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
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