On riemannian connections and semi-simplicity of a Lie algebra

Abstract

Using a almost product structure defined by a spray, we give a necessary and sufficient condition, for a linear connection with vanishing torsion to be Riemannian and, for the semi-simplicity of Lie algebra of projectable vector fields which commute with a spray. We show the equivalence of the semi-simplicity of a finite dimensional Lie algebra to the coincidence of the derived ideal with its algebra, to the interiority of any derivation of the algebra and, to the semi-simplicity of its adjoint representation.