DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 121–126

THE RICCI IDENTITY FOR GENERAL CONNECTIONS

ANTONELLA CABRAS

Abstract. Given an arbitrary fibered manifold Y → M, we introduce the second order absolute differentiation with respect to a general connection Γ on Y and a classical linear connection Λ on M. Then we generalize the Ricci identity to such a situation. The problem requires a new geometric operation on a subclass of the iterated 2-jets.

Introduction

It was pointed out by many authors,[KoMiSl],[MaMo],[MaMo],[Mo],[Sau], that the general connections on an arbitrary fibered manifold Y → M have several properties of the classical principal or linear connections. In the present paper we deduce that even the Ricci identity for the second order absolute differential can be generalized to any fibered manifold. The second order absolute differentiation of sections of Y is based on a general connection Γ on Y and on a classical linear connection Λ on M. In Section 2 we discuss an arbitrary connection Γ on a vector bundle. In this case the tensor alternation can be used for formulating the Ricci identity. In the case of a linear connection Γ we rededuce a classical result. However, on an arbitrary fibered manifold we need a new geometric operation, which is defined on a subclass of the iterated 2–jets. The basic properties of iterated 2–jets are described in Section 3 and the difference tensor is introduced in Section 4. This concept enables us to formulate and prove the Ricci identity for general connections in Section 5. All manifolds and maps are assumed to be infinitely differentiable.

Acknowledgements. The author acknowledges Prof. Ivan Kolář for suggesting the problem and for several stimulating discussions.

1. The iterated absolute differentiation

Let p : Y → M be an arbitrary fibered manifold and Γ : Y → J^{1}Y be a connection on Y, [KoMiSl]. The first absolute differentiation of the sections of Y can be based on the fact that J^{1}Y → Y is an affine bundle with derived vector bundle V Y ⊗T^{∗}M, where V Y is the vertical tangent bundle of Y. The tensor product is over Y, i.e. we construct the pullback p^{∗}T^{∗}M and then we tensorize over Y. The absolute differential of a section s : M → Y is a section ∇s : M → V Y ⊗T^{∗}M defined by

1991 Mathematics Subject Classification. 53C05, 58A20.

Key words and phrases. General connection, absolute differentiation, Ricci identity, iterated 2-jet, difference tensor.

This paper is in final form and no version of it will be submitted for publication elsewhere.

This work is partially supported by MURST (by national and local funds) and GNSAGA of Consiglio Nazionale delle Ricerche.

∇s(x) : j_{x}^{1}s − Γ(s(x)) , x ∈ M (1)

Let x^{i}, y^{p} be some local fiber coordinates on Y and let Γ be expressed by

dy^{p} = F_{i}^{p}(x, y)dx^{i} (2)

and s by y^{p} = s^{p}(x). Then the coordinate form of (1) is

\frac{∂s^{p}}{∂x^{i}} − F_{i}^{p}(x, s(x)) (3)

There is a canonical isomorphism i_{Y} : V (J^{1}Y → M) → J^{1}(V Y → M), [KoMiSl, p.255]. If we compose the vertical tangent map V Γ : V Y → V J^{1}Y with i_{Y}, we obtain a connection VΓ := i_{Y} ∘V Γ on V Y → M, which is called the vertical prolongation of Γ. If Y^{p} are the additional coordinates on V Y, then the equations of VΓ are (2) and

dY^{p} = \frac{∂F_{i}^{p}}{∂y^{q}} Y^{q}dx^{i} (4)

In general, let q : Z → Y be a fibered manifold, so that we have the total projection p∘q : Z → M. A connection γ : Z → J^{1}Z = J^{1}(Z → M) is called projectable, if there exists a connection Γ : Y → J^{1}Y such that J^{1}q ∘ γ = Γ ∘ q, where J^{1}q : J^{1}Z → J^{1}Y is the induced map. Moreover, if Z → Y is a vector bundle, then J^{1}Z is a vector bundle over J^{1}Y. A projectable connection γ : Z → J^{1}Z over Γ is called semilinear, if γ is a vector bundle morphism of Z → Y into J^{1}Z → J^{1}Y over the base map Γ : Y → J^{1}Y. If Z̄ → Y is another vector bundle and γ̄ is another semilinear connection over the same Γ, then one can construct the tensor product γ⊗γ̄ analogously to [KoMiSl, p.381]. Also γ⊗γ̄ is a semilinear connection on Z⊗Z̄ → Y → M over Γ.

