"This established reference work continues to lead its readers to some of the hottest topics of contemporary mathematical research. This new edition introduces and explains the ideas of the parabolic methods that have recently found such a spectacular success in the work of Perelman at the examples of closed geodesics and harmonic forms. It also discusses further examples of geometric variational problems from quantum field theory, another source of profound new ideas and methods in geometry."--Jacket

zlibzh/Mathematics/Geometry and Topology/Jürgen Jost/Riemannian Geometry and Geometric Analysis_29590745.pdf

JURGEN JOST著

Springer Spektrum. in Springer-Verlag GmbH

Steinkopff. in Springer-Verlag GmbH

Springer Nature (Textbooks & Major Reference Works), Berlin, Heidelberg, 2008

Germany, Germany

January 2008

2008.03

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Includes bibliographical references (p. [561]-576) and index.

Riemannian geometry is characterized, and research is oriented towards and shaped by concepts (geodesics, connections, curvature,...) andobjectives,inparticularto understand certain classes of (compact) Riemannian manifolds de?ned by curvature conditions (constant or positive or negative curvature,...). Bywayofcontrast,g- metric analysis is a perhaps somewhat less systematic collection of techniques, for solving extremal problems naturally arising in geometry and for investigating and characterizing their solutions. It turns out that the two?elds complement each other very well; geometric analysis o?ers tools for solving di?cult problems in geometry, and Riemannian geometry stimulates progress in geometric analysis by setting am- tious goals. It is the aim of this book to be a systematic and comprehensive introduction to Riemannian geometry and a representative introduction to the methods of geometric analysis. It attempts a synthesis of geometric and analytic methods in the study of Riemannian manifolds. The present work is the?fth edition of my textbook on Riemannian geometry and geometric analysis. It has developed on the basis of several graduate courses I taught at the Ruhr-University Bochum and the University of Leipzig. The main new features of the present edition are the systematic inclusion of?ow equations and a mathematical treatment of the nonlinear sigma model of quantum?eld theory. These new topics also led to a systematic reorganization of the other material. Naturally, I have also included several smaller additions and minor corrections (for which I am grateful to several readers).

This established reference work continues to lead its readers to some of the hottest topics of contemporary mathematical research. The previous edition already introduced and explained the ideas of the parabolic methods that had found a spectacular success in the work of Perelman at the examples of closed geodesics and harmonic forms. It also discussed further examples of geometric variational problems from quantum field theory, another source of profound new ideas and methods in geometry. The 6th edition includes a systematic treatment of eigenvalues of Riemannian manifolds and several other additions. Also, the entire material has been reorganized in order to improve the coherence of the book. From the reviews: "This book provides a very readable introduction to Riemannian geometry and geometric analysis. ... With the vast development of the mathematical subject of geometric analysis, the present textbook is most welcome." Mathematical Reviews "...the material ... 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." Zentralblatt MATH

2024-06-13