Lectures on the Geometry of Manifolds 3rd Edition by Liviu Nicolaescu 9789811215957 9811215952

1. Manifolds

1.1 Preliminaries

1.1.1 Space and Coordinatization

1.1.2 The implicit functiontheorem

1.2 Smoothmanifolds

1.2.1 Basic definitions

1.2.2 Partitions of unity

1.2.3 Examples

1.2.4 How many manifolds are there?

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 products

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 fields

2.3.1 Operations with vector bundles

2.3.2 Tensor fields

2.3.3 Fiber bundles

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 Orientability 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

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 property 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 Cartan’s moving frame method

4.2.4 The geometry of submanifolds

4.2.5 Correlators and their geometry

4.2.6 The Gauss–Bonnet theorem for oriented surfaces

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

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 property

6.2.3 Homotopy lifting property

6.2.4 On the existence of lifts

6.2.5 The universal cover and the fundamental group

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 Kunneth 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 a complex Grassmannian

7.5 Čech cohomology

7.5.1 Sheaves and presheaves

7.5.2 Čech cohomology

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 Tn

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

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 submanifolds of an Euclidean space

9.3.5 Curvature measures of domains in an Euclidean space

9.3.6 Crofton formulae for domains of an Euclidean space

9.3.7 Crofton formulae for submanifolds of an Euclidean space

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 RN

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 RN

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

11. Spectral Geometry

11.1 Generalized functions and currents

11.1.1 Generalized functions and operations withy them

11.1.2 Currents

11.1.3 Temperated is tributions 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

12. Dirac Operators

12.1 The structure of Dirac operators

12.1.1 Basic 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 spinc Dirac operator