Since G is real valued, D_{M}G(x⃗) is in the tangent space Tx⃗, and the expression at the right hand side is in Ty⃗. Similarly the second term at the right hand side is in Ty⃗, and the remaining terms are either both in Ty⃗, or not. Hence, if there is one g, vanishing almost nowhere, such that D_{ϕ}g(y⃗) is in the tangent space, then D_{ϕ}1(y⃗) is in the tangent space, and so D_{ϕ}g is in the tangent space for all real g, and D_{ϕ} is the Dirac operator of ϕ(M).

2.3 Examples of Dirac operators

2.3.1 Euclidean space

In the case where the m-dimensional manifold M is embedded in IR^{n}, m = n, and the orientation of M coincides with the orientation given by the orthonormal basis e_{1}, …, e_{n}, Stokes’ formula can be written in a different form. Fix C, let e_{N} = e_{1} … e_{n} be the pseudoscalar of IR_{0n} and let ny⃗ be the outer unit normal on ∂C. Then the equalities

dM_{n}(x⃗) = \frac{e_{N}dx⃗}{α^{n}(x⃗)}
dM_{n−1}(y⃗) = \frac{−ny⃗e_{N}dσ(y⃗)}{α^{n−1}(y⃗)} = (−1)^{n} \frac{e_{N}ny⃗dσ(y⃗)}{α^{n−1}(y⃗)},

hold, where again N/α is the metric of M, dx⃗ is the Lebesgue measure on IR^{n} and dσ(y⃗) is the n − 1-dimensional surface measure. For real valued f we have that D_{M}fe_{N} = (−1)^{n−1}e_{N}D_{M}f and Stokes’ formula becomes

∫_{∂C} \overline{f(y⃗)}\frac{ny⃗dσ(y⃗)}{α^{n−1}(y⃗)}g(y⃗)

= − ∫_{C} \overline{D_{M}f(x⃗)}\frac{dx⃗}{α^{n}(x⃗)}g(x⃗) + ∫_{C} \overline{f(x⃗)}\frac{dx⃗}{α^{n}(x⃗)}D_{M}g(x⃗)

which still is IR_{0n}-linear in f and so is equivalent to Stokes’ formula even for f which are not real. This is (almost) the form of Stokes’ theorem which is used e.g. in [3] for Euclidean space itself, i.e. for α(x⃗) = 1. The proof there that the Dirac operator defined there satisfies Stokes’ theorem thus can be considered as a proof that this operator is the Dirac operator as we have defined it:

Theorem 2.6 The Dirac operator in Euclidean space is defined by

Df(x⃗) = ∑_{i=1}^{n} e_{i}∂_{i}f(x⃗).

¿From this operator it easy to derive the Dirac operator for the more general metric N/α:

Theorem 2.7 Let the m-dimensional manifold M be embedded in IR^{n}, with m = n, as a domain with metric N/α. Then the Dirac operator D_{M} is given as

D_{M}f(x⃗) = α(x⃗)^{\frac{1+n}{2}} D (α(x⃗)^{\frac{1−n}{2}} f(x⃗)) .

Proof. It is immediately clear that if f is real valued, then D_{M}f is vector valued, so we only have to prove that D_{M} is a pre-Dirac operator. We put F (x⃗) = α(x⃗)^{(1−n)/2}f(x⃗) and G(x⃗) = α(x⃗)^{(1−n)/2}g(x⃗) and obtain

∫_{∂C} \overline{f(y⃗)}dM_{n−1}(y⃗)g(y⃗)
= ∫_{∂C} \overline{F (y⃗)}dN_{n−1}(y⃗)G(y⃗)

= ∫_{C} \overline{DF (x⃗)}dN_{n}(x⃗)G(x⃗) + (−1)^{n} ∫_{C} \overline{F (x⃗)}dN_{n}(x⃗)DG(x⃗)
= ∫_{C} \overline{α(x⃗)^{(−1−n)/2)}D_{M}f(x⃗)}dN_{n}(x⃗)α(x⃗)^{(1−n)/2)}g(x⃗)
+(−1)^{n} ∫_{C} \overline{α(x⃗)^{(1−n)/2)}f(x⃗)}dN_{n}(x⃗)α(x⃗)^{(−1−n)/2)}D_{M}g(x⃗)
= ∫_{C} \overline{D_{M}f(x⃗)}dM_{n}(x⃗)g(x⃗) + (−1)^{n} ∫_{C} \overline{f(x⃗)}dM_{n}(x⃗)D_{M}g(x⃗).

2.3.2 The sphere

We consider here the sphere S = S^{n−1} embedded in the natural way in IR^{n}. Let C be a cycle in S and let T be the n-dimensional cycle T = {rξ : ξ ∈ C and r_{1} ≤ r ≤ r_{2}} for some 0 < r_{1} < 1 < r_{2}. ∂T consists of three parts: r_{2}C, r_{1}C (but with opposite orientation), and ∂C × [r_{1}, r_{2}]. We choose the orientation of the sphere defined by the outer unit normal to the sphere and obtain, for dN_{n−1} on the sphere, explicitly that

dN_{n}(x⃗) = ξr^{n−1}drdN_{n−1}(ξ)

If one calculates ∫_{∂T} \overline{F (y⃗)}dN_{n−1}(y⃗)G(y⃗), where F is defined in T by F (rξ) = ξr^{1−n}f(ξ) while G(rξ) = g(ξ), one sees that the terms over the first two parts annihilate each other while the integral over the third part can be written as an integral over ∂C

