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2016 Trying to Explicit Proofs of Some Veys Theorems in Linear Connections
LS Lantonirina
J. Gen. Lie Theory Appl. 10(S2): 1-4 (2016). DOI: 10.4172/1736-4337.1000S2-009


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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LS Lantonirina. "Trying to Explicit Proofs of Some Veys Theorems in Linear Connections." J. Gen. Lie Theory Appl. 10 (S2) 1 - 4, 2016.


Published: 2016
First available in Project Euclid: 16 November 2016

zbMATH: 1376.53031
MathSciNet: MR3663978
Digital Object Identifier: 10.4172/1736-4337.1000S2-009

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

Rights: Copyright © 2016 Ashdin Publishing (2009-2013) / OMICS International (2014-2016)

Vol.10 • No. S2 • 2016
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