Okay, here is the translation following your instructions: Differential Manifolds: Foundations

Author: Song Weidong
Publisher:
Publish Date: 2006-01-01
Features: This book is prepared for undergraduate elective courses in university mathematics majors and for foundational mathematics and applied mathematics graduate public core courses. It primarily introduces the basic concepts of differentiable manifolds, tangent spaces, tangent bundles and vector fields, tensor analysis, linear connections, Riemannian manifolds, etc. The book is concise and comprehensive, providing space for further study, bridging the gap between elementary and classical differential geometry and modern differential geometry. It serves as an introductory textbook on differentiable manifolds, covering topics such as an introduction to differentiable manifolds, tensor analysis, the integration of exterior differential forms and Stokes' theorem, affine connections, and various differential operators on manifolds. At the end of each chapter, there are problems and exercises. This book can be used as a textbook for graduate public core courses in foundational mathematics-related fields at advanced normal universities and as an elective course for senior undergraduate students majoring in mathematics education. It can also serve as a reference for researchers in related fields such as mechanics and theoretical physics.