A Comprehensive Introduction to Differential Geometry, Vol. 4, Third Edition

About The Book

Volume Four of Michael Spivak’s “A Comprehensive Introduction to Differential Geometry” focuses on higher dimensions and codimensions, curves in Riemannian manifolds, and fundamental equations for submanifolds. The text begins with an exploration of constant curvature manifolds, discussing models like the n-sphere S n S n and hyperbolic space H n H n , and their properties such as stereographic projection and conformal mappings. It covers geodesic mappings, horospheres, equidistant hypersurfaces, and Beltrami’s theorem, which states that a connected Riemannian manifold where every point has a neighborhood that can be mapped geodesically to Euclidean space has constant curvature. The section on curves generalizes Serret-Frenet formulas, introducing curvature functions and Frenet frames, and proving that curves with certain vanishing curvature functions lie in j-dimensional planes. The fundamental equations for submanifolds include Gauss’ formulas, Weingarten equations, Codazzi-Mainardi equations, and Ricci equations, with detailed derivations and applications to moving frames. The text concludes with the fundamental theorem for submanifolds, stating that second fundamental forms and normal connections determine submanifolds up to Euclidean motion, and generalizing this to manifolds of constant curvature. The appendix includes problems and additional results, such as flat ruled surfaces, modifications for surfaces in different spaces, and hypersurfaces of constant curvature in higher dimensions.