About this book
This established reference work continues to provide its readers with a gateway to some of the most interesting developments in contemporary geometry. It offers insight into a wide range of topics, including fundamental concepts of Riemannian geometry, such as geodesics, connections and curvature; the basic models and tools of geometric analysis, such as harmonic functions, forms, mappings, eigenvalues, the Dirac operator and the heat flow method; as well as the most important variational principles of theoretical physics, such as Yang-Mills, Ginzburg-Landau or the nonlinear sigma model of quantum field theory. The present volume connects all these topics in a systematic geometric framework. At the same time, it equips the reader with the working tools of the field and enables her or him to delve into geometric research.
The 7th edition has been systematically reorganized and updated. Almost no page has been left unchanged. It also includes new material, for instance on symplectic geometry, as well as the Bishop-Gromov volume growth theorem which elucidates the geometric role of Ricci curvature.
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
“For readers familiar with the basics of differential geometry and some acquaintance with modern analysis, the book is reasonably self-contained … The book succeeds very well in laying out the foundations of modern Riemannian geometry and geometric analysis. It introduces a number of key techniques and provides a representative overview of the field.” Monatshefte für Mathematik
53B21, 53L20, 32C17, 35I60, 49-XX, 58E20, 57R15 Riemannian geometry Riemannian manifolds Lie groups symplectic geometry vector bundles Laplace operator harmonic functions harmonic maps curvature Dirac operator geometry of submanifolds geodesics Jacobi fields symmetric spaces Kähler manifolds Morse theory Floer homology quantum field theory variational problems theoretical physics variational principles
- DOI https://doi.org/10.1007/978-3-319-61860-9
- Copyright Information Springer International Publishing AG 2017
- Publisher Name Springer, Cham
- eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
- Print ISBN 978-3-319-61859-3
- Online ISBN 978-3-319-61860-9
- Series Print ISSN 0172-5939
- Series Online ISSN 2191-6675
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