◎编辑推荐 《现代几何学——方法和应用》是莫斯科大学数学力学系对几何课程现代化改革的成果,作者之一的诺维科夫是1970年菲尔兹奖和2005年沃尔夫奖得主。 ◎内容简介 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。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。 ◎作者简介 鲍里斯·杜布罗文(B. A. Dubrovin)、阿纳托利·福缅科(A. T. Fomenko)和谢尔盖·诺维科夫(S. P. Novikov)都是苏联/俄罗斯著名数学家,三位数学家都毕业于莫斯科大学,并在莫斯科大学任教。杜布罗文在苏联解体后去了意大利国际高等研究院任教,在他2019年去世后,意大利国际高等研究院在莫斯科数学会等组织的支持下设立了杜布罗文奖章,授予那些在数学物理和几何领域做出突出贡献的青年数学家,每两年颁奖一次。福缅科和诺维科夫都是俄罗斯科学院院士,福缅科获得过1996年俄罗斯联邦国家奖,诺维科夫获得过1970年菲尔兹奖和2005年沃尔夫奖。 ◎图书目录 第一卷:
Preface. Chapter 1: Geometry in Regions of a Space. Basic Concepts. Chapter 2: The Theory of Surfaces. Chapter 3: Tensors: The Algebraic Theory. Chapter 4: The Differential Calculus of Tensors. Chapter 5: The Elements of the Calculus of Variations. Chapter 6: The Calculus of Variations in Several Dimensions. Fields and Their Geometric Invariants. Bibliography. 第二卷:
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. 第三卷:
Preface. Chapter 1: Homology and Cohomology. Computational Recipes. Chapter 2: Critical Points of Smooth Functions and Homology Theory. Chapter 3: Cobordisms and Smooth Structures. Bibliography. Appendix 1: An Analogue of Morse Theory for Many-Valued Functions. Certain Properties of Poisson Brackets. Appendix 2: Plateau's Problem. Spectral Bordisms and Globally Minimal Surfaces in Riemannian Manifolds. Errata to Parts I and II. '