In particular, consider a linear connection Λ on T M, which induces the dual connection Λ^{∗} on T^{∗}M. The product Γ×Λ^{∗} is a connection on Y ×_{M}T^{∗}M = p^{∗}T^{∗}M,

which a semilinear connection on p^{∗}T^{∗}M → Y → M over Γ. The tensor product of VΓ and Γ × Λ^{∗} is a semilinear connection on V Y ⊗T^{∗}M, which will be denoted by VΓ⊗Λ^{∗}. If Y_{i}^{p} are the induced coordinates on V Y ⊗T^{∗}M and Λ_{jk}^{i}(x) are the Christoffel symbols of Λ, then the coordinate expression of VΓ⊗Λ^{∗} is (2) and

dY_{i}^{p} = (\frac{∂F_{j}^{p}}{∂y^{q}} Y_{i}^{q} − Λ_{jk}^{i}Y_{k}^{p}) dx^{j} (5)

Definition 1.1. The absolute differential ∇_{Λ}^{2}s of ∇s with respect to VΓ⊗Λ^{∗} is called the second absolute differential of a section s : M → Y with respect to Γ and Λ.

Thus, the coordinate form of ∇_{Λ}^{2}s is (3) and

\frac{∂}{∂x^{j}} (\frac{∂s^{p}}{∂x^{i}} − F_{i}^{p}) − \frac{∂F_{j}^{p}}{∂y^{q}} (\frac{∂s^{q}}{∂x^{i}} − F_{i}^{q}) + Λ_{ij}^{k} (\frac{∂s^{p}}{∂x^{k}} − F_{k}^{p}) (6)

We recall that the curvature R of Γ is a map Y ×_{M} ∧^{2}T M → V Y. Its coordinate

expression is

(\frac{∂F_{i}^{p}}{∂x^{j}} − \frac{∂F_{i}^{p}}{∂y^{q}} F_{j}^{q}) dx^{i} ∧ dx^{j} (7)

On the other hand, the torsion S of Λ is a section of T M ⊗∧^{2}T^{∗}M.

2. The Ricci identity for non linear connections

Given a vector bundle p : E → M, a general connection Γ on E can be called a non linear connection. Since V E = E ×_{M} E and similarly for E ⊗T^{∗}M, (6) can be

considered as a section (denoted by the same symbol)

∇_{Λ}^{2}s : M → E⊗T^{∗}M ⊗T^{∗}M (8)

Hence we have defined the alternation Alt(∇_{Λ}^{2}s) : M → E⊗∧^{2}T^{∗}M. On the other hand, the curvature R of Γ can be interpreted as a map

R : E → E⊗∧^{2}T^{∗}M (9)

Clearly, the contraction ⟨S, ∇s⟩ of the torsion of Λ with ∇s is a section M →

E⊗∧^{2}T^{∗}M. From (6) and (7) we obtain easily

Proposition 2.1. (The Ricci identity for non linear connections.) It holds

Alt (∇_{Λ}^{2}s) = −R∘s + ⟨S, ∇s⟩ (10)

In particular, consider a linear connection Γ on E. Its coordinate form is

dy^{p} = Γ_{qi}^{p}(x)y^{q}dx^{i} (11)

By (4) we have VΓ = Γ × Γ, so that ∇_{Λ}^{2}s coincides with the absolute differential of ∇s : M → E⊗T^{∗}M with respect to the classical tensor product Γ⊗Λ^{∗}. The curvature

of a linear connection can be interpreted as a section M → E ⊗ E^{∗}⊗ ∧^{2}T^{∗}M. Hence

the contraction ⟨R, s⟩ is a section of E⊗∧^{2}T^{∗}M and (10) reduces to

Alt (∇_{Λ}^{2}s) = −⟨R, s⟩ + ⟨S, ∇s⟩ (12)

This is the general Ricci identity for linear connections in the sense of [KoMiSl, p.235].

