Connections and Curvature

Part of the Applied Mathematical Sciences book series (AMS, volume 116)


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 Chap. 1, but here we also study the curvature of a connection. Material in §2 on second covariant derivatives is connected with material in Chap. 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 Chap. 5, and the uniformization theorem for compact Riemann surfaces and other problems involving nonlinear, elliptic PDE, arising from studies of curvature, treated in Chap. 14. Section 5 on the Gauss–Bonnet theorem is useful both for estimates related to the proof of the uniformization theorem and for applications to the Riemann–Roch theorem in Chap. 10. Furthermore, it serves as a transition to more advanced material presented in §§6–8.


Vector Field Riemannian Manifold Vector Bundle Fundamental Form Gauss Curvature 
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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of North CarolinaChapel HillUSA

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