by Michael Spivak (Author)

A Comprehensive Introduction to Differential Geometry, Volume One, Third Edition by Michael Spivak is an essential resource for anyone interested in the field of differential geometry. Published by Publish or Perish, Inc. in Houston, Texas, this volume is part of a five-volume series that has been meticulously crafted to bridge the gap between classical and modern differential geometry. This third edition includes significant updates and corrections, along with the inclusion of Gauss' paper in Volume 2. The book provides a thorough introduction to the theory of differentiable manifolds, emphasizing both historical context and modern language. It covers a wide range of topics, including immersion and functions, the tangent bundle, integral curves and vector fields, differential equations, integration of differential forms, inner product on vector spaces, critical points and conditions, exact sequence of complexes and cohomology vector spaces, and Euler characteristic and related theorems. Unique features of this book include detailed explanations and propositions related to integral curves, vector fields, and coordinate systems, comprehensive coverage of solving differential equations and integration of differential forms, in-depth discussion on inner product on vector spaces, including Euclidean and Riemannian metrics, exploration of critical points, variational approach, and geodesics, extensive treatment of exact sequences, cohomology vector spaces, and Euler characteristic, and application of Poincaré duality theorem, Thom class, Euler class, vector fields, and indices. With its rigorous approach and clear presentation, A Comprehensive Introduction to Differential Geometry, Volume One, Third Edition is an invaluable addition to the library of mathematicians, physicists, and anyone with a keen interest in differential geometry.