ON THE LIFTS OF SEMI-RIEMANNIAN METRICS

In this paper, we extend Sasaki metric for tangent bundle of a Riemannian manifold and Sasaki-Mok metric for the frame bundle of a Riemannian manifold [I] to the case of a semi-Riemannian vector bundle over a semi- Riemannian manifold. In fact, if E is a semi-Riemannian vector bundle over a semi-Riemannian manifold M, then by using an arbitrary (linear) connection on E, we can make E, as a manifold, into a semi-Riemannian manifold. When the metric of the vector bundle E is parallel with respect to the chosen connection, we compute the Levi-Civita connection of E, its geodesics, and its curvature tensors. We also show that the sphere and pseudo-sphere bundles of E are nondegenerate submanifolds of E, and we shall compute their second fundamental forms. We shall also prove some results on the metric of E