A Panoramic View of Riemannian Geometry

Cover
Copyright
Preface
Contents
1 Old and New Euclidean Geometry and Analysis
1.1 Preliminaries
1.2 Distance Geometry
1.2.1 A Basic Formula
1.2.2 The Length of a Path
1.2.3 The First Variation Formula and Application to Billiards
1.3 Geometry of Plane Curves and Two Dimensional Point Kinematics
1.3.1 Length
1.3.2 Curvature
1.4 Global Theory of Closed Plane Curves
1.4.1 “Obvious” Truths About Curves Which are Hard to Prove
1.4.2 The Four Vertex Theorem
1.4.3 Convexity with Respect to Arc Length
1.4.4 Umlaufsatz with Corners
1.4.5 Heat Shrinking of Plane Curves
1.4.6 Arnold’s Revolution in Plane Curve Theory
1.5 The Isoperimetric Inequality for Curves
1.6 The Geometry of Surfaces Before and After Gauß
1.6.1 Inner Geometry: a First Attempt
Note 1.6.1.1 (Space forms)
1.6.2 Looking for Shortest Curves: Geodesics
1.6.3 The Second Fundamental Form and Principal Curvatures
1.6.4 The Meaning of the Sign of K
1.6.5 Global Surface Geometry
Note 1.6.5.1
Note 1.6.5.2
Note 1.6.5.3
1.6.6 Minimal Surfaces
Note 1.6.6.1
1.6.7 The Hartman-Nirenberg Theorem for Inner Flat Surfaces
1.6.8 The Isoperimetric Inequality in E^3 à la Gromov
Note 1.6.8.1
1.6.8.1 Notes
1.7 Generic Surfaces
1.8 Heat and Wave Analysis in E^2
1.8.1 Classical equations of physics for a plane domain
1.8.1.1 Bibliographical Note
1.8.2 Why the Eigenvalue Problem?
1.8.3 The First Way: the Minimax Technique
1.8.4 Direct and Inverse Problems: Can One Hear the Shape of a Drum?
1.8.4.1 A Few Direct Problems
1.8.4.2 The Faber–Krahn Inequality
1.8.4.3 Inverse Problems
1.8.5 Second Way: the Heat Equation
1.8.5.1 Eigenfunctions
1.8.6 Relations Between the Two Spectra
1.9 Heat and Waves in E^3, E^d and on the Sphere
1.9.1 Euclidean Spaces
1.9.2 Spheres
1.9.3 Billiards in Higher Dimensions
1.9.4 The Wave Equation Versus the Heat Equation
2 Transition: The Need for a More General Framework
3 Surfaces from Gauß to Today
3.1 Gauß
3.1.1 Theorema Egregium
3.1.1.1 The First Proof of Gauß’s Theorema Egregium; the Concept of ds^2
Note 3.1.1.1
Note 3.1.1.2
3.1.1.2 Second Proof of the Theorema Egregium
3.1.2 The Gauß-Bonnet Formula and the Rodrigues-Gauß Map
3.1.3 Another Look at the Gauß-Bonnet Formula: Parallel Transport
3.1.4 Inner Geometry
3.2 Alexandrov’s Theorems and Gauß’s Angle Correction
Note 3.2.0.1
3.2.1 Angle Corrections of Legendre and Gauß in Geodesy
Note 3.2.1.1
3.3 Back to Metric Questions: Cut Loci and Injectivity Radius
Note 3.3.0.2
3.4 Global Results for Surfaces in E^3
3.4.1 Bending Surfaces
3.4.1.1 Bending Polyhedra
3.4.1.2 Bending and Wrinkling with Little Smoothness
3.4.2 Mean Curvature Rigidity of the Sphere
3.4.3 Negatively Curved Surfaces
3.4.4 The Willmore Conjecture
3.4.5 The Global Gauß–Bonnet Theorem for Surfaces
