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El. knyga: Riemannian Geometry and Geometric Analysis

4.00/5 (9 ratings by Goodreads)
  • Formatas: PDF+DRM
  • Serija: Universitext
  • Išleidimo metai: 13-Oct-2017
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783319618609
Kitos knygos pagal šią temą:
Riemannian Geometry and Geometric AnalysisDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Serija: Universitext
  • Išleidimo metai: 13-Oct-2017
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783319618609
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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 newmaterial, 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

Recenzijos

The present volume ends with two appendices (on linear elliptic partial differential equations and topological results about fundamental groups and covering spaces) and a rich bibliography of 454 items, including some classical books and papers. All the material, written in a clear and precise style, is carefully developed, many examples supporting the understanding. In the reviewers opinion, this is an excellent book, a very useful addition to any good library. (Gabriel Eduard Vilcu, zbMATH 1380.53001, 2018)

1 Riemannian Manifolds
1(50)
1.1 Manifolds and Differentiable Manifolds
1(7)
1.2 Tangent Spaces
8(5)
1.3 Submanifolds and Foliations
13(3)
1.4 Riemannian Metrics
16(20)
1.5 Existence of Geodesies on Compact Manifolds
36(3)
1.6 The Heat Flow and the Existence of Geodesies
39(4)
1.7 Existence of Geodesies on Complete Manifolds
43(8)
Basic Exercises for Chap. 1
46(2)
Further Exercises for Chap. 1
48(3)
2 Lie Groups and Vector Bundles
51(64)
2.1 Vector Bundles
51(10)
2.2 Complex and Holomorphic Vector Bundles: Almost Complex and Complex Manifolds
61(4)
2.3 Integral Curves of Vector Fields: Lie Algebras
65(11)
2.4 Symplectic Structures
76(6)
2.5 Lie Groups
82(6)
2.6 Spin Structures
88(27)
Basic Exercises for Chap. 2
112(2)
Further Exercises for Chap. 2
114(1)
3 The Laplace Operator and Harmonic Differential Forms
115(48)
3.1 The Laplace Operator on Functions
115(6)
3.2 The Spectrum of the Laplace Operator
121(10)
3.3 The Laplace Operator on Forms
131(12)
3.4 Representing Cohomology Classes by Harmonic Forms
143(10)
3.5 The Heat Flow and Harmonic Forms
153(10)
Basic Exercises for Chap. 3
159(2)
Further Exercises for Chap. 3
161(2)
4 Connections and Curvature
163(70)
4.1 Connections in Vector Bundles
163(13)
4.2 Metric Connections. The Yang--Mills Functional
176(17)
4.3 The Levi-Civita Connection
193(20)
4.4 Connections for Spin Structures and the Dirac Operator
213(7)
4.5 The Bochner Method
220(7)
4.6 Eigenvalue Estimates by the Method of Li--Yau
227(6)
Basic Exercises for Chap. 4
232(1)
Further Exercises for Chap. 4
232(1)
5 Geometry of Submanifolds
233(18)
5.1 The Second Fundamental Form
233(3)
5.2 The Curvature of Submanifolds
236(4)
5.3 The Volume of Submanifolds
240(3)
5.4 Minimal Submanifolds
243(8)
Basic Exercises for Chap. 5
248(1)
Further Exercises for Chap. 5
248(3)
6 Geodesies and Jacobi Fields
251(66)
6.1 First and Second Variation of Arc Length and Energy
251(6)
6.2 Jacobi Fields
257(10)
6.3 Conjugate Points and Distance Minimizing Geodesies
267(9)
6.4 Riemannian Manifolds of Constant Curvature
276(3)
6.5 The Rauch Comparison Theorems and Other Jacobi Field Estimates
279(6)
6.6 The Hessian of the Squared Distance Function
285(3)
6.7 Volume Comparison
288(5)
6.8 Approximate Fundamental Solutions and Representation Formulas
293(2)
