In his papers (Tohoku, Math. J., 10, 338-354; 14, 146-155), Sasaki, S. has studied the differential geometry of tangent bundles of Riemannian manifolds, in this paper, we extended Sasaki's work to vector bundles on Riemannian manifolds with connections, and investigated the Riemannian connections, geodesics and totally geodesic submanifolds of the bundle spaces.