Note 3.4.5.1
Note 3.4.5.2
Note 3.4.5.3
3.4.6 The Hopf Index Formula
4 Riemann’s Blueprints for Architecture in Myriad Dimensions
4.1 Smooth Manifolds
4.1.1 Introduction
Note 4.1.1.1
4.1.2 The Need for Abstract Manifolds
4.1.3 Examples
4.1.3.1 Submanifolds
4.1.3.2 Products
4.1.3.3 Lie Groups
4.1.3.4 Homogeneous Spaces
4.1.3.5 Grassmannians over Various Algebras
4.1.3.6 Gluing
Note 4.1.3.1
4.1.4 The Classification of Manifolds
4.1.4.1 Surfaces
Note 4.1.4.1 (Riemann surfaces)
4.1.4.2 Higher Dimensions
4.1.4.3 Embedding Manifolds in Euclidean Space
4.2 Calculus on Manifolds
4.2.1 Tangent Spaces and the Tangent Bundle
4.2.2 Differential Forms and Exterior Calculus
Note 4.2.2.1
Note 4.2.2.2
4.3 Examples of Riemann’s Definition of a Geometric Object
4.3.1 Riemann’s Definition
4.3.2 The Most Famous Example: Hyperbolic Geometry
Note 4.3.2.1
Note 4.3.2.2
Note 4.3.2.3 (Space forms again)
Note 4.3.2.4
4.3.3 Products, Coverings and Quotients
4.3.3.1 Products
4.3.3.2 Coverings
4.3.4 Homogeneous Spaces
4.3.5 Symmetric Spaces
4.3.5.1 Classification
4.3.5.2 Rank
4.3.6 Riemannian Submersions
Note 4.3.6.1 (Warped products)
4.3.7 Gluing and Surgery
4.3.7.1 Gluing of Hyperbolic Surfaces
4.3.7.2 Higher Dimensional Gluing
4.3.8 Classical Mechanics
4.4 The Riemann Curvature Tensor
4.4.1 Discovery and Definition
4.4.2 The Sectional Curvature
4.4.3 Curvature of Some Standard Examples
Note 4.4.3.1
4.4.3.1 Constant Sectional Curvature
4.4.3.2 Projective Spaces KP^n
4.4.3.3 Products
4.4.3.4 Homogeneous Spaces
Note 4.4.3.2
4.4.3.5 Hypersurfaces in Euclidean Space
4.5 A Naive Question: Does the Curvature Determine the Metric?
4.5.1 Surfaces
4.5.2 Any Dimension
4.6 What are Abstract Riemannian Manifolds?
4.6.1 Isometrically Embedding Surfaces in E^3
4.6.2 Local Isometric Embedding of Surfaces in E^3
4.6.3 Isometric Embedding in Higher Dimensions
5 A One Page Panorama
6 Riemannian Manifolds as Metric Spaces and the Geometric Meaning of Sectional and Ricci Curvature
6.1 First Metric Properties
6.1.1 Local Properties and the First Variation Formula
6.1.2 Hopf–Rinow and de Rham Theorems
6.1.2.1 Products
6.1.3 Convexity and Small Balls
Note 6.1.3.1
6.1.4 Totally Geodesic Submanifolds
6.1.5 The Center of Mass
6.1.6 Explicit Calculation of Geodesics of Certain Riemannian Manifolds
6.1.7 Transition
6.2 The First Technical Tools: Parallel Transport, Second Variation, and First Appearance of the Ricci Curvature
Note 6.2.0.1
6.3 The Second Technical Tools: The Equation for Jacobi Vector Fields
6.3.1 The Exponential Map and its Derivative: The Philosophy of Élie Cartan
6.3.1.1 Rank
6.3.2 Spaces of Constant Sectional Curvature: Space Forms
Note 6.3.2.1
6.3.3 Nonpositive Curvature: The von Mangoldt–Hadamard–Cartan Theorem
