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Metrizability of affine connections on analytic manifolds. (English) Zbl 0699.53038
Let M be a connected, simply connected and analytic manifold provided with a torsion-free analytic connection $$\nabla$$, let $$h^*(x)$$ be the holonomy algebra of $$\nabla$$ at $$x\in M$$. Consider $$H(x)\subset S^ 2T_ xM$$ the subspace of all symmetric bilinear forms $$G_ x$$ on the tangent space $$T_ xM$$ satisfying the condition $$G_ x(Au,v)+G_ x(u,Av)=0,$$ for all $$A\in H^*(x)$$ and all $$u,v\in T_ xM$$. The author proves the theorem: If the subspace H(x) for some $$x\in M$$ contains a positive definite form, then $$\nabla$$ is a Riemannian connection on M and conversely, if $$\nabla$$ is Riemannian, then each subspace H(x) contains a positive definite form. The purpose of this article is to give an interesting algorithm for deciding whether a torsion-free connection $$\nabla$$ on M is Riemannian, or not. The description of the algorithm is essentially based on this theorem and on linear algebra.
Reviewer: C.Apreutesei

MSC:
 53C05 Connections (general theory) 53B05 Linear and affine connections 53B20 Local Riemannian geometry