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

Michael Spivak

Hardcover

ISBN 10: 091409873X ISBN 13: 9780914098737

Publisher: Publish or Perish, 1999

Synopsis

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.

About this title

About the Author

Michael David Spivak, born on May 25, 1940, is an American mathematician with a profound impact on the field of differential geometry. As an expositor of mathematics, he has skillfully bridged the gap between complex theory and accessible explanations. Spivak’s legacy extends beyond academia—he is also the visionary behind Publish-or-Perish Press. 📚 Author of the Five-Volume Masterpiece: Spivak’s magnum opus, “A Comprehensive Introduction to Differential Geometry,” stands as a testament to his mathematical prowess. This five-volume work is a treasure trove for those seeking a deep understanding of geometric concepts. 🔍 Why Choose Spivak? 🌟 Clarity and Precision: Spivak’s writing style demystifies intricate topics, making them approachable for learners at all levels. 🌟 Problem-Solving Focus: Dive into challenging problems guided by a master mathematician. 🌟 Mathematical Rigor: Immerse yourself in proofs, theorems, and applications—the hallmark of Spivak’s approach. 📖 About the Mathematician: Hailing from Queens, New York, Spivak earned his Ph.D. from Princeton University. His contributions have left an indelible mark on mathematics, and his passion for clarity continues to inspire generations of learners. 🛒 Explore Spivak’s Work: Whether you’re a student, educator, or lifelong learner, delve into the intricacies of calculus, geometry, and mathematical reasoning with Michael Spivak. 📚🔢

Excerpt. © Reprinted by permission. All rights reserved.

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.

Edition details

• Publisher: Publish or Perish
• Publication date: 1999
• Language: English
• ISBN 10: 091409873X
• ISBN 13: 9780914098737
• Binding: Hardcover
• Edition number: 3
• Number of pages: 390