Metrizability of affine connections on analytic manifolds. (English) Zbl 0699.53038

Kowalski, Oldřich

Note Mat. 8, No. 1, 1-11 (1988).

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

Keywords:

analytic manifold; analytic connection; holonomy algebra