Global Riemannian Geometry: Curvature and Topology

Description

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. Contents Preface to the first edition, 2003 Preface to the second edition, 2020 Distance Geometric Analysis on Manifolds 1. Appetizer and Introduction 2. The Comparison Setting and Preliminaries 3. Analysis of Riemannian Distance Functions 4. Analysis of Lorentzian Distance Functions 5. Concerning the Riemannian Setting and Notation 6. Green's Formulae and the Co-area Formula 7. The First Dirichlet Eigenvalue Comparison Theorem 8. Isoperimetric Relations 9. A Consequence of the Co-area Formula 10. The Fundamental Differential Equation 11. Isoperimetric Comparison 12. Mean Exit Times and Moment Spectra 13. The Poisson Hierarchy 14. Capacity Comparison 15. The Kelvin–Nevanlinna–Royden Criteria for Transience 16. Surfaces of Revolution 17. Warped Products 18. Answering the Questions in the Appetizer 19. Sufficient Conditions for Parabolicity and Hyperbolicity 20. Hyperbolicity of Spacelike Hypersurfaces 21. Weighted Riemannian Manifolds 22. Weighted Capacities 23. Weighted Rotationally Symmetric Spaces and the Ahlfors Criterion for Weighted Parabolicity 24. Weighted Curvatures 25. Analysis of Restricted Distance Functions in Weighted Submanifolds 26. Extrinsic Criteria for Weighted Parabolicity 27. The Grigor'yan–Fernandez Criterion for Weighted Hyperbolicity 28. Graphs and Flows 29. Scherk's Graph is Transient References The Dirac Operator in Geometry and Physics Foreword Foreword to the Second Edition 1. Spinors and the Dirac Operator 1.1. Introduction to Spinors Examples 1.2. The Dirac Operator 1.3. The Lichnerowicz Formula 2. Gromov's K-Area 2.1. Definition of K-Area 2.2. The Fundamental Estimate in Terms of Scalar Curvature 2.3. Connections with Symplectic Invariants 2.4. The Vafa–Witten Inequality 3. Positive Mass Theorems 3.1. Description of Results 3.2. Main Ideas behind the Proofs 3.3. Some Mathematical Aspects of the AdS/CFT Correspondence 4. Epilogue 4.1. Scalar curvature on the hemisphere 4.2. Gromov's work on scalar curvature 4.3. Some speculations and musings References