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Riemannian geometry and geometric analysis. 4th ed. (English) Zbl 1083.53001
Universitext. Berlin: Springer (ISBN 3-540-25907-4/pbk). xiii, 566 p. (2005).
The book is divided into eight chapters. At the end of the each chapter, some exercises for readers are given. The first chapter introduces basic geometric concepts such as differentiable manifolds, tangent spaces, vector bundles, vector fields, Lie groups, Lie algebras etc. The second chapter introduces de Rham cohomology groups and essential tools from elliptic PDE’s for treating these groups. The 3-rd chapter treats the general theory of connections and curvatures. In the 4-th chapter, Jacobi fields are introduced. The Ranch Theorems for Jacobi fields are proved. The 5-th chapter treats symmetric spaces. The 6-th chapter is devoted to classical Morse Theory and Floer homology. The 7-th chapter treats some problems on quantum fields theory, spin geometry and Dirac operators. In the 8-th chapter, harmonic maps between Riemannian manifolds are considered. Some existence theorems are proved and applied to classical Riemannian geometry. The sections entitled “Perspectives” include some historical remarks and references.

MSC:
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
53C22 Geodesics in global differential geometry
53C55 Global differential geometry of Hermitian and Kählerian manifolds
58-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to global analysis
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