Trying to Explicit Proofs of Some Vey’s Theorems in Linear Connections

Let Χ a diferentiable paracompact manifold. Under the hypothesis of a linear connection r with free torsion Τ on Χ, we are going to give more explicit the proofs done by Vey for constructing a Riemannian structure. We proposed three ways to reach our object. First, we give a sufficient and necessary condition on all of holonomy groups of the connection ∇ to obtain Riemannian structure. Next, in the analytic case of Χ, the existence of a quadratic positive definite form g on the tangent bundle ΤΧ such that it was invariant in the infinitesimal sense by the linear operators ∇kR, where R is the curvature of ∇, shows that the connection ∇ comes from a Riemannian structure. At last, for a simply connected manifold Χ, we give some conditions on the linear envelope of the curvature R to have a Riemannian structure.


Preliminary and Introduction
In 1978 Vey was the examinator of Anona's phd at Institut Fourier Grenoble. In this time, Vey was written some theorems in linear connections. The title of this unpublished paper was: "Sur les connexions riemanniennes" means "On the Riemannian connections". They had discussions, as said Vey, this result was not well explicit. So, one of the motivations of this paper is to explicit some of them about linear connections. First, we consider a paracompact manifoldΧ, we are looking for a Riemannian metric g which is invariant by parallel transport such that it produces a linear connection∇ with ∇g = 0. Next, we assume that Χ is a simply connected real analytic manifold accompanied by a real analytic connection∇, these new assumptions construct us a positive definite quadratic form g (satisfying ∇g = 0) infinitesimally preserved by the infinitesimal holonomy group. This is obtaining by the fact that the Lie algebra of holonomy groups coincides with the Lie algebra of the infinitesimal holonomy group. Finally, we present our problem so as to consider the linear envelope of the curvature, in the case where X is a smooth manifold. However, under these conditions that the linear envelope of the curvature of constant dimension coincides with the Lie algebra of the holonomy group, make the quadratic form g parallel to∇. Thus builds a Riemannian structure. Recently, several authors treat the same questions as R. Feres in and A. Vanzurava [1,2]. This last author gave an algorithm for constructing Riemannian structure which is similar as Veydone in the end of his paper. This redaction gives an interest for a next one where we use some results and idea of the present paper. So, let us recall some useful definitions in linear connections.

Definition 1
First, let X is a smooth manifold,Τ x X, x ∈ X the vector space of tangent vectors on x and Τ X the tangent bundle defined by Τ = x X ∈ ∪ Τ x X. Recall that a Riemannian manifold X is a smooth manifold equipped with a Riemannian structure g on X which is defined by the morphism of bundles g : ΤΧ × ΤΧ → such that ∀x∈X

Definition 2
Let X be a smooth manifold, let Γ(X) be the module of vector fields on X and ( ) { : the ring of real smooth functions. We recall that a linear connection on X is the application

Theorem 3
(Fundamental Theorem of Riemannian Geometry) Let X be a Riemannian manifold. Any Riemannian structure g produces an unique linear connection ∇ called Levi-Civita's connection on X with free torsion such that ∇g=0.

Proposition 6
Let X be a smooth manifold, and x, y ∈ X [3]. If X is connected and γ a path x to y, the holonomy groups of the point x, y are isomorphic and we have

Definition 7
Let K be a compact group. We call that an Haar measure on K denoted µ is the unique invariant measure by right and left translation in K, then such that f∈L(K) a ring of all functions f: K→ .

Definition 8
Let X be a Riemannian manifold, g 0 a metric on X, K a compact group. We define an invariant metric g on X which normalizes any metric g 0 of X on the compact group K. Let g=⎰ K k * g 0 dk Where k * is the "pull-back "of k.
This integration follows the Haar measure property.

Proposition 9
Recall that all paracompact manifolds X admit a Riemannian metric [4].

Proposition 10
Let X be a smooth manifold and the linear connection on X with free torsion. ∇ proceeds from a Riemannian structure if and only if its holonomy groups are relatively compact.

Proof.
On the one hand, suppose ∇ comes from a Riemannian structure (X, g).Then by the Fundamental Theorem of Riemannian Geometry, we have the Levi-Civita's connection ∇ on X. Consequently, the Riemannian holonomy is well defined by the holonomy group of Levi-Civita's connection on X. But, the holonomy group is a subgroup of the orthogonal group O (n) which is compact, so this subgroup is relatively compact.
On the other hand, let us suppose that the holonomy group on each point of x € X is relatively compact. Let g 0 be a Riemannian metric on the paracompact manifold X, and x € X. Then, it leaves invariant a positive definite quadratic form g on a point x of X on TX. Indeed, we have Let ,υ α ∈Η οι(x),then we have, Since X a smooth manifold, then X is locally homeomorphic to an open set of  n . Otherwise,  n is locally connected, and locally connected is preserved by homeomorphism. Consequently, under the Proposition 6, the holonomy is independent of the selected base point. Then, the parallel transport relatively to ∇ of the quadratic form g along the path from x to y is independent of the path, by definition of holonomy. Thus, it constructs a Riemannian structure g on X preserved by parallel transport, with ∇ g = 0. Since ∇ is of free torsion, and uniqueness of ∇, and then ∇ is a Riemannian connection.

Real analytic manifold and Riemannian structure
For the following, we suppose X a simply connected manifold to have an explicit result.