6.4 Triangle Comparison Theorems
6.4.1 Using Upper and Lower Bounds on Sectional Curvature
Note 6.4.1.1
Note 6.4.1.2
6.4.2 Using only a Lower Bound on Ricci Curvature
Note 6.4.2.1 (Busemann functions)
Note 6.4.2.2
6.4.3 Philosophy Behind These Bounds
6.5 Injectivity, Convexity Radius and Cut Locus
6.5.1 Definition of Cut Points and Injectivity Radius
6.5.2 Klingenberg and Cheeger Theorems
Note 6.5.2.1
Note 6.5.2.2
Note 6.5.2.3
Note 6.5.2.4
Note 6.5.2.5
6.5.3 Convexity Radius
6.5.4 Cut Locus
Note 6.5.4.1 (Poinsot motions)
Note 6.5.4.2
6.5.5 Simple Question Scandalously Unsolved: Blaschke Manifolds
6.6 The Geometric Hierarchy and Generalized Space Forms
6.6.1 The Geometric Hierarchy
6.6.1.1 Space Forms
6.6.1.2 Rank 1 Symmetric Spaces
6.6.1.3 Measure Isotropy
6.6.1.4 Symmetric Spaces
6.6.1.5 Homogeneous Spaces
6.6.2 Space Forms of Type (i): Constant Sectional Curvature
6.6.2.1 Negatively Curved Space Forms in Three and Higher Dimensions
6.6.2.2 Mostow Rigidity
6.6.2.3 Classification of Arithmetic and Nonarithmetic Negatively Curved Space Forms
6.6.2.4 Volumes of Negatively Curved Space Forms
6.6.3 Space Forms of Type (ii): Rank 1 Symmetric Spaces
6.6.4 Space Forms of Type (iii): Higher Rank Symmetric Spaces
6.6.4.1 Superrigidity
6.6.5 Homogeneous Spaces
7 Volumes and Inequalities on Volumes of Cycles
7.1 Curvature Inequalities
7.1.1 Bounds on Volume Elements and First Applications
7.1.1.1 The Canonical Measure and Computing it with Jacobi Fields
Note 7.1.1.1
7.1.1.2 Volumes of Standard Spaces
7.1.1.3 The Isoperimetric Inequality for Spheres
Note 7.1.1.2 (On the volumes of balls in spheres)
7.1.1.4 Sectional Curvature Upper Bounds
7.1.1.5 Ricci Curvature Lower Bounds
Note 7.1.1.3
Note 7.1.1.4 (About Milnor’s result)
Note 7.1.1.5
7.1.2 Bounding the Isoperimetric Profile with the Diameter and Ricci Curvature
7.1.2.1 Definition and Examples
7.1.2.2 The Gromov–Bérard–Besson–Gallot Bound
7.1.2.3 Nonpositive Curvature on Noncompact Manifolds
Note 7.1.2.2
7.2 Curvature Free Inequalities on Volumes of Cycles
7.2.1 Systolic Inequalities for Curves in Surfaces
7.2.1.1 Loewner, Pu and Blatter–Bavard Theorems
Note 7.2.1.1
7.2.1.2 Higher Genus Surfaces
Note 7.2.1.2
Note 7.2.1.3
Note 7.2.1.4
7.2.1.3 The Sphere
7.2.1.4 Homological Systoles
7.2.2 Systolic Inequalities for Curves in Higher Dimensional Manifolds
7.2.2.1 The Problem, and Standard Manifolds
7.2.2.2 Filling Volume and Filling Radius
Note 7.2.2.1
7.2.2.3 Gromov’s Theorem and Sketch of the Proof
Note 7.2.2.2
7.2.3 Higher Dimensional Systoles: Systolic Freedom Almost Everywhere
Note 7.2.3.1 (Stable systoles)
Note 7.2.3.2
Note 7.2.3.3
Note 7.2.3.4 (Besicovitch’s results 1952 [181])
7.2.4 Embolic Inequalities
7.2.4.1 Introduction, Questions and Answers
Note 7.2.4.1 (Best metrics)
