Lectures on the Geometry of Manifolds

✍ Scribed by Liviu I. Nicolaescu

• Publisher: World Scientific Publishing Company
• Year: 2020
• Tongue: English
• Leaves: 701
• Edition: 3
• Category: Library

✦ Synopsis

The goal of this book is to introduce the reader to some of the main techniques, ideas and concepts frequently used in modern geometry. It starts from scratch and it covers basic topics such as differential and integral calculus on manifolds, connections on vector bundles and their curvatures, basic Riemannian geometry, calculus of variations, DeRham cohomology, integral geometry (tube and Crofton formulas), characteristic classes, elliptic equations on manifolds and Dirac operators. The new edition contains a new chapter on spectral geometry presenting recent results which appear here for the first time in printed form.

✦ Table of Contents

Preface
Preface to the second edition
Preface to the third edition
Contents
Chapter 1. Manifolds
1.1 Preliminaries
1.1.1 Space and Coordinatization
1.1.2 The implicit function theorem
1.2 Smooth manifolds
1.2.1 Basic definitions
1.2.2 Partitions of unity
1.2.3 Examples
1.2.4 How many manifolds are there?
Chapter 2. Natural Constructions on Manifolds
2.1 The tangent bundle
2.1.1 Tangent spaces
2.1.2 The tangent bundle
2.1.3 Transversality
2.1.4 Vector bundles
2.1.5 Some examples of vector bundles
2.2 A linear algebra interlude
2.2.1 Tensor produets
2.2.2 Symmetric and skew-symmetric tensors
2.2.3 The "super" slang
2.2.4 Duality
2.2.5 Some complex linear algebra
2.3 Tensor Iields
2.3.1 Operations with vector bundles
2.3.2 Tensor fields
2.3.3 Fiber bundles
Chapter 3. Calculus on Manifolds
3.1 The Lie derivative
3.1.1 Flows on manifolds
3.1.2 The Lie derivative
3.1.3 Examples
3.2 Derivations of Ω^●(M)
3.2.1 The exterior derivative
3.2.2 Examples
3.3 Connections on vector bundles
3.3.1 Covariant derivatives
3.3.2 Parallel transport
3.3.3 The curvature of a connection
3.3.4 Holonomy
3.3.5 The Bianchi identities
3.3.6 Connections on tangent bundles
3.4 Integration on manifolds
3.4.1 Integration of 1-densities
3.4.2 Orientabilily and integration of differential forms
3.4.3 Stokes' formula
3.4.4 Representations and characters of compact Lie groups
3.4.5 Fibered calculus
Chapter 4. Riemannian Geometry
4.1 Metric properties
4.1.1 Definitions and examples
4.1.2 The Levi-Civita connection
4.1.3 The exponential map and normal coordinates
4.1.4 The length minimizing properly of geodesics
4.1.5 Calculus on Riemann manifolds
4.2 The Riemann curvature
4.2.1 Definitions and properties
4.2.2 Examples
4.2.3 Carton's moving frame method
4.2.4 The geometry of submanfolds
4.2.5 Correlators and their geometry
4.2.6 The Gauss-Bonnet theorem for oriented surfaces
Chapter 5. Elements of the Calculus of Variations
5.1 The least action principle
5.1.1 The 1-dimensional Euler-Lagrange equations
5.1.2 Noether's conservation principle
5.2 The variational theory of geodesics
5.2.1 Variational formulae
5.2.2 Jacobi fields
5.2.3 The Hamilton-Jacobi equations
Chapter 6. The Fundamental Group and Covering Spaces
6.1 The fundamental group
6.1.1 Basic notions
6.1.2 Of categories and functors
6.2 Covering Spaces
6.2.1 Definitions and examples
6.2.2 Unique lifting properly
6.2.3 Homotopy lifting properly
6.2.4 On the existence of lifts
6.2.5 The universal cover and the fundamental group
Chapter 7. Cohomology
7.1 DeRham cohomology
7.1.1 Speculations around the Poincaré lemma
7.1.2 Čech vs. DeRham
7.1.3 Very little homological algebra
7.1.4 Functorial properties of the DeRham cohomology
7.1.5 Some simple examples
7.1.6 The Mayer-Vietoris principle
7.1.7 The Künneth formula
7.2 The Poincaré duality
7.2.1 Cohomology with compact supports
7.2.2 The Poincaré duality
