On Naturally reductive left-invariant metrics of SL(2,R)

On any real semisimple Lie group we consider a one parameter family of left-invariant naturally reductive metrics. Their geodesic flow in terms of Killing curves, the Levi Civita connection and the main curvature properties are explicitly computed. Furthermore we present a group theoretical revisitation of a classical realization of all simply connected 3-dimensional manifolds with a transitive group of isometries due to L. Bianchi and \'E. Cartan. As a consequence one obtains a characterization of all naturally reductive left-invariant Riemannian metrics of $\,SL(2,\R)$