Abstract

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 ∇k R, 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

Highlights

  • We assume that Χ is a connected real analytic manifold accompanied by a real analytic connection∇, these new assumptions construct us a positive definite quadratic form g 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

  • Suppose ∇ comes from a Riemannian structure (X, g). by the Fundamental Theorem of Riemannian Geometry, we have the Levi-Civita's connection ∇ on X

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Summary

Preliminary and Introduction

Vey was written some theorems in linear connections The title of this unpublished paper was: "Sur les connexions riemanniennes" means "On the Riemannian connections". We assume that Χ is a 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. 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 one where we use some results and idea of the present paper. Any Riemannian structure g produces an unique linear connection ∇ called Levi-Civita's connection on X with free torsion such that ∇g=0

John Nash
Real analytic manifold and Riemannian structure
Lie algebras of holonomy groups
Vi t
TxX infinitesimally preserved by r

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