7.3 Intersection theory
7.3.1 Cycles and their duals
7.3.2 Intersection theory
7.3.3 The topological degree
7.3.4 The Thom isomorphism theorem
7.3.5 Gauss-Bonnet revisited
7.4 Symmetry and topology
7.4.1 Symmetric spaces
7.4.2 Symmetry and cohomology
7.4.3 The cohomology of compact Lie groups
7.4.4 Invariant forms on Grassmannians and Weyl's integral formula
7.4.5 The Poincaré polynomial of complex Grassmannian
7.5 Čech cohomology
7.5.1 Sheaves and presheaves
7.5.2 Čech cohomology
Chapter 8. Characteristic Classes
8.1 Chern-Weil theory
8.1.1 Connections in principal G-bundles
8.1.2 G-vector bundles
8.1.3 Invariant polynomials
8.1.4 The Chern-Weil Theory
8.2 Important examples
8.2.1 The invariants of the torus T^n
8.2.2 Chern classes
8.2.3 Pontryagin classes
8.2.4 The Euler class
8.2.5 Universal classes
8.3 Computing characteristic classes
8.3.1 Reductions
8.3.2 The Gauss-Bonnet-Chern theorem
Chapter 9. Classical Integral Geometry
9.1 The integral geometry of real Grassmannians
9.1.1 Co-area formulae
9.1.2 Invariant measures on linear Grassmannians
9.1.3 Affine Grassmannians
9.2 Gauss-Bonnet again?!?
9.2.1 The shape operator and the second fundamental form
9.2.2 The Gauss-Bonnet theorem for hypersurfaces of an Euclidean space
9.2.3 Gauss-Bonnet theorem for domains in an Euclidean space
9.3 Curvature measures
9.3.1 Tame geometry
9.3.2 Invariants of the orthogonal group
9.3.3 The tube formula and curvature measures
9.3.4 Tube formula → Gauss-Bonnet formula for arbitrary submamfolds of an Euclidean space
9.3.5 Curvature measures of domains in an Euclidean space
9.3.6 Crofton formula for domains of an Euclidean space
9.3.7 Crofton formulae for submanifolds of an Euclidean space
Chapter 10. Elliptic Equations on Manifolds
10.1 Partial differential operators: algebraic aspects
10.1.1 Basic notions
10.1.2 Examples
10.1.3 Formal adjoints
10.2 Functional framework
10.2.1 Sobolev spaces in R^N
10.2.2 Embedding theorems: integrability properties
10.2.3 Embedding theorems: differentiability properties
10.2.4 Functional spaces on manifolds
10.3 Elliptic partial differential operators: analytic aspects
10.3.1 Elliptic estimates in R^N
10.3.2 Elliptic regularity
10.3.3 An application: prescribing the curvature of surfaces
10.4 Elliptic operators on compact manifolds
10.4.1 Fredholm theory
10.4.2 Spectral theory
10.4.3 Hodge theory
Chapter 11. Spectral Geometry
11.1 Generalized functions and currents
11.1.1 Generalizedfunetions and operations withy them
11.1.2 Currents
11.1.3 Temperate distnlbutions and the Fourier transform
11.1.4 Linear differential equations with distributional data
11.2 Important families of generalized functions
11.2.1 Some classical generalized functions on the real axis
11.2.2 Homogeneous generalized functions
11.3 The wave equation
11.3.1 Fundamental solutions of the wave
11.3.2 The wave family
11.3.3 Local parametrices for the wave equation with variable coefficients
11.4 Spectral geometry
11.4.1 The spectral function of the Laplacian on a compact manifold
11.4.2 Short time asymptotics for the wave kernel
11.4.3 Spectral function asymptotics
11.4.4 Spectral estimates of smoothing operators
11.4.5 Spectral perestroika
Chapter 12. Dirac Operators
12.1 The structure of Dirac operators
12.1.1 Basie definitions and examples
12.1.2 Clifford algebras
12.1.3 Clifford modules: the even case
12.1.4 Clifford modules: the odd case
12.1.5 A look ahead
12.1.6 The spin group
12.1.7 The complex spin group
12.1.8 Low dimensional examples
12.1.9 Dirac bundles
12.2 Fundamental examples
12.2.1 The Hodge-DeRham operator
12.2.2 The Hodge-Dolbeault operator
12.2.3 The spin Dirac operator
12.2.4 The spin^c Dirac operator
Bibliography
1-21
22-49
50-79
80-107
108-137
Index