本书是莫斯科大学数学力学系经典教材《现代几何学——方法和应用》三卷本的第２卷。全书力求以直观的和物理的视角阐述，是一本难得的现代几何方面的佳作。整套书内容包括张量分析、曲线和曲面几何、一维和高维变分法（第１卷），微分流形的拓扑和几何（第２卷），以及同调与上同调理论（第３卷）。本书可用作数学和理论物理专业高年级和研究生的教学用书，对从事几何和拓扑研究的工作者也极具参考价值。

Preface. Chapter 1: Examples of Manifolds. Chapter 2: Foundational Questions. Essential Facts Concerning Functions on a Manifold. Typical Smooth Mappings. Chapter 3: The Degree of a Mapping. The Intersection Index of Submanifolds. Applications. Chapter 4: Orientability of Manifolds. The Fundamental Group. Covering Spaces (Fibre Bundles with Discrete Fibre). Chapter 5: Homotopy Groups. Chapter 6: Smooth Fibre Bundles. Chapter 7: Some Examples of Dynamical Systems and Foliations on Manifolds. Chapter 8: The Global Structure of Solutions of Higher-Dimensional Variational Problems. Bibliography.