A generalization of the Levi-Civitae connection on Riemannian manifolds - Petrova, Irina (Ann. Sofia Univ. Fac. Math. and Inf., 1994)
Let M be a Riemannian manifold with a metric tensor g. A linear connection abla on M with properties: abla_XY- abla_YX-[X,Y]=0 2cm X,Y,Zσ( abla_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.