Review of Tensors, Manifolds, and Vector Bundles

Abstract

Most of the technical machinery of Riemannian geometry is built up using tensors; indeed, Riemannian metrics themselves are tensors. Thus we begin by reviewing the basic definitions and properties of tensors on a finite-dimensional vector space. When we put together spaces of tensors on a manifold, we obtain a particularly useful type of geometric structure called a “vector bundle,” which plays an important role in many of our investigations. Because vector bundles are not always treated in beginning manifolds courses, we include a fairly complete discussion of them in this chapter. The chapter ends with an application of these ideas to tensor bundles on manifolds, which are vector bundles constructed from tensor spaces associated with the tangent space at each point.