Contents -- Preface -- 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? -- 2. Natural Constructions on Manifolds -- 2.1 The tangent bundle -- 2.1.1 Tangent spaces -- 2.1.2 The tangent bundle -- 2.1.3 Sard's Theorem -- 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 omega(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 di erential 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 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 formulæ -- 5.2.2 Jacobi fields -- 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 Cech 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 Poincare duality -- 7.2.1 Cohomology with compact supports -- 7.2.2 The Poincare 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 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 Cech cohomology -- 7.5.1 Sheaves and presheaves -- 7.5.2 Cech cohomology -- 8. Characteristic Classes -- 8.1 Chern-Weil Theory -- 8.1.1 Connections on 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 formulæ -- 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 of a submanifold in Rn -- 9.2.2 The Gauss-Bonnet theorem for hypersurfaces of a Euclidean space -- 9.2.3 Gauss-Bonnet theorem for domains of a 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 -- 9.3.5 Curvature measures of domains of a Euclidean space -- 9.3.6 Crofton formulæ for domains of a Euclidean space -- 9.3.7 Crofton formula for submanifolds of a Euclidean space -- 10. Elliptic Equations on Manifolds -- 10.1 Partial di erential 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. Dirac Operators -- 11.1 The structure of Dirac operators -- 11.1.1 Basic definitions and examples -- 11.1.2 Clifford algebras.

11.1.3 Clifford modules: the even case -- 11.1.4 Clifford modules: the odd case -- 11.1.5 A look ahead -- 11.1.6 Spin -- 11.1.7 Spinc -- 11.1.8 Low dimensional examples -- 11.1.9 Dirac bundles -- 11.2 Fundamental examples -- 11.2.1 The Hodge-DeRham operator -- 11.2.2 The Hodge-Dolbeault operator -- 11.2.3 The spin Dirac operator -- 11.2.4 The spinc Dirac operator -- Bibliography -- Index.