A generalization of the Levi-Civitae connection on Riemannian manifolds

Abstract

Let $M$ be a Riemannian manifold with a metric tensor $g$. A linear connection $\widetilde{\nabla}$ on $M$ with properties:

$\widetilde{\nabla}_XY-\widetilde{\nabla}_YX-[X,Y]=0 \hspace{2cm} \underset{X,Y,Z}{\sigma}(\widetilde{\nabla}_X^g)(Y,Z)=0$

is called a Generalised Levi-Civitae connection for $g$. In the paper are given examples of such connections. One of them is $(0)$-connection of Cartan and Schouten on a Lie group for any left invariant metric. A relation is described between these connections and the metric connections. It is proved that tehre is a Generalised Levi-Civitae connection that is not a Levi-Civitae connection on Riemannian manifolds.