Journal of Generalized Lie Theory and Applications

Trying to Explicit Proofs of Some Veys Theorems in Linear Connections

LS Lantonirina

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.

Article information

Source
J. Gen. Lie Theory Appl., Volume 10, Number S2 (2016), 4 pages.

Dates
First available in Project Euclid: 16 November 2016

Permanent link to this document
https://projecteuclid.org/euclid.jglta/1479265228

Digital Object Identifier
doi:10.4172/1736-4337.1000S2-009

Mathematical Reviews number (MathSciNet)
MR3663978

Zentralblatt MATH identifier
1376.53031

Citation

Lantonirina, LS. Trying to Explicit Proofs of Some Veys Theorems in Linear Connections. J. Gen. Lie Theory Appl. 10 (2016), no. S2, 4 pages. doi:10.4172/1736-4337.1000S2-009. https://projecteuclid.org/euclid.jglta/1479265228