Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 39 (1998), No. 2, 283-290.

On Holonomicity Criteria in Second Order Geometry

Ivan Kolář^{∗}

Department of Algebra and Geometry Faculty of Science, Masaryk University Janáčkovo nám. 2a, 662 95 Brno, Czech Republic

Abstract. We introduce the intrinsic exterior differential of the contact morphism on the first jet prolongation of a fibered manifold Y. Using this concept, we deduce a holonomicity criterion for the second semi-holonomic prolongation of Y and we clarify that this is closely related to several classical procedures. The same idea is applied to the bundle of all smooth maps between the fibers over the same base point of two fibered manifolds over the same base.

MSC 1991: 58A20, 53C05, 53C10

Keywords: contact morphism, intrinsic exterior differential, semi-holonomic 2-jet, connection, G-structure, bundle of smooth fiber maps

The contact forms on the first jet prolongation π_{0}^{1} : J^{1}Y → Y of a fibered manifold π : Y → M represent an important tool for several branches of differential geometry and global analysis. They are generated by the components of a map ψ_{Y} : T J^{1}Y → V Y, which is called the contact morphism. It is well known that a section s : M → J^{1}Y is holonomic, i.e. s = j^{1}(π_{0}^{1}s) if and only if s^{∗}ψ_{Y} = 0. In the present paper, we first deduce a more subtle holonomicity criterion, in which an original concept of intrinsic exterior differential Dψ_{Y} of ψ_{Y} is used. We recall that every element A of the second semi-holonomic prolongation \overline{J}^{2}Y defines a deviation ΔA ∈ V Y ⊗Λ^{2}T^{∗}M, which vanishes for the elements of the second jet prolongation J^{2}Y ⊂ \overline{J}^{2}Y. Then we prove that the deviation of A coincides with the restriction of Dψ_{Y} to the horizontal m-plane σA ⊂ T J^{1}Y determined by A. In particular, this implies that A is holonomic if and only if Dψ_{Y} |σA = 0.

∗The author was supported by the grant No. 201/96/0079 of the GA ČR 0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1998 Heldermann Verlag

Next we show how this general idea is related with some more concrete procedures from the classical theories of connections and G-structures. In the last section, we develop an infinite dimensional modification of our main idea. (A physical motivation for such a research goes from [3].) We consider two fibered manifolds Y_{1} → M, Y_{2} → M over the same base and study the functional bundle F(Y_{1}, Y_{2}) of all smooth maps between the fibers over the same base point. We introduce the contact morphism of F(Y_{1}, Y_{2}) and deduce an analogous holonomicity criterion for the second semi-holonomic prolongation of F(Y_{1}, Y_{2}). All manifolds and maps are assumed to be infinitely differentiable.

1. The intrinsic exterior differential

Given a vector bundle p : E → M, an E-valued 1-form is a base-preserving linear morphism φ : T M → E. If γ is a linear connection on E, the exterior differential d_{γ}φ : Λ^{2}T M → E of φ with respect to γ is defined by

(1)  (d_{γ}φ) (X, Y) = ∇_{X}φ(Y) − ∇_{Y} φ(X) − φ([X, Y])

for any vector fields X, Y on M, [9]. If v^{p} = φ_{i}^{p}(x) dx^{i} is the coordinate expression of φ in some local coordinates x^{i} on M and some linear fiber coordinates v^{p} on E, then (1) yields the following coordinate expression of d_{γ}φ

(2)  dφ_{i}^{p} ∧ dx^{i} + φ_{i}^{q}dx^{i} ∧ Γ_{qj}^{p}dx^{j} ,

where Γ_{qi}^{p}(x) are the Christoffel symbols of γ.

Assume φ has constant rank, so that its kernel K_{φ} is a vector bundle over M. Then the restriction Dφ of d_{γ}φ to Λ^{2}K_{φ} does not depend on γ.

Definition 1. The map Dφ : Λ^{2}K_{φ} → E is called the intrinsic exterior differential of φ.

By (2), the coordinate form of Dφ on K_{φ} is

(3)  dφ_{i}^{p} ∧ dx^{i} .

