Tangent Bundles and Tangent Sphere Bundles

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. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.