by Bo-Yu Hou (Northwest University, China) & Bo-Yuan Hou (Academia Sinica, China)

This book is divided into fourteen chapters, with 18 appendices as introduction to prerequisite topological and algebraic knowledge, etc. The first seven chapters focus on local analysis. This part can be used as a fundamental textbook for graduate students of theoretical physics. Chapters 8–10 discuss geometry on fibre bundles, which facilitates further reference for researchers. The last four chapters deal with the Atiyah-Singer index theorem, its generalization and its application, quantum anomaly, cohomology field theory and noncommutative geometry, giving the reader a glimpse of the frontier of current research in theoretical physics.

• Differentiable Manifolds and Differential Forms
• Transformation of Manifold, Manifolds with Given Vector Fields and Lie Group Manifold
• Affine Connection and Covariant Differentiation
• Riemannian Manifold
• Symplectic Manifold and Contact Manifold
• Complex Manifolds
• Homology of Manifolds
• Homotopy of Manifold, Fibre Bundle, Classification of Fibre Bundles
• Differential Geometry of Fibre Bundle, Yang–Mills Gauge Theory
• Characteristic Classes
• The Atiyah–Singer Index Theorem
• Index Theorem on Manifold with Boundary and on Open Infinite Manifold
• Family Index Theorem, Topological Properties of Quantum Gauge Theory
• Noncommutative Geometry, Quantum Group, and q-Deformation of Chern-Characters