7.2.4.2 Starting the Proof and Introducing the Unit Tangent Bundle
7.2.4.3 The Core of the Proof
7.2.4.4 Croke’s Three Results
Note 7.2.4.2
7.2.4.5 Infinite Injectivity Radius
7.2.4.6 Using Embolic Inequalities and Local Contractibility
8 Transition: The Next Two Chapters
8.1 Spectral Geometry and Geodesic Dynamics
8.2 Why are Riemannian Manifolds So Important?
8.3 Positive Versus Negative Curvature
9 Riemannian Manifolds as Quantum Mechanical Worlds: The Spectrum and Eigenfunctions of the Laplacian
9.1 History
Note 9.1.0.3 (On the bibliography)
9.2 Motivation
9.3 Setting Up
9.3.1 Xdefinition
9.3.2 The Hodge Star
9.3.3 Facts
Note 9.3.3.1
9.3.4 Heat, Wave and Schrödinger Equations
9.4 The Cheapest (But Most Robust) Method to Obtain Eigenfunctions: The Minimax Principle
9.4.1 The Principle
Note 9.4.1.1
9.4.2 An Application
9.5 Some Extreme Examples
9.5.1 Square Tori, Alias Several Variable Fourier Series
9.5.2 Other Flat Tori
9.5.3 Spheres
9.5.4 KP^n
9.5.5 Other Space Forms
9.6 Current Questions
9.6.1 Direct Questions About the Spectrum
9.6.2 Direct Problems About the Eigenfunctions
9.6.3 Inverse Problems on the Spectrum
9.7 First Tools: The Heat Kernel and Heat Equation
9.7.1 The Main Result
9.7.2 Great Hopes
Note 9.7.2.1 (Spectra of space forms)
Note 9.7.2.2 (Futility of the U_k)
9.7.3 The Heat Kernel and Ricci Curvature
9.8 The Wave Equation: The Gaps
9.9 The Wave Equation: Spectrum and Geodesic Flow
Note 9.9.0.1 (Quasimodes)
Note 9.9.0.2
9.10 The First Eigenvalue
9.10.1 λ1 and Ricci Curvature
9.10.2 Cheeger’s Constant
9.10.3 λ1 and Volume; Surfaces and Multiplicity
9.10.4 Kähler Manifolds
9.11 Results on Eigenfunctions
9.11.1 Distribution of the Eigenfunctions
9.11.2 Volume of the Nodal Hypersurfaces
9.11.3 Distribution of the Nodal Hypersurfaces
9.12 Inverse Problems
9.12.1 The Nature of the Image
Note 9.12.1.1
9.12.2 Inverse Problems: Nonuniqueness
9.12.3 Inverse Problems: Finiteness, Compactness
9.12.4 Uniqueness and Rigidity Results
9.12.4.1 Vignéras Surfaces
9.13 Special Cases
9.13.1 Riemann Surfaces
9.13.2 Space Forms
9.13.2.1 Scars
9.14 The Spectrum of Exterior Differential Forms
10 Riemannian Manifolds as Dynamical Systems: the Geodesic Flow and Periodic Geodesics
10.1 Introduction: Motivation, Problems and Structure of this Chapter
10.2 Some Well Understood Examples
10.2.1 Surfaces of Revolution
10.2.1.1 Zoll Surfaces
10.2.1.2 Weinstein Surfaces
10.2.2 Ellipsoids and Morse Theory
10.2.3 Flat and Other Tori: Influence of the Fundamental Group
10.2.3.1 Flat Tori
10.2.3.2 Manifolds Which are not Simply Connected
10.2.3.3 Tori, not Flat
10.2.4 Space Forms
10.2.4.1 Space Form Surfaces
10.2.4.2 Higher Dimensional Space Forms
10.3 Geodesics Joining Two Points
10.3.1 Birkhoff ’s Proof for the Sphere
10.3.2 Morse Theory
10.3.3 Discoveries of Morse and Serre
10.3.4 Computing with Entropy
10.3.5 Rational Homology and Gromov’s Work
Note 10.3.5.1 (Topology of positive Ricci curvature manifolds)
10.4 Periodic Geodesics
10.4.1 The Difficulties
10.4.2 General Results
10.4.2.1 Gromoll and Meyer
10.4.2.2 Results for the Generic (“Bumpy”) Case
10.4.3 Surfaces
10.4.3.1 The Lusternik–Schnirelmann Theorem
10.4.3.2 The Bangert–Franks–Hingston Results
10.5 The Geodesic Flow
10.5.1 Review of Ergodic Theory of Dynamical Systems
10.5.1.1 Ergodicity and Mixing
10.5.1.2 Notions of Entropy
10.6 Negative Curvature
10.6.1 Distribution of Geodesics
10.6.2 Distribution of Periodic Geodesics
10.7 Nonpositive Curvature
10.8 Entropies on Various Space Forms
10.8.1 Liouville Entropy
10.9 From Osserman to Lohkamp
10.10 Inverse Problems: Manifolds All of Whose Geodesics are Closed
10.10.1 Definitions and Caution
10.10.2 Bott and Samelson Theorems
10.10.3 The Structure on a Given S^d and KP^n
10.11 Inverse Problems: Conjugacy of Geodesic Flows
Note 10.11.0.1 (Different metrics with the same geodesics)
