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Trying to explicit proofs of some Vey’s theorems in linear connections. (English) Zbl 1376.53031

Summary: Let \(X\) be a differentiable paracompact manifold. Under the hypothesis of a linear connection \(r\) with free torsion \(T\) on \(X\), 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 \(\nabla\) to obtain Riemannian structure. Next, in the analytic case of \(X\), the existence of a quadratic positive definite form \(g\) on the tangent bundle \(TX\) such that it was invariant in the infinitesimal sense by the linear operators \(\nabla^k R\), where \(R\) is the curvature of \(\nabla\) shows that the connection \(\nabla\) comes from a Riemannian structure. At last, for a simply connected manifold \(X\), we give some conditions on the linear envelope of the curvature \(R\) to have a Riemannian structure.

MSC:

53B05 Linear and affine connections
53C20 Global Riemannian geometry, including pinching
53C29 Issues of holonomy in differential geometry
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Full Text: Euclid