Riemannian geometry and geometric analysis
The first four chapters of the book deal with the following basic geometric notions: differentiable manifold, tangent space, vector bundle, vector field, one-parameter group of diffeomorphisms, Lie algebras and Lie groups, Riemannian metrics, geodesics (Chapter 1); de Rham cohomology groups, harmonic forms, heat flow method (Chapter 2); general theory of connections and curvature (Chapter 3); Jacobi fields, conjugate points, Rauch comparison theorems, spaces of nonpositive sectional curvature (Chapter 4). In Chapter 5, symmetric spaces and Kähler manifolds are analyzed in detail. Chapter 6 is devoted to Morse theory and Floer homology. Chapter 7 deals with harmonic maps between Riemanian manifolds, while the case of harmonic maps from Riemannian surfaces is studied in Chapter 8. The last chapter treats variational problems from quantum field theory, in particular the Ginzburg-Landau, Seibert-Witten equations, and a mathematical version of the nonlinear supersymmetric sigma model. In two final appendices results concerning linear elliptic partial differential equations, fundamental groups, and covering spaces are included. With few exceptions, the material of this book is self-contained. Each chapter ends with a set of exercises. Most of the paragraphs have a section ``Perspectives'', written with the aim to place the material in a broader context and explain further results and directions. The main new features of this fifth edition are the systematic inclusion of flow equations and a mathematical treatment of the nonlinear sigma model of quantum field theory. For previous editions of this book see Zbl 0828.53002, Zbl 0997.53500, Zbl 1034.53001 and Zbl 1083.53001.
scientific article; zbMATH DE number 5233828