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Riemannian geometry and geometric analysis. 3rd ed. (English) Zbl 1034.53001
Universitext. Berlin: Springer (ISBN 3-540-42627-2/pbk). xiii, 532 p. EUR 32.05; sFr. 85.07; £ 33.00; $ 54.95 (2002).
This third edition (Zbl 0828.53002, Zbl 0997.53500) gives a new presentation of Morse theory and Floer homology that emphasises the geometric aspects and integrates it into the context of Riemannian geometry and geometric analysis. It also gives a new presentation of the geometric aspects of harmonic maps: This uses geometric methods from the theory of geometric spaces of nonpositive curvature and, at the same time, sheds light on these, as an excellent example of the integration of deep geometric insights and powerful analytical tools.
Ch. 1, Foundational Material, introduces the basic geometric concepts, like differentiable manifolds, tangent spaces, vector bundles, vector fields and one-parameter groups of diffeomorphisms, Lie algebras and groups and in particular Riemannian metrics. Some elementary results about geodesics are also derived.
Ch. 2, De Rham Cohomology and Harmonic Differential Forms, introduces de Rham cohomology groups and the essential tools from elliptic PDE for treating these groups. In later chapters, the reader will encounter nonlinear versions of the methods presented here.
Ch. 3, Parallel Transport, Connections, and Covariant Derivatives, treats the general theory of connections and curvature.
In Ch. 4, Geodesies and Jacobi Fields are introduced, the Rauch comparison theorems for Jacobi fields are proved and applied to geodesics.
These first four chapters treat the more elementary and basic aspects of the subject. Their results are used in the remaining, more advanced chapters that are essentially independent of each other.
Ch. 5, Symmetric Spaces and Kähler Manifolds, treats in detail symmetric spaces as important examples of Riemannian manifolds.
Ch. 6, Morse Theory and Floer Homology, attempts to explain the relevant ideas and concepts in an elementary manner and with detailed examples.
Ch. 7, treats Variational Problems from Quantum Field Theory, in particular the Ginzburg-Landau and Seiberg-Witten equations. The background material on spin geometry and Dirac operators is developed in earlier chapters.
Ch. 8, treats Harmonic Maps between Riemannian manifolds. Several existence theorems are proved and applied to Riemannian geometry. A guiding principle for this textbook is that the material in the main body should be self contained. The essential exception is that the author uses material about Sobolev spaces and linear elliptic PDEs without giving proofs.
This material is collected in Appendix A, Linear Elliptic Partial Differential Equation. Appendix B, Fundamental Groups and Covering Spaces, collects some elementary topological results about fundamental groups and covering spaces.
As appendices to most of the paragraphs, sections with the title “Perspectives” are given. The aim of those sections is to place the material in a broader context and explain further results and directions without detailed proofs. At the end of each chapter, some exercises for the reader are given.

53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
53C20 Global Riemannian geometry, including pinching
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58E20 Harmonic maps, etc.
53C22 Geodesics in global differential geometry
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53B21 Methods of local Riemannian geometry
58-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to global analysis