11 What is the Best Riemannian Metric on a Compact Manifold?
11.1 Introduction and a Possible Approach
11.1.1 An Approach
11.2 Purely Geometric Functionals
11.2 Purely Geometric Functionals
11.2.1 Systolic Inequalities
11.2.2 Counting Periodic Geodesics
11.2.3 The Embolic Constant
11.2.4 Diameter and Injectivity
11.3 Which Metric is Less Curved?
11.3.1 Definitions
11.3.1.1 inf ||R||_{L^{d/2}}
11.3.1.2 Minimal Volume
11.3.1.3 Minimal Diameter
Note 11.3.1.1 (The best pinching problem)
11.3.2 The Case of Surfaces
11.3.3 Generalities, Compactness, Finiteness and Equivalence
11.3.4 Manifolds with inf Vol(resp. inf ||R||_{L^{d/2}} , inf diam) = 0
11.3.4.1 Circle Fibrations and Other Examples
11.3.4.2 Allof–Wallach’s Type of Examples
11.3.4.3 Nilmanifolds and the Converse: Almost Flat Manifolds
11.3.4.4 The Examples of Cheeger and Rong
11.3.5 Some Manifolds with inf Vol > 0 and inf ||R||_{L^{d/2}} > 0
11.3.5.1 Using Integral Formulas
11.3.5.2 The Simplicial Volume of Gromov
11.3.6 inf ||R||_{L^{d/2}} in Four Dimensions
11.3.7 Summing up Questions on inf Vol, inf ||R||_{L^{d/2}}
11.4 Einstein Manifolds
11.4.1 Hilbert’s Variational Principle and Great Hopes
Note 11.4.1.1 (Connes’ description of the Hilbert functional)
11.4.2 The Examples from the Geometric Hierarchy
11.4.2.1 Symmetric Spaces
11.4.2.2 Homogeneous Spaces and Others
11.4.3 Examples from Analysis: Evolution by Ricci Flow
11.4.4 Examples from Analysis: Kähler Manifolds
11.4.5 The Sporadic Examples
Fact 281
11.4.6 Around Existence and Uniqueness
11.4.6.1 Existence
11.4.6.2 Uniqueness
11.4.6.3 Moduli
11.4.6.4 The Set of Constants, Ricci Flat Metrics
11.4.7 The Yamabe Problem
11.5 The Bewildering Fractal Landscape of RS (M) According to Nabutovsky
Note 11.5.0.1
Note 11.5.0.2 (Low dimensions)
12 From Curvature to Topology
12.1 Some History, and Structure of the Chapter
12.1.1 Hopf ’s Inspiration
12.1.2 Hierarchy of Curvatures
12.1.2.1 Control via Curvature
12.1.2.2 Other Curvatures
12.1.2.3 The Problem of Rough Classification
12.1.2.4 References on the Topic, and the Significance of Noncompact Manifolds
12.2 Pinching Problems
12.2.1 Introduction
12.2.2 Positive Pinching
12.2.2.1 The Sphere Theorem
12.2.2.2 Sphere Theorems Invoking Bounds on Other Invariants
12.2.2.3 Homeomorphic Pinching
12.2.2.4 The Sphere Theorem with Lower Bound on Diameter, and no Upper Bound on Curvature
12.2.2.5 Topology at the Diameter Pinching Limit
12.2.2.6 Pointwise Pinching
12.2.2.7 Cutting Down the Hypotheses
12.2.3 Pinching Near Zero
12.2.4 Negative Pinching
Note 12.2.4.1 (The philosophy of negative curvature)
Note 12.2.4.2 (Most geometries are negatively curved)
12.2.5 Ricci Curvature Pinching
12.3 Curvature of Fixed Sign
12.3.1 The Positive Side: Sectional Curvature
12.3.1.1 The Known Examples
12.3.1.1.1 Positive Curvature
12.3.1.1.2 Nonnegative Curvature
Note 12.3.1.1 (Hopf ’s questions)
12.3.1.2 Homology Type and the Fundamental Group
12.3.1.2.1 Homology Type