6.9 The Geometry of Manifolds of Nonpositive Sectional Curvature
295(22)
Basic Exercises for Chap. 6
313(1)
Further Exercises for Chap. 6
313(4)
A Short Survey on Curvature and Topology
317(340)
7 Symmetric Spaces and Kahler Manifolds
327(64)
7.1 Complex Projective Space
327(7)
7.2 Kahler Manifolds
334(12)
7.3 The Geometry of Symmetric Spaces
346(12)
7.4 Some Results About the Structure of Symmetric Spaces
358(7)
7.5 The Space Sl(n, R)/SO(n, R)
365(20)
7.6 Symmetric Spaces of Noncompact Type as Examples of Nonpositively Curved Riemannian Manifolds
385(6)
Basic Exercises for Chap. 7
390(1)
Further Exercises for Chap. 7
390(1)
8 Morse Theory and Floer Homology
391(98)
8.1 Preliminaries: Aims of Morse Theory
391(5)
8.2 Compactness: The Palais--Smale Condition and the Existence of Saddle Points
396(3)
8.3 Local Analysis: Nondegeneracy of Critical Points, Morse Lemma, Stable and Unstable Manifolds
399(16)
8.4 Limits of Trajectories of the Gradient Flow
415(8)
8.5 The Morse--Smale--Floer Condition: Transversality and Z2-Cohomology
423(7)
8.6 Orientations and Z-homology
430(4)
8.7 Homotopies
434(5)
8.8 Graph Flows
439(4)
8.9 Orientations
443(17)
8.10 The Morse Inequalities
460(12)
8.11 The Palais--Smale Condition and the Existence of Closed Geodesies
472(17)
Exercises for Chap. 8
486(3)
9 Harmonic Maps Between Riemannian Manifolds
489(84)
9.1 Definitions
489(8)
9.2 Formulas for Harmonic Maps: The Bochner Technique
497(12)
9.3 Definition and Lower Semicontinuity of the Energy Integral
509(12)
9.4 Higher Regularity
521(12)
9.5 Harmonic Maps into Manifolds of Nonpositive Sectional Curvature: Existence
533(7)
9.6 Harmonic Maps into Manifolds of Nonpositive Sectional Curvature: Regularity
540(23)
9.7 Harmonic Maps into Manifolds of Nonpositive Curvature: Uniqueness and Other Properties
563(10)
Basic Exercises for Chap. 9
569(1)
Further Exercises for Chap. 9
570(3)
10 Harmonic Maps from Riemann Surfaces
573(58)
10.1 Two-dimensional Harmonic Mappings and Holomorphic Quadratic Differentials
573(15)
10.2 The Existence of Harmonic Maps in Two Dimensions
588(25)
10.3 Regularity Results
613(18)
Basic Exercises for Chap. 10
628(1)
Further Exercises for Chap. 10
629(2)
11 Variational Problems from Quantum Field Theory
631(26)
11.1 The Ginzburg--Landau Functional
631(9)
11.2 The Seiberg--Witten Functional
640(7)
11.3 Dirac-Harmonic Maps
647(10)
Exercises for Chap. 11
655(2)
A Linear Elliptic Partial Differential Equations
657(12)
A.1 Sobolev Spaces
657(5)
A.2 Existence and Regularity Theory for Solutions of Linear Elliptic Equations
662(4)
A.3 Existence and Regularity Theory for Solutions of Linear Parabolic Equations
666(3)
B Fundamental Groups and Covering Spaces
669(6)
Bibliography 675(16)
Index 691
Jürgen Jost is Codirector of the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany, an Honorary Professor at the Department of Mathematics and Computer Sciences at Leipzig University, and an External Faculty Member of the Santa Fe Institute for the Sciences of Complexity, New Mexico, USA.





He is the author of a number of further Springer textbooks including Postmodern Analysis (1997, 2002, 2005), Compact Riemann Surfaces (1997, 2002, 2006), Partial Differential Equations (2002, 2007, 2013), Differentialgeometrie und Minimalflächen (1994, 2007, 2014, with J. Eschenburg), Dynamical Systems (2005), Mathematical Concepts (2015), as well as several research monographs, such as Geometry and Physics (2009), and many publications in scientific journals.
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