A Comprehensive Introduction to Differential Geometry, Vol. 1, Third Edition

About The Book

The third edition of A Comprehensive Introduction to Differential Geometry, Volume One by Michael Spivak includes significant updates and corrections, along with the inclusion of Gauss’ paper translation in Volume 2. The book begins with a preface that discusses the evolution from mimeographed notes to a comprehensive book, highlighting the challenges in understanding differential geometry due to the gap between classical and modern treatments. The first volume focuses on differentiable manifolds, while the second volume covers fundamental papers by Gauss and Riemann. The content includes chapters on manifolds, differential structures, the tangent bundle, tensors, vector fields and differential equations, integral manifolds, differential forms, integration, Riemannian metrics, Lie groups, and an excursion in algebraic topology. The text emphasizes the importance of understanding the geometric aspect of differential geometry through historical development and presents major approaches in the field. The document section discusses advanced concepts in differential geometry, focusing on vector bundles, particularly the tangent bundle (TM) and cotangent bundle (TM) of a manifold M. Key points include the definition and properties of vector bundles, including the tangent bundle TM and cotangent bundle TM, construction of dual bundles and their properties, local triviality and equivalence of vector bundles, and sections of TM (vector fields) and their properties. The document also covers the properties and applications of differential forms, focusing on alternating tensors, wedge products, and the exterior derivative. It provides a detailed exploration of differential forms, their algebraic properties, and their significance in differential geometry and topology. The document concludes with a discussion on the integration in polar coordinates and the calculation of de Rham cohomology vector spaces for specific manifolds.

The emphasis on this First Volume is the study of differential forms and de Rham Cohomology Theory. Spivak also covers two ‘bonus’ topics: integral manifolds & foliations and Lie groups.