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...

fibre is the re-dimensional complex projective space (complex projective co-tangent bundle). This bundle of complex dimension 2re + l is our M. Considering the fibre of T(V) as the (2w+2)-dimensional real vector space, we take as P the co-tangent sphere bundle over V (i.e., the fibre of P is a sphere in the fibre of T(V)). T(V) — V is the principal fibre bundle associated with a line bundle L o...