alexa Trying to Explicit Proofs of Some Veys Theorems in Linear Connections
ISSN: 1736-4337

Journal of Generalized Lie Theory and Applications
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Research Article

Trying to Explicit Proofs of Some Veys Theorems in Linear Connections

Lantonirina LS*

Department of Mathematics and Computer Sciences, University of Antananarivo, Madagascar

*Corresponding Author:
Lantonirina LS
Department of Mathematics and Computer Sciences,
University of Antananarivo, Madagascar
Tel: 261334651397
E-mail: [email protected]

Received Date: January 23,2016; Accepted Date: February 15,2016; Published Date: March 15,2016

Citation: Lantonirina LS (2016) Trying to Explicit Proofs of Some Vey’s Theorems in Linear Connections. J Generalized Lie Theory Appl S2:009. doi:10.4172/1736- 4337.S2-009

Copyright: © 2016 Lantonirina LS. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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

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