Let \overline{p} : \overline{E} → \overline{M} be another vector bundle, \overline{φ} : T \overline{M} → \overline{E} be an \overline{E}-valued 1-form of constant rank and f : E → \overline{E} be a linear morphism over f_{0} : M → \overline{M}. If φ and \overline{φ} are f-related, i.e. f ∘ φ = \overline{φ} ∘ T f_{0}, then T f_{0}(K_{φ}) ⊂ K\overline{φ} and we denote by K_{f} : K_{φ} → K\overline{φ} the restriction and corestriction of T f_{0}. We have D\overline{φ} : Λ^{2}K\overline{φ} → \overline{E}.

Proposition 1. If φ and \overline{φ} are f-related, then Dφ and D\overline{φ} are also f-related, i.e.

(4)  f ∘ Dφ = (D\overline{φ}) ∘ Λ^{2}K_{f} .

Proof. Let z^{a} be some local coordinates on \overline{M} and w^{s} be some linear fiber coordinates on \overline{E}. Hence the coordinate expression of \overline{φ} is w^{s} = \overline{φ}_{a}^{s}(z) dz^{a}. If z^{a} = f^{a}(x), w^{s} = f_{p}^{s}(x)y^{p} is the coordinate form of f, then the condition for φ and \overline{φ} to be f-related reads

(5)  f_{p}^{s}(x)φ_{i}^{p}(x) dx^{i} = \overline{φ}_{a}^{s}(f(x)) df^{a} .

The exterior differentiation of (5) yields (4). □

Remark 1. The simpliest situation, in which the intrinsic exterior differential appears, is the case of an arbitrary connection Γ on a fibered manifold π : Y → M. Write V Y for the vertical tangent bundle of Y. The connection morphism ω_{Γ} : T Y → V Y is a V Y -valued 1-form, [4]. Its kernel K_{ω_{Γ}} is identified with Y ×_{M} T M, so that Dω_{Γ} : Y ×_{M} Λ^{2}T M → V Y. If we rewrite the curvature CΓ : Y ×_{M} Λ^{2}T M → V Y in the form from [7], p. 158, we obtain immediately

(6)  Dω_{Γ} = −CΓ ,

i.e. the intrinsic exterior differential of the connection morphism coincides with the curvature up to sign.

2. The case of the contact morphism

Consider the first jet prolongation π_{0}^{1} : J^{1}Y → Y. Every element A ∈ J_{y}^{1}Y is identified with an m-dimensional subspace σA ⊂ T_{y}Y, m = dim M. The contact morphism ψ_{Y} : T J^{1}Y → V Y transforms every B ∈ T_{A}J^{1}Y into the projection of T π_{0}^{1}(B) ∈ T_{y}Y into V_{y}Y in the direction of σA. Clearly, ψ_{Y} is a 1-form with values in the pullback bundle π_{0}^{1∗}V Y → J^{1}Y. Write K_{Y} = Ker ψ_{Y} ⊂ T J^{1}Y. Using Definition 1, we obtain Dψ_{Y} : Λ^{2}K_{Y} → π_{0}^{1∗}V Y. If there is no danger of confusion, we shall simplify the notation to

Dψ_{Y} : Λ^{2}K_{Y} → V Y .

If x^{i}, y^{p}, y_{i}^{p} are the standard coordinates on J^{1}Y, then the coordinate expression of ψ_{Y} is

(7)  dy^{p} − y_{i}^{p} dx^{i} .

By (3), the coordinate form of Dψ_{Y} on K_{Y} is

(8)  −dy_{i}^{p} ∧ dx^{i} .

We recall that FM_{m} means the category of fibered manifolds with m-dimensional bases and fiber respecting maps with local diffeomorphisms as base maps, [7]. By definition, we have T J^{1}f(K_{Y}) ⊂ K\overline{Y} for every FM_{m}-morphism f : Y → \overline{Y}. Write K_{f} : K_{Y} → K\overline{Y} for the restriction and corestriction of T J^{1}f. By Proposition 1, we have

Proposition 2. For every FM_{m}-morphism f : Y → \overline{Y},

(9)  V f ∘ Dψ_{Y} = (Dψ\overline{Y}) ∘ Λ^{2}K_{f} .