Moreover, if E = ⊗^{r}T M ⊗ ⊗^{s}T^{∗}M is a classical tensor bundle over M, then one starts from a classical linear connection Λ on M and put Γ = ⊗^{r}Λ ⊗ ⊗^{s}Λ^{∗}. Then (12) is the Ricci identity from the classical tensor analysis.

However, in the case of an arbitrary fibered manifold Y → M one must replace the alternation in (10) by a more sophisticated geometric operation. It will be defined in Section 4 below.

3. Iterated 2–jets

Let J^{1}(M, N) be the space of all 1–jets between two manifolds M, N and α : J^{1}(M, N) → M or β : J^{1}(M, N) → N be the source or target projection, respectively. According to Virsik, [Vir], the space J̌^{2}(M, N) of iterated 2–jets of M into N is defined by the iteration J̌^{2}(M, N) = J^{1}(M, J^{1}(M, N)). Denote by α_{1} : J̌^{2}(M, N) → M or β_{1} : J̌^{2}(M, N) → J^{1}(M, N) the source or target projection of J^{1}(M, J^{1}(M, N)). By [KoMiSl, p.118], J^{1} is a bundle functor on the product category Mf_{m} × Mf, where Mf is the category of manifolds and smooth maps and Mf_{m} is the category of m–dimensional manifolds and local diffeomorphisms. Thus, if 1_{M} denotes the identity of M, we can construct J^{1}(1_{M}, α) : J̌^{2}(M, N) → J^{1}(M, M) and J^{1}(1_{M}, β) : J̌^{2}(M, N) → J^{1}(M, N). Beside α_{1} we have another projection α_{2} := α∘β_{1} : J̌^{2}(M, N) → M and a projection β_{0} = β∘β_{1} : J̌^{2}(M, N) → N. The set J̃^{2}(M, N) of non holonomic 2–jets by Ehresmann, [Eh], is the subset of all A ∈ J̌^{2}(M, N) satisfying J^{1}(1_{M}, α)(A) = j_{x}^{1}1_{M}, x = α_{1}A = α_{2}A. The subset J̄^{2}(M, N) ⊂ J̃^{2}(M, N) of semiholonomic 2–jets is further characterised by β_{1}(A) = J^{1}(1_{M}, β)(A).

Let x^{i} or y^{p} be some local coordinates on M or N. The additional coordinates on J^{1}(M, N) are y_{i}^{p} = ∂y^{p}/∂x^{i}, so that the corresponding coordinates on J^{1}(M, N) are (y^{p}, y_{i}^{p}, x^{i}). By iteration, the induced coordinates on J̌^{2}(M, N) are

(y^{p}, y_{i}^{p}, x^{i}, y_{0i}^{p}, y_{ij}^{p}, x_{j}^{i}, u^{i}) (13)

For an element A with coordinates (13) we have α_{1}A = (u^{i}), α_{2}A = (x^{i}), β_{0}A = (y^{p}), β_{1}A = (y^{p}, y_{i}^{p}, x^{i}), J^{1}(1_{M}, β)(A) = (y^{p}, y_{0i}^{p}, u^{i}), J^{1}(1_{M}, α)(A) = (x^{i}, x_{j}^{i}, u^{i}).

Let [x] denote the constant map of M into x ∈ M.

Definition 3.1. An element A ∈ J̌^{2}(M, N) will be called distinguished, if J^{1}(1_{M}, α)(A) = j_{x}^{1}[x] for some x ∈ M and β_{1}(A) = J^{1}(1_{M}, β)(A).

The set of all distinguished iterated 2–jets will be denoted by Ĵ^{2}(M, N). Since J^{1}(1_{M}, α)(A) = j_{x}^{1}[x] implies α_{1}A = α_{2}A, we have only one projection α_{0} : Ĵ^{2}(M, N) → M. Analogously to the semiholonomic and nonholonomic jets, we shall denote by Ĵ_{x}^{2}(M, N)_{y} the subspace of all A’s satisfying α_{0}A = x, β_{0}A = y.

