Riemannian Geometry and Geometric Analysis

Beschreibung

This textbook introduces techniques from nonlinear analysis at an early stage. Such techniques have recently become an indispensable tool in research in geometry, and they are treated here for the first time in a textbook. Topics treated include: Differentiable and Riemannian manifolds, metric properties, tensor calculus, vector bundles; the Hodge Theorem for de Rham cohomology; connections and curvature, the Yang-Mills functional; geodesics and Jacobi fields, Rauch comparison theorem and applications; Morse theory (including an introduction to algebraic topology), applications to the existence of closed geodesics; symmetric spaces and Kähler manifolds; the Palais-Smale condition and closed geodesics; Harmonic maps, minimal surfaces.
The present textbook is a somewhat expanded version of the material of a three-semester course I gave in Bochum. It attempts a synthesis of geometric and analytic methods in the study of Riemannian manifolds. In the first chapter, we introduce the basic geometric concepts, like dif ferentiable manifolds, tangent spaces, vector bundles, vector fields and one parameter groups of diffeomorphisms, Lie algebras and groups and in par ticular Riemannian metrics. We also derive some elementary results about geodesics. The second chapter introduces de Rham cohomology groups and the es sential tools from elliptic PDE for treating these groups. In later chapters, we shall encounter nonlinear versions of the methods presented here. The third chapter treats the general theory of connections and curvature. In the fourth chapter, we introduce Jacobi fields, prove the Rauch com parison theorems for Jacobi fields and apply these results to geodesics. These first four chapters treat the more elementary and basic aspects of the subject. Their results will be used in the remaining, more advanced chapters that are essentially independent of each other. In the fifth chapter, we develop Morse theory and apply it to the study of geodesics. The sixth chapter treats symmetric spaces as important examples of Rie mannian manifolds in detail.
Geometric analysis, in particular nonlinear methods, is combined with Riemannian geometry for the first time in a textbook. Detailed proofs of basic results are introduced at an early stage. The student will be introduced to Riemannian geometry from a local and global, mathematical and physical point of view. In particular the reader will get a working knowledge of methods of nonlinear analysis in geometry.

Autor*in

Jürgen Jost

Themen in »Riemannian Geometry and Geometric Analysis«

Harmonische Abbildungen Hodge theorem Levi-Civita connection Lovi-Civita-Zusammenhang Minimalflächen Morse theory Morse-Theorie Rauch comparison theorem Rauchscher Vergleichssatz Riemannian m curvature geodesics harmonic maps minimal surfaces

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