12.3.1.2.2 Fundamental group
12.3.1.3 The Noncompact Case
12.3.1.4 Positivity of the Curvature Operator
12.3.1.4.1 Positive Curvature Operator
12.3.1.4.2 Nonnegative curvature operator
12.3.1.5 Possible Approaches, Looking to the Future
12.3.1.5.1 Rong’s Work on Pinching
12.3.1.5.2 Widths
Note 12.3.1.2 (Rational homotopy)
12.3.1.5.3 The best pinched metric
Note 12.3.1.3 (Moduli)
12.3.2 Ricci Curvature: Positive, Negative and Just Below
Note 12.3.2.1 (Three Dimensions)
Note 12.3.2.2 (Positive Ricci Versus Positive Sectional Curvature)
Note 12.3.2.3 (Simply connected compact positive Ricci curvature manifolds)
12.3.2.0.4 Just below zero Ricci curvature
Note 12.3.2.4 (Structure at infinity)
Note 12.3.2.5 (Thin triangles)
12.3.3 The Positive Side: Scalar Curvature
12.3.3.1 The Hypersurfaces of Schoen & Yau
12.3.3.2 Geometrical Descriptions
Note 12.3.3.1 (Volumes of Balls and Divergence of Geodesics)
12.3.3.3 Gromov’s Quantization of K-theory and Topological Implications of Positive Scalar Curvature
12.3.3.4 Trichotomy
12.3.3.5 The Proof
12.3.3.6 The Gromov–Lawson Torus Theorem
12.3.4 The Negative Side: Sectional Curvature
12.3.4.1 Introduction
12.3.4.2 Literature
12.3.4.3 Quasi-isometries
12.3.4.3.1 The fundamental groups
12.3.4.3.2 Tools
12.3.4.4 Volume and Fundamental Group
12.3.4.4.1 Volumes
12.3.4.4.2 Fundamental Groups
12.3.4.5 Negative Versus Nonpositive Curvature
12.3.4.5.1 Sectional Curvature Just Above Zero
12.3.5 The Negative Side: Ricci Curvature
12.4 Finiteness, Compactness, Collapsing and the Space of Riemannian Metrics
12.4.1 Finiteness
12.4.1.1 Cheeger’s Finiteness Theorems
12.4.1.1.1 Improving Cheeger’s Proof
12.4.1.2 More Finiteness Theorems
12.4.1.2.1 Homeomorphism Finiteness
12.4.1.2.2 Integral Curvature Bounds
12.4.1.3 Ricci Curvature
Note 12.4.1.1 (Scalar Curvature and Finiteness)
12.4.2 Compactness and Convergence
12.4.2.1 Motivation
12.4.2.2 History
12.4.2.3 Contemporary Definitions and Results
12.4.2.3.1 The Lipschitz Topology
12.4.2.3.2 Compactness with Ricci curvature bounds
Note 12.4.2.1 (Noncompact manifolds)
12.4.3 Collapsing and the Space of Riemannian Metrics
12.4.3.1 Collapsing
12.4.3.1.1 Dropping the volume constraint
12.4.3.1.2
12.4.3.1.3 Collapsing down to a compact manifold
12.4.3.1.4 Alexandrov Spaces
12.4.3.1.5 Ricci Curvature Control
12.4.3.1.6 Integral curvature bounds
12.4.3.2 Closures on a Compact Manifold
13 Holonomy Groups and Kähler Manifolds
13.1 Definitions and Philosophy
13.2 Examples
13.3 General Structure Theorems
13.4 Classification
13.5 The Rare Cases
13.5.1 G2 and Spin(7)
13.5.2 Quaternionic Kähler Manifolds
13.5.2.1 The Bérard Bergery/Salamon Twistor Space of Quaternionic Kähler Manifolds
13.5.2.2 The Konishi Twistor Space of a Quaternionic Kähler Manifold
13.5.2.3 Other Twistor Spaces
Note 13.5.2.1 (Historical mistake)
13.5.3 Ricci Flat Kähler and Hyper-Kähler Manifolds
13.5.3.1 Hyperkähler Manifolds
Note 13.5.3.1
13.6 Kähler Manifolds
13.6.1 Symplectic Structures on Kähler Manifolds
