Journal of Generalized Lie Theory and Applications
- J. Gen. Lie Theory Appl.
- Volume 10, Number S2 (2016), 4 pages.
Trying to Explicit Proofs of Some Veys 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 ∇$^k$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.
J. Gen. Lie Theory Appl., Volume 10, Number S2 (2016), 4 pages.
First available in Project Euclid: 16 November 2016
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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