Definition 11
Let x a point of X, denoted by C(x) the set of closed curves on x, and C 0 (x) the set of the contractible curves on point x. Take a point u∈ p -1 (x), where p is the projection of the linear structure bundle L(X) at X cf. [5]. We recall that the holonomy group on x denoted by ψ( x) is the subgroup of the diffeomorphism p -1 (x) whose the element are obtained by parallel transport of the curves in C(x), that is to say, the element of ψ( x) are of the form τ : p -1 (x)→ψ 0 (x), →τ∈C(x). In the similar ways, we call holonomy group restricted on point x denoted ψ 0 ( x) the subgroup of diffeomorphism of p -1 (x) whose the element are of the form τ : p -1 (x), ∀τ∈C 0 (x).

Proposition 12
Let X be a smooth manifold, L(X) a linear structure fiber on X, H a connection on L(X) [6].
Then, all tangent vectors U of T u L(X) is called vertical (resp. horizontal) when it belong to V u (resp. H u ), u ∈ L(X), where V u (resp. H u ) indicate the vertical (resp. horizontal) subspace of T u L(X).

Definition 13
Let m 0 (u) a subspace of g l (n,) generated by Ω u (U; V )(the curvature form on L(X) cf. [6] p.152) the horizontal vectors U; V ∈ T u (L(X)) with u ∈ L(X). By recurrence on k, m k (u) defined a subspace of gl(n.) generated by the elements of m k-1 (u) and the element of the form V 1 _ _ _ _ _ V k (Ω(U; V )), where U; V; V 1 ,…, V k are the horizontal is a sub-algebra of gl(n;) and the connected Lie subgroup generated by g ' (u) is called infinitesimal holonomy groups on point u of L(X). We denote it by ψ ' (u).
For a real analytic manifold, denoted by Η ο l (x) resp. Η ο l (x), resp. Η ο l (x) the holonomy groups (resp. the restricted holonomy group, resp. the infinitesimal holonomy groups) on point x in X.

Proposition 14
Let X be a real analytic manifold, the linear real analytic connection [2]. Denoted by the respective Lie algebras of holonomy groups Η οl (x), Η οl * (x), Η οl ' (x). The two groups Η οl * (x), Η οl ' (x) constitute a Lie subgroups of Η οι (x) namely Since X is a real analytic manifold, we have the reverse inclusion for the holonomy groups and the equality between ( ) ( ) ' .

Theorem 15
Let X be a real analytic manifold,∇ the linear real analytic connection associated to X [6].

Proposition 16
Let (X;∇) be a real analytic simply connected, with free torsion, x a point of X, g a symmetric bilinear quadratic form on T x X [7]. Then the invariance of g by Η ο l (x) is characterized by:

Proposition 17
Let X be a real analytic manifold, simply connected, ∇ an analytic connection with free torsion on X and R its curvature. Let x be point of X. If there exist a positive definite quadratic form g on T x X, preserved in the infinitesimal sens by all of linear operators∇ k R(W,U 1 ,U 2 ,…..,U k ) (k ≥0,W∈Λ 2 Τ x Χ,U i ∈Τ x Χ), then ∇ proceeds from a Riemannian structure.

Proof.
Let x ∈ X, g a positive definite quadratic form on T x X, ∇ the analytic connection associated X , R the curvature of the connection ∇. Suppose that g is preserved in the infinitesimal sense by all linear operators ∇ k R(W,U 1 ,U 2 ,…..,U k ) of ( ) therefore ∇ k g = 0 for all k ≥ 0, it follows that ∇g = 0. Since the torsion is supposed null, then ∇ comes from a Riemannian structure.

Linear envelope and Riemannian structure Lemma 18
Let E be a real vector space of finite dimension, and I an interval of , and V i (t) a sequence of vectors which depend differentially on t ∈ I. Let L(t) the vector space spanned by V i (t). Suppose that L(t) of constant dimension, and for all i and for all

Proof.
We proceed like the arguments of proof in [8] pp. 943-944.

Proposition 20
Let X be a smooth manifold, and a linear connection of X. If the tensor ∇R takes all values in the linear envelope of the curvature, and if it has a constant dimension, then it coincides with the Lie algebra of the holonomy groups in each point.

Proof.
Let x a point of X. Let us indicate by  Let X be a compact smooth manifold and we have an action on X by the Lie algebra of finite dimension, then this is a lift of an action of Lie group of finite dimension.

Proposition 22
Let X be a simply connected manifold, ∇ a linear connection on X with free torsion, and R its curvature. Supposed that the free conditions which follow are verified: on T x X infinitesimally preserved by x r − ,then the connection∇ proceeds from a Riemannian structure on X.

Proof.
Let x a point of X, suppose verified the above hypothesis. Since (a) and (b) are true by hypothesis, then according to the Proposition 20, the linear envelope coincides with the Lie algebra of the holonomy group. Consequently, the linear envelope is stable by the Lie bracket and then admits a Lie algebra structure. Now according to the condition (c), there exist a quadratic positive definite form on T x X preserved infinitesimally by x r − It follows that the linear envelope is compact. We deduce there that the Lie algebra of holonomy groups is of finite dimension see condition (a). Since X is locally compact, then according to Lie-Palais theorem, the Lie algebra of the group is the lift of the holonomy group whose the topologic structure compact is preserved. Therefore, the holonomy group is relatively compact. Now the Proposition 10 says that for X be a smooth manifold, ∇ its connection with free torsion, ∇ comes from a Riemanniann structure if and only if its holonomy group are relatively compact. Then we have the result.