Tangent Bundles and Tangent Sphere Bundles

Abstract

In the first two sections of this chapter we discuss the geometry of the tangent bundle and the tangent sphere bundle. In Section 3 we briefly present a more general construction on vector bundles and in Section 4 specialize to the case of the normal bundle of a submanifold. The formalism for the tangent bundle and the tangent sphere bundle is of sufficient importance to warrant its own development, rather than specializing from the vector bundle case. In Section 5 we discuss briefly a contact structure on the projectivized tangent bundle of the hyperbolic plane and the geodesic flow on this bundle rather than on the unit tangent bundle, which is its usual setting.