4. The difference tensor

First of all, we recall how a semiholonomic 2–jet A ∈ J̄_{x}^{2}(M, N)_{y} defines the

deviation ΔA ∈ T_{y}N ⊗∧^{2}T_{x}^{∗}M, [CaKo], [MaMo], [Pra], [Sau]. We have A = j_{x}^{1}s, where s is a section of α : J^{1}(M, N) → M. Every s(u) ∈ J_{u}^{1}(M, N) can be interpreted as a map T_{u}M → T N, so that s determines a map T M → T N. Its tangent map λ(A) : T T_{x}M → T T_{y}N over x ∈ M is determined by A. If κ_{M,x} is the restriction of the canonical involution of the second tangent bundle over x ∈ M, then there exists a unique element κA ∈ J̄_{x}^{2}(M, N)_{y} such that λ(κA) = κ_{N,y}∘λ(A)∘κ_{M,x}. Since J̄^{2}(M, N) → J^{1}(M, N) is an affine vector bundle with derived vector bundle

T N ⊗ ⊗^{2}T^{∗}M, the vector ΔA = A − κA belongs to T_{y}N⊗∧^{2}T_{x}^{∗}M. If (x^{i}, y^{p}, y_{i}^{p}, y_{ij}^{p})

are the essential coordinates of A, then ΔA = y_{[ij]}^{p}.

This procedure can be modified to an element A ∈ Ĵ_{x}^{2}(M, N)_{y} as follows. We have A = j_{x}^{1}s, where s(u) ∈ J_{x}^{1}(M, N). Hence s defines a map σ : M ×T_{x}M → T N,

σ(u, ξ) = s(u)(ξ), ξ ∈ T_{x}M (14)

The tangent map is T σ : T M × T (T_{x}M) → T T N. If s(u) = (y^{p}(u), y_{i}^{p}(u), x^{i}) and ξ = (ξ^{i}), then the coordinate form of (14) is

y^{p} = y^{p}(u), η^{p} = y_{i}^{p}(u)ξ^{i} (15)

provided η^{p} are the additional coordinates on T N. Let Y^{p} = dy^{p}, H^{p} = dη^{p} be the induced coordinates on T T N. On the other hand, since T_{x}M is a vector space, we have T (T_{x}M)) = T_{x}M × T_{x}M. Let ξ^{i}, Ξ^{i} = dξ^{i} be the corresponding coordinate on T (T_{x}M) and X^{i} = du^{i} be the induced coordinates on T M. Then the restriction of T σ to T_{x}M × T_{x}M × T_{x}M is

Y^{p} = y_{i}^{p}X^{i},  η^{p} = y_{i}^{p}ξ^{i},  H^{p} = y_{ij}^{p}ξ^{i}X^{j} + y_{i}^{p}Ξ^{i} (16)

If we apply κ_{N,y} and the echange map between the first and the second factor of T_{x}M × T_{x}M × T_{x}M, we obtain

Y^{p} = y_{i}^{p}X^{i},  η^{p} = y_{i}^{p}ξ^{i},  H^{p} = y_{ij}^{p}X^{i}ξ^{j} + y_{i}^{p}Ξ^{i} (17)

For every X ∈ T_{x}M and ξ ∈ T_{x}M take any Ξ ∈ T_{x}M. Then the strong difference (see [KoMiSl, p.391]) of (16) and (17) does not depend on the choice of Ξ and defines an antisymmetric map T_{x}M × T_{x}M → T_{y}N of the form

(y_{ij}^{p} − y_{ji}^{p})ξ^{i}X^{j} (18)

This will be denoted by δ(A) ∈ T_{y}N ⊗∧^{2}T_{x}^{∗}M.

Definition 4.1. δ(A) ∈ T_{y}N ⊗ ∧^{2}T_{x}^{∗}M will be called the difference tensor of A ∈ Ĵ_{x}^{2}(M, N)_{y}.