∫_{∂T} \overline{F (y⃗)}dN_{n−1}(y⃗)G(y⃗) = ∫_{∂C} \overline{ξf(ξ)} (∫_{r_{1}}^{r_{2}} \frac{−ξ}{r} dr) dN_{n−2}(ξ)g(ξ)
= − ∫_{∂C} \overline{f(ξ)}dN_{n−2}(ξ)g(ξ) (\frac{1}{r_{1}} − \frac{1}{r_{2}} )

The right hand side of Stokes’ equation for T becomes

∫_{T} \overline{\frac{Γf(ξ)}{r}}dN_{n}(x⃗)\frac{1}{r^{n−1}} g(ξ) + (−1)^{n} ∫_{T} \overline{ξf(ξ)}dN_{n}(x⃗)\frac{ξΓg(ξ)}{r^{n}}

where we have calculated DF (x⃗) by

DF (x⃗) = ξ (∂_{r} + \frac{Γ}{r}) \frac{ξ}{r^{n−1}} f(ξ)
= \frac{1 − n}{r^{n}}ξ^{2}f(ξ) + \frac{ξΓξf(ξ)}{r^{n}}
= \frac{(n − 1) + (Γ − n + 1)}{r^{n}}f(ξ) = \frac{Γf(ξ)}{r^{n}}

using the formula ξΓξ = Γ − n + 1. If we write the integrals over T as a double integral, first for r going from r_{1} to r_{2} and then over C we obtain

− (\frac{1}{r_{1}} − \frac{1}{r_{2}} ) ∫_{∂C} \overline{f(ξ)}dN_{n−2}(ξ)g(ξ)

= (\frac{1}{r_{1}} − \frac{1}{r_{2}} )
(∫_{C} \overline{Γf(ξ)}ξdN_{n−1}(ξ)g(ξ) + (−1)^{n} ∫_{C} \overline{ξf(ξ)}ξdN_{n−1}(ξ)ξΓg(ξ))
= − (\frac{1}{r_{1}} − \frac{1}{r_{2}} )
(∫_{C} \overline{ξΓf(ξ)}dN_{n−1}(ξ)g(ξ) + (−1)^{n−1} ∫_{C} \overline{f(ξ)}dN_{n−1}(ξ)ξΓg(ξ))

This proves that ξΓ is a pre-Dirac operator on S. However, from the explicit expression from the Γ operator, Γ = −ξ ∧ D = − ∑_{i}(ξ ∧ e_{i})∂_{i} it follows that in the expression

D_{S} = ξΓ = − ∑_{i} ξ(ξ ∧ e_{i})∂_{i}

the coefficients of each ∂_{i} are orthogonal to ξ and hence parallel to the sphere, and so D_{S} is the Dirac operator of the sphere.

2.3.3 The nondefinite space IR^{pq}

The Dirac operator here is given by

Df(x⃗) = − ∑_{i=1}^{n} e_{i}^{3}∂_{i}f(x⃗)
= − ∑_{i=1}^{p} e_{i}∂_{i}f(x⃗) + ∑_{i=p+1}^{n} e_{i}∂_{i}f(x⃗)

where n = p + q. Indeed, for f real valued this obviously leads to values Df(x⃗) which are vector valued and parallel to IR^{pq}, and D is IR_{pq}-linear. The only thing left to prove is that D is a pre-Dirac operator, i.e. that Stokes’ theorem holds. This is done in the same way as in the case of IR^{n}. Notice, that one cannot rewrite Stokes’ theorem here in the same way as was done for IR^{n} without caution. Indeed, while it is always possible to write

dM_{n−1}(y⃗) = \frac{−ny⃗e_{N}dσ(y⃗)}{α^{n−1}(y⃗)}

for some vector ny⃗, which is normal to ∂_{C}, this needs not be the outer unit normal. Indeed ∂_{C} may have an isotropic normal vector (for which no exterior direction can even be defined). Moreover, even if ny⃗ is not isotropic, the sign of n2 ; y⃗ must be taken into account.

The Dirac operator is a linearisation of the wave operator:

D^{2} = = ∑_{i} e_{i}^{2}∂_{i}^{2}.

REFERENCES

[ 1 ]H. Blaine Lawson, M-L. Michelsohn: Spin geometry, Princeton Mathematical Series 38, Princeton University Press, 1989.

[ 2 ]C. J. S. Clarke: ‘On the global isometric imbedding of pseudo-Riemannian manifolds’, Proc. Roy. Soc. London, ser. A, 314(1970), pp.417-428.

[ 3 ]R. Delanghe, F. Sommen, V. Souček:Clifford analysis and spinor valued functions, Kluwer Acad. Publ., Dordrecht, 1992.

[ 4 ]G. de Rham:Differentiable manifolds, Springer, Heidelberg, 1984.

[ 5 ]J. Gilbert and M. Murray: Clifford algebras and Dirac operators in harmonic analysis, Cambridge University Press, 1991.

[ 6 ]J. Nash: ‘The imbedding problem for Riemannian manifolds’,Annals, 63(1956), pp. 20-63.

[ 7 ] N. Théodoresco: La dérivée aréolaire, Annales Roumaines de Mathé- matiques cahier 3, Bucarest 1936.

[ 8 ]A. Trautman: ‘Spinors and the Dirac operator on hypersurfaces. I. General theory’, J. Math. Phys., 33/12(1992), pp. 4011-4019.