We are going to describe a direct construction of Dψ_{Y}. We shall need the operation of strong difference, [7], p. 391. Write p_{M} : T M → M for the tangent bundle of a manifold M. On the iterated tangent bundle T T M, we have two projections p_{T M} : T T M → T M and T p_{M} : T T M → T M. If W, Z ∈ T T M satisfy

(10)  p_{T M}(W) = T p_{M}(Z) , p_{T M}(Z) = T p_{M}(W) ,

they determine a well defined element W ÷ Z ∈ T M. In coordinates, if W = (x^{i}, u^{i}, v^{i}, w^{i}), then (10) implies Z = (x^{i}, v^{i}, u^{i}, z^{i}) and the coordinate expression of W ÷ Z is

(11)  (x^{i}, w^{i} − z^{i}) .

Consider the lifting map λ : J^{1}Y ×_{M} T M → T Y, which transforms every (A, X) into the lift of X ∈ T M into σA. Hence T λ : T J^{1}Y ×_{T M} T T M → T T Y. For every B ∈ T J^{1}Y we denote by B_{0} its projection into T M. Let κ : T T M → T T M be the canonical involution.

Proposition 3. Let B, C ∈ (K_{Y})_{A} and D ∈ T T M be any element satisfying p_{T M}(D) = B_{0} and T p_{M}(D) = C_{0}. Then T λ(B, D) and T λ(C, κD) satisfy the condition (10) for the existence of the strong difference and

(12)  Dψ_{Y} (B, C) = T λ(C, D) ÷ T λ(B, κD) .

Proof. We shall use the coordinate expressions B =(x^{i}, y^{p}, y_{i}^{p}, b^{i}, b^{p}, b_{i}^{p}), C =(x^{i}, y^{p}, y_{i}^{p}, c^{i}, c^{p}, c_{i}^{p}), D = (x^{i}, b^{i}, c^{i}, d^{i}). By (8), we have

(13)  Dψ_{Y} (B, C) = −b_{i}^{p}c^{i} + c_{i}^{p}b^{i} .

The condition B, C ∈ K_{Y} reads

(14)  b^{p} − y_{i}^{p}b^{i} = 0, c^{p} − y_{i}^{p}c^{i} = 0 .

Since dx^{i} = dx^{i}, dy^{p} = y_{i}^{p} dx^{i} is the coordinate form of λ, we have

T λ(C, D) = (x^{i}, y^{p}, b^{i}, y_{i}^{p}b^{i}, c^{i}, c^{p}, d^{i}, c_{i}^{p}b^{i} + y_{i}^{p}d^{i}) ,
T λ(B, κD) = (x^{i}, y^{p}, c^{i}, y_{i}^{p}c^{i}, b^{i}, b^{p}, d^{i}, b_{i}^{p}c^{i} + y_{i}^{p}d^{i}) .

By (14), the condition for the existence of the strong difference is fulfilled. Then (11) implies our claim. □

3. The deviation from holonomicity

The second nonholonomic prolongation π_{1}^{2} : J̃^{2}Y → J^{1}Y is defined by the iteration J̃^{2}Y = J^{1}(J^{1}Y → M), [2]. Every A ∈ J̃^{2}Y determines an m-dimensional subspace σA ⊂ T J^{1}Y. The jet prolongation of π_{0}^{1} : J^{1}Y → Y is another projection J^{1}π_{0}^{1} : J̃^{2}Y → J^{1}Y. An element A ∈ J̃^{2}Y is called semi-holonomic if

(15)  π_{1}^{2}(A) = J^{1}π_{0}^{1}(A) .

The subset \overline{J}^{2}Y ⊂ J̃^{2}Y of all such elements is called the second semi-holonomic prolongation of Y. The next lemma follows directly from definitions.

Lemma 1. A ∈ \overline{J}^{2}Y if and only if σA ⊂ K_{Y}. □

Thus, if x^{i}, y^{p}, y_{i}^{p} are the standard coordinates on J^{1}Y, every A ∈ \overline{J}^{2}Y is characterized by dy^{p} = y_{i}^{p} dx^{i} and

(16)  dy_{i}^{p} = y_{ij}^{p} dx^{i}

with arbitrary y_{ij}^{p}. These y_{ij}^{p} are the additional coordinates on \overline{J}^{2}Y. The second holonomic prolongation J^{2}Y of Y is identified with the subset of all elements of \overline{J}^{2}Y satisfying y_{ij}^{p} = y_{ji}^{p}.

