A Comprehensive Introduction to Differential Geometry, Vol. 4, 3rd Edition

A Comprehensive Introduction to Differential Geometry, Volume Four, Third Edition” by Michael Spivak, published by Publish or Perish, Inc. in 1999, is an advanced mathematical text focusing on differential geometry. This volume, part of a five-volume series, delves into higher dimensions and codimensions, curves in Riemannian manifolds, and the fundamental equations for submanifolds. Key features of this volume include a detailed exploration of constant curvature manifolds, comprehensive treatment of curves in Riemannian manifolds, and in-depth discussion of the fundamental equations for submanifolds. The book extensively uses orthonormal moving frames and connection forms, providing rigorous proofs and theorems such as Beltrami’s Theorem and the Ricci Equations. This volume is ideal for advanced students and researchers in mathematics, providing a thorough understanding of differential geometry’s higher-dimensional aspects. It includes detailed mathematical formulations and proofs to support the presented concepts, making it an essential resource for those studying or working in the field of differential geometry.