Connections and Curvature

Abstract

In this appendix we present results in differential geometry that serve as a useful background for material in the main body of the book. Material in §1 on connections is somewhat parallel to the study of the natural connection on a Riemannian manifold made in §11 of Chapter 1, but here we also study the curvature of a connection. Material in §2 on second covariant derivatives is connected with material in Chapter 2 on the Laplace operator. Ideas developed in §§3 and 4, on the curvature of Riemannian manifolds and submanifolds, make contact with such material as the existence of complex structures on two-dimensional Riemannian manifolds, established in Chapter 5, and the uniformization theorem for compact Riemann surfaces and other problems involving nonlinear, elliptic PDE, arising from studies of curvature, treated in Chapter 14. Section 5 on the Gauss-Bonnet theorem is useful both for estimates related to the proof of the unif ormization theorem and for applications to the Riemann-Roch theorem in Chapter 10. Furthermore, it serves as a transition to more advanced material presented in §§6–8.KeywordsVector FieldRiemannian ManifoldVector BundleCovariant DerivativeFundamental FormThese 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.