J^{1}Y → Y is an affine bundle with associated vector bundle V Y ⊗ T^{∗}M, see e.g. [7], p. 125. By iteration, J̃^{2}Y → J^{1}Y is also an affine bundle, whose associated vector bundle is V J^{1}Y ⊗ T^{∗}M. The subset \overline{J}^{2}Y is an affine subbundle with associated vector bundle V Y ⊗_{J^{1}Y} ⨂^{2} T^{∗}M. Even J^{2}Y ⊂ \overline{J}^{2}Y is an affine subbundle with associated vector bundle V Y ⊗_{J^{1}Y} S^{2}T^{∗}M.

Taking into account the direct sum decomposition ⨂^{2} T^{∗}M = S^{2}T^{∗}M ⊕ Λ^{2}T^{∗}M, we can project every A ∈ \overline{J}^{2}Y in the direction of V Y ⊗ Λ^{2}T^{∗}M into an element sA ∈ J^{2}Y.

Definition 2. The vector ΔA = A −sA ∈ V Y ⊗Λ^{2}T^{∗}M is called the deviation of A ∈ \overline{J}^{2}Y.

Obviously, A ∈ J^{2}Y if and only if ΔA = 0. In coordinates, if A = (x^{i}, y^{p}, y_{i}^{p}, y_{ij}^{p}) then

(17)  ΔA = (y_{[ij]}^{p}) .

where the square bracket means antisymmetrization, [12].

Consider the contact morphism ψ_{Y}. Clearly, for every A ∈ \overline{J}^{2}Y, the restriction Dψ_{Y} |σA is identified with an element of V Y ⊗Λ^{2}T^{∗}M. The following assertion, which follows directly from (8) and (17), represents the “pure idea” for some holonomicity criteria in second order geometry.

Proposition 4. ΔA = Dψ_{Y} |σA. □

In particular, A ∈ J^{2}Y if and only if Dψ_{Y} |σA = 0.

4. Examples

Consider first a connection Γ : Y → J^{1}Y. Clearly, the composition ψ_{Y} ∘ T Γ : T Y → V Y coincides with the connection morphism ω_{Γ} of Γ. On the other hand, we can construct the jet prolongation J^{1}Γ : J^{1}Y → J̃^{2}Y. The composition Γ′ = J^{1}Γ ∘ Γ, whose values lie in \overline{J}^{2}Y, is called the Ehresmann prolongation of Γ. Since the jet prolongation can be interpreted in terms of the tangent map, (6) and Propositions 1 and 4 yield directly a very conceptual proof of the following result, [7], [11], [13].

Corollary 1. The curvature of Γ differs from the deviation of its Ehresmann prolongation only by sign, i.e. CΓ = −Δ ∘ Γ′. □

Remark 2. Let Y be a principal fiber bundle P (M, G). Since V P is canonically identified with P × 𝔤, its pullback over J^{1}P is J^{1}P × 𝔤, so that ψ_{P} is a 𝔤-valued 1-form on J^{1}Y. The intrinsic exterior differential Dψ_{P} is defined in terms of an auxiliary connection. If we use the zero connection of the product bundle J^{1}P × 𝔤 → J^{1}P, we obtain the classical exterior differential dψ_{P} of a vector valued 1-form. This leads to a simple holonomicity criterion for the elements of \overline{J}^{2}P, which is somewhat stronger than Proposition 1 of [5].

Remark 3. The solder form Θ : T P^{1}M → ℝ^{m} of the first order frame bundle π : P^{1}M → M of an m-dimensional manifold M is defined analogously to the contact morphism ψ_{Y}. Every u ∈ P_{x}^{1}M is interpreted as a linear isomorphism ũ : ℝ^{m} → T_{x}^{1}M. For every X ∈ T_{u}P^{1}M, one defines

Θ(X) = ũ^{−1}(T π(X)) .

Taking into account this analogy, it is interesting to express explicitely the relation of the exterior differential dΘ of this ℝ^{m}-valued 1-form on P^{1}M to the deviation of certain semiholonomic 2-jets.

