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E-grāmata: Global Riemannian Geometry: Curvature and Topology

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Global Riemannian Geometry: Curvature and TopologyPalielināt
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This book contains a clear exposition of two contemporary topics in modern differential geometry:









distance geometric analysis on manifolds, in particular, comparison theory for distance functions in spaces which have well defined bounds on their curvature



the study of scalar curvature rigidity and positive mass theorems using spinors and the Dirac operator

It is intended for both graduate students and researchers.





This second edition has been updated to include recent developments such as promising results concerning the geometry of exit time moment spectra and potential analysis in weighted Riemannian manifolds, as well as results pertaining to an early conjecture on the geometry of the scalar curvature and speculations on new geometric approaches to the Index Theorem.
Preface to the first edition, 2003 vii
Preface to the second edition, 2020 vii
1 Distance Geometric Analysis on Manifolds
Ana Hurtado
Steen Markvorsen
Vicente Palmer
1 Appetizer and Introduction
1(1)
2 The Comparison Setting and Preliminaries
2(1)
3 Analysis of Riemannian Distance Functions
3(5)
4 Analysis of Lorentzian Distance Functions
8(5)
5 Concerning the Riemannian Setting and Notation
13(3)
6 Green's Formulae and the Co-area Formula
16(1)
7 The First Dirichlet Eigenvalue Comparison Theorem
17(6)
8 Isoperimetric Relations
23(1)
9 A Consequence of the Co-area Formula
23(1)
10 The Fundamental Differential Equation
24(1)
11 Isoperimetric Comparison
25(4)
12 Mean Exit Times and Moment Spectra
29(3)
13 The Poisson Hierarchy
32(5)
14 Capacity Comparison
37(5)
15 The Kelvin-Nevanlinna-Royden Criteria for Transience
42(2)
16 Surfaces of Revolution
44(3)
17 Warped Products
47(2)
18 Answering the Questions in the Appetizer
49(2)
19 Sufficient Conditions for Parabolicity and Hyperbolicity
51(3)
20 Hyperbolicity of Spacelike Hypersurfaces
54(2)
21 Weighted Riemannian Manifolds
56(1)
22 Weighted Capacities
57(1)
23 Weighted Rotationally Symmetric Spaces and the Ahlfors Criterion for Weighted Parabolicity
58(2)
24 Weighted Curvatures
60(1)
25 Analysis of Restricted Distance Functions in Weighted Submanifolds
61(2)
26 Extrinsic Criteria for Weighted Parabolicity
63(3)
27 The Grigor'yan-Fernandez Criterion for Weighted Hyperbolicity
66(2)
28 Graphs and Flows
68(3)
29 Scherk's Graph is Transient
71(3)
References
74(9)
2 The Dirac Operator in Geometry and Physics
Maung Min-Oo
Foreword
83(1)
Foreword to the Second Edition
83(1)
1 Spinors and the Dirac Operator
84(8)
1.1 Introduction to Spinors
84(4)
Examples
88(1)
1.2 The Dirac Operator
88(2)
1.3 The Lichnerowicz Formula
90(2)
2 Gromov's K-Area
92(13)
2.1 Definition of K-Area
92(3)
2.2 The Fundamental Estimate in Terms of Scalar Curvature
95(6)
2.3 Connections with Symplectic Invariants
101(1)
2.4 The Vafa-Witten Inequality
102(3)
3 Positive Mass Theorems
105(10)
3.1 Description of Results
105(4)
3.2 Main Ideas behind the Proofs
109(5)
3.3 Some Mathematical Aspects of the AdS/CFT Correspondence
114(1)
4 Epilogue
115(3)
4.1 Scalar curvature on the hemisphere
116(1)
4.2 Gromov's work on scalar curvature
117(1)
4.3 Some speculations and musings
117(1)
References 118
Ana Hurtado is a professor at the Universidad de Granada.





Steen Markvorsen is a professor at the Technical University of Denmark.





Maung Min-Oo is emeritus professor at the McMaster University.





Vicente Palmer is a professor of Geometry at the University Jaume I.
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