5. The case of a general connection

In order to show that the difference tensor can be applied to ∇_{Λ}^{2}s from Section 1, we discuss a more general situation. Let γ be any connection in V Y ⊗T^{∗}M → Y → M over the connection Γ on Y. Write ∇_{Γ}s for ∇s, so that ∇_{γ} (∇_{Γ}s) means the absolute differential of ∇_{Γ}s with respect to γ. We have V_{y}Y ⊗T_{x}^{∗}M = T_{y}(Y_{x})⊗ T_{x}^{∗}M = J_{x}^{1}(M, Y_{x})_{y} for all y ∈ Y, x = p(y), which implies V Y ⊗T^{∗}M ⊂ J^{1}(M, Y). By iteration, V (V Y ⊗T^{∗}M)⊗T^{∗}M ⊂ J^{1}(M, J^{1}(M, Y)) = J̌^{2}(M, Y).

Proposition 5.1. For every section s of Y and every x ∈ M, we have

∇_{γ} (∇_{Γ}s) (x) ∈ Ĵ^{2}(M, Y) and δ (∇_{γ} (∇_{Γ}s) (x)) ∈ V Y ⊗∧^{2}T^{∗}M ⊂ T Y ⊗∧^{2}T^{∗}M.

Proof. We shall use the coordinate expressions. If (x^{i}, y^{p}, Y_{i}^{p}) are the coordinates on V Y ⊗T^{∗}M from Section 1 and (x^{i}, y^{p}, x_{i}^{k}, y_{i}^{p}, u^{i}) are the coordinates on J^{1}(M, Y) from Section 3, then the inclusion V Y ⊗T^{∗}M ⊂ J^{1}(M, Y) is expressed by

(x^{i}, y^{p}, Y_{i}^{p}) ↦ (x^{i}, y^{p}, 0, Y_{i}^{p}, x^{i}) (19)

By iteration, let

(x^{i}, y^{p}, Y_{i}^{p}, Y_{0i}^{p}, Y_{ij}^{p}) (20)

be the analogous coordinates on V (V Y ⊗T^{∗}M)⊗T^{∗}M and let

(x^{i}, y^{p}, x_{i}^{k}, y_{i}^{p}, u^{i}, x_{j}^{i}, y_{0j}^{p}, x_{ij}^{k}, y_{ij}^{p}, u_{j}^{i}, v^{i}) (21)

be the corresponding coordinates on J̌^{2}(M, Y). Then the image of (20) in J̌^{2}(M, Y) has the form

(x^{i}, y^{p}, 0, Y_{i}^{p}, x^{i}, 0, Y_{0i}^{p}, 0, Y_{ij}^{p}, 0, x^{i}) (22)

If y^{p} = s^{p}(x) is the coordinate expression of s, then the coordinate form of ∇_{Γ}s is y^{p} = s^{p}(x) and Y_{i}^{p} = \frac{∂s^{p}}{∂x^{i}} −F_{i}^{p}(x, s(x)) =: s_{i}^{p}. Since γ is over Γ, the additional coordinate expression of ∇_{γ} (∇_{Γ}s) is Y_{0i}^{p} = s_{i}^{p} and Y_{ij}^{p} are some functions of x^{i}, which we do not need. Then (22) proves that each iterated 2–jet ∇_{γ} (∇_{Γ}s) (x) is distinguished. The zero at the eighth place in (22) indicates that the values of

δ (∇_{γ} (∇_{Γ}s)) lie in V Y ⊗∧^{2}T^{∗}M.  QED

Consider now the case γ = VΓ⊗Λ^{∗}, so that ∇_{VΓ⊗Λ^{∗}} (∇_{Γ}s) = ∇_{Λ}^{2}s. Then we have

three sections δ (∇_{Λ}^{2}s), R∘s and ⟨S, ∇s⟩ of V Y ⊗∧^{2}T^{∗}M.

Proposition 5.2. (The Ricci identity for general connections.) It holds

δ (∇_{Λ}^{2}s) = −R∘s + ⟨S, ∇s⟩ (23)

Proof. By (18), the coordinate form of (23) is the same as in Proposition 2.1.

QED

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[Vir] G. Virsik: Total connections in Lie groupoids, to appear in Arch. Math. (Brno).

Department of Applied Mathematics “G. Sansone” Via S. Marta 3, 50139 Florence, Italy

E–mail: cabras@ingfi1.ing.unifi.it