Let \overline{P}^{2}M be the second order semi-holonomic frame bundle of M, i.e. the set of all invertible semi-holonomic 2-jets from ℝ^{m} into M with source 0. Every U ∈ \overline{P} _{x}^{2}M determines the deviation ΔU ∈ T_{x}M ⊗ Λ^{2}ℝ^{m∗} analogously to Section 3. P. Libermann established an identification i : J^{1}P^{1}M → \overline{P}^{2}M, [10], [8]. On one hand, for every A ∈ J^{1}P_{x}^{1}M, dΘ|σA is identified with a linear map Λ^{2}T_{x}M → ℝ^{m}. On the other hand, Δ(iA) ∈ T_{x}M ⊗ Λ^{2}ℝ^{m∗} can be interpreted as a linear map Λ^{2}ℝ^{m} → T_{x}M. The evaluations from [8], pp. 278 and 281, can be summarized as follows. For every A ∈ J_{u}^{1}P^{1}M , π(u) = x, the following diagram commutes

(18)  \begin{matrix}Λ^{2}ℝ^{m} & ⟶^{Δ(iA)} & T_{x}M \\ _{Λ^{2}ũ}↓ & & ↓_{u^{−1}} \\ Λ^{2}T_{x}M & ⟶^{dΘ|σA} & ℝ^{m} \end{matrix}

We remark that (18) gives an interesting approach to the structure function τ : Q → H^{0,2}(𝔤) of a G-structure Q on M, where H^{0,2} means the Spencer cohomology class of bidegree (0, 2). Since Q ⊂ P^{1}M is a reduction to G ⊂ GL(m, ℝ), we have J^{1}Q ⊂ J^{1}P^{1}M. Then we can define

(19)  τ(u) = {ũ^{−1} ∘ Δ(iA); A ∈ J_{u}^{1}Q}

for every u ∈ Q, [8]. Such an approach clarifies properly that the structure function is an obstruction for the prolongability of Q, [8], [14].

5. The contact morphism in the functional case

Consider two locally trivial fibered manifolds Y_{1} → M, Y_{2} → M over the same base. It was clarified in [1] and [6] that several ideas from the finite dimensional differential geometry can be extended to the functional bundle of all smooth fiber maps

F(Y_{1}, Y_{2}) := ⋃_{x∈M} C^{∞}(Y_{1x}, Y_{2x}) → M .

We are going to discuss the concept of contact morphism of F(Y_{1}, Y_{2}).

According to [1], we can construct the jet prolongations of F(Y_{1}, Y_{2}) as well as the tangent and vertical tangent bundles. In particular, every A ∈ T J^{1}F(Y_{1}, Y_{2}) is identified with an m-dimensional subspace σA ⊂ T F(Y_{1}, Y_{2}), so that we can define the contact morphism

(20)  ψ : T J^{1}F(Y_{1}, Y_{2}) → V F(Y_{1}, Y_{2})

in the same way as in Section 2. Write K for the kernel of ψ. Even in the functional case it holds that an element A ∈ J̃^{2}F(Y_{1}, Y_{2}) of the second nonholonomic prolongation is semiholonomic if and only if σA ⊂ K. Furthermore, every A ∈ \overline{J}^{2}F(Y_{1}, Y_{2}) defines the deviation ΔA ∈ V F(Y_{1}, Y_{2}) ⊗ Λ^{2}T^{∗}M, [1].

On the other hand, it is not clear whether the idea of intrinsic exterior differential can be applied to (20), for the pullback bundle in question has infinite dimensional base. However, the lifting map

λ : J^{1}F(Y_{1}, Y_{2}) ×_{M} T M → T F(Y_{1}, Y_{2})

is defined in the same way as in Section 2, so that we can use the idea of Proposition 3. Write again B_{0} for the canonical projection of B ∈ K into T M. By [1], the tangent map

T λ : T J^{1}F(Y_{1}, Y_{2}) ×_{T M} T T M → T T F(Y_{1}, Y_{2})

is well defined and has the following property. If B, C ∈ K are over the same element of J^{1}F(Y_{1}, Y_{2}) and D ∈ T T M is any element satisfying p_{T M}(D) = B_{0} and T p_{M}(D) = C_{0}, then T λ(B, D) and T λ(C, κD) satisfy the condition for the existence of the strong difference. Hence we can introduce formally a map Ψ as follows.

Definition 3. For every B, C ∈ K over the same element of J^{1}F(Y_{1}, Y_{2}), we define

(21)  Ψ(B, C) = T λ(C, D) ÷ T λ(B, κD) ∈ V F(Y_{1}, Y_{2}) .

One verifies directly that (21) is independent of the choice of D. Using the procedures from [1], one deduces easily the following analogy of Proposition 4.

Proposition 5. For every A ∈ \overline{J}^{2}F(Y_{1}, Y_{2}), we have

ΔA = Ψ|σA .

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Received January 30, 1997