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. As we saw in Chapter 1, the cotangent bundle of a manifold has a natural symplectic structure and we will see here that the same is true of the tangent bundle of a Riemannian manifold.