13.6.2 Imitating Complex Algebraic Geometry on Kähler Manifolds
14 Some Other Important Topics
14.1 Noncompact Manifolds
14.1.1 Noncompact Manifolds of Nonnegative Ricci Curvature
14.1.2 Finite Volume
14.1.3 Bounded Geometry
14.1.4 Harmonic Functions
14.1.5 Structure at Infinity
14.1.6 Chopping
14.1.7 Positive Mass
14.1.8 Cohomology and Homology Theories
14.2 Bundles over Riemannian Manifolds
14.2.1 Differential Forms and Related Bundles
14.2.1.1 The Hodge Star
14.2.1.2 A Variational Problem for Differential Forms and the Laplace Operator
Note 14.2.1.1 (Intrinsically harmonic forms)
14.2.1.3 Calibration
14.2.1.4 Harmonic Analysis of Other Tensors
14.2.2 Spinors
14.2.2.1 Algebra of Spinors
14.2.2.2 Spinors on Riemannian Manifolds
14.2.2.3 History of Spinors
14.2.2.4 Applying Spinors
14.2.2.5 Warning: Beware of Harmonic Spinors
14.2.2.6 The Half Pontryagin Class
14.2.2.7 Reconstructing the Metric from the Dirac Operator
14.2.2.8 Spin^c Structures
14.2.3 Various Other Bundles
14.2.3.1 Secondary Characteristic Classes
14.2.3.2 Yang–Mills Theory
14.2.3.3 Twistor Theory
14.2.3.4 K-theory
14.2.3.5 The Atiyah–Singer Index Theorem
14.2.3.6 Supersymmetry and Supergeometry
14.3 Harmonic Maps Between Riemannian Manifolds
14.4 Low Dimensional Riemannian Geometry
14.5 Some Generalizations of Riemannian Geometry
14.5.1 Boundaries
14.5.2 Orbifolds
14.5.3 Conical Singularities
14.5.4 Spectra of Singular Spaces
14.5.5 Alexandrov Spaces
14.5.6 CAT Spaces
14.5.6.1 The CAT (k) Condition
14.5.7 Carnot–Carathéodory Spaces
14.5.7.1 Example: the Heisenberg Group
14.5.8 Finsler Geometry
14.5.9 Riemannian Foliations
14.5.10 Pseudo-Riemannian Manifolds
14.5.11 Infinite Dimensional Riemannian Geometry
14.5.12 Noncommutative Geometry
14.6 Gromov’s mm Spaces
14.7 Submanifolds
14.7.1 Higher Dimensions
14.7.2 Geometric Measure Theory and Pseudoholomorphic Curves
15 The Technical Chapter
15.1 Vector Fields and Tensors
15.2 Tensors Dual via the Metric: Index Aerobics
15.3 The Connection, Covariant Derivative and Curvature
15.4 Parallel Transport
15.4.1 Curvature from Parallel Transport
15.5 Absolute (Ricci) Calculus and Commutation Formulas: Index Gymnastics
Note 15.5.0.1 (Riemannian invariants)
15.6 Hodge and the Laplacian, Bochner’s Technique
15.6.1 Bochner’s Technique for Higher Degree Differential Forms
15.7 Generalizing Gauß–Bonnet, Characteristic Classes and Chern’s Formulas
15.7.1 Chern’s Proof of Gauß–Bonnet for Surfaces
15.7.2 The Proof of Allendoerfer and Weil
15.7.3 Chern’s Proof in all Even Dimensions
15.7.4 Chern Classes of Vector Bundles
15.7.5 Pontryagin Classes
15.7.6 The Euler Class
15.7.7 The Absence of Other Characteristic Classes
15.7.8 Applying Characteristic Classes
15.7.9 Characteristic Numbers
15.8 Two Examples of Riemannian Manifolds and Calculation of their Curvatures
15.8.1 Homogeneous Spaces
15.8.2 Riemannian Submersions
References
Acknowledgements
List of Notation
List of Authors
Subject Index