Journal of Generalized Lie Theory and Applications

Trying to Explicit Proofs of Some Veys Theorems in Linear Connections

LS Lantonirina

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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

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

First available in Project Euclid: 16 November 2016

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Zentralblatt MATH identifier

Linear connections Riemannian connection Levi-civita connection Holonomy groups Linear envelope Kth derivations Lie algebras


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.

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