Riemannian Geometry

1 Differential manifolds.- 1.A From submanifolds to abstract manifolds.- 1.A.1 Submanifolds of Euclidean spaces.- 1.A.2 Abstract manifolds.- 1.A.3 Smooth maps.- 1.B The tangent bundle.- 1.B.1 Tangent space to a submanifold of Rn+k.- 1.B.2 The manifold of tangent vectors.- 1.B.3 Vector bundles.- 1.B.4 Tangent map.- 1.C Vector fields.- 1.C.1 Definitions.- 1.C.2 Another definition for the tangent space.- 1.C.3 Integral curves and flow of a vector field.- 1.C.4 Image of a vector field by a diffeomorphism.- 1.D Baby Lie groups.- 1.D.1 Definitions.- 1.D.2 Adjoint representation.- 1.E Covering maps and fibrations.- 1.E.1 Covering maps and quotients by a discrete group.- 1.E.2 Submersions and fibrations.- 1.E.3 Homogeneous spaces.- 1.F Tensors.- 1.F.1 Tensor product (a digest).- 1.F.2 Tensor bundles.- 1.F.3 Operations on tensors.- 1.F.4 Lie derivatives.- 1.F.5 Local operators, differential operators.- 1.F.6 A characterization for tensors.- 1.G. Differential forms.- 1.G.1 Definitions.- 1.G.2 Exterior derivative.- 1.G.3 Volume forms.- 1.G.4 Integration on an oriented manifold.- 1.G.5 Haar measure on a Lie group.- 1.H Partitions of unity.- 2 Riemannian metrics.- 2.A Existence theorems and first examples.- 2.A.1 Basic definitions.- 2.A.2 Submanifolds of Euclidean or Minkowski spaces.- 2.A.3 Riemannian submanifolds, Riemannian products.- 2.A.4 Riemannian covering maps, flat tori.- 2.A.5 Riemannian submersions, complex projective space.- 2.A.6 Homogeneous Riemannian spaces.- 2.B Covariant derivative.- 2.B.1 Connections.- 2.B.2 Canonical connection of a Riemannian submanifold.- 2.B.3 Extension of the covariant derivative to tensors.- 2.B.4 Covariant derivative along a curve.- 2.B.5 Parallel transport.- 2.B.6 natural metric on the tangent bundle.- 2.C Geodesies.- 2.C.1 Definition, first examples.- 2.C.2 Local existence and uniqueness for geodesies, exponential map.- 2.C.3 Riemannian manifolds as metric spaces.- 2.C.4 An invitation to isosystolic inequalities.- 2.C.5 Complete Riemannian manifolds, Hopf-Rinow theorem.- 2.C.6 Geodesies and submersions, geodesies of PnC.- 2.C.7 Cut-locus.- 2.C.8 The geodesic flow.- 2.D A glance at pseudo-Riemannian manifolds.- 2.D.1 What remains true?.- 2.D.2 Space, time and light-like curves.- 2.D.3 Lorentzian analogs of Euclidean spaces, spheres and hegeode spaces.- 2.D.4 (In)completeness.- 2.D.5 The Schwarzschild model.- 2.D.6 Hyperbolicity versus ellipticity.- 3 Curvature.- 3.A. The curvature tensor.- 3.A.1 Second covariant derivative.- 3.A.2 Algebraic properties of the curvature tensor.- 3.A.3 Computation of curvature: some examples.- 3.A.4 Ricci curvature, scalar curvature.- 3.B. First and second variation.- 3.B.1 Technical preliminaries.- 3.B.2 First variation formula.- 3.B.3 Second variation formula.- 3.C. Jacobi vector fields.- 3.C.1 Basic topics about second derivatives.- 3.C.2 Index form.- 3.C.3 Jacobi fields and exponential map.- 3.C.4 Applications.- 3.D. Riemannian submersions and curvature.- 3.D.1 Riemannian submersions and connections.- 3.D.2 Jacobi fields of PnC.- 3.D.3 O’Neill’s formula.- 3.D.4 Curvature and length of small circles. Application to Riemannian submersions.- 3.E. The behavior of length and energy in the neighborhood of a geodesic.- 3.E.1 Gauss lemma.- 3.E.2 Conjugate points.- 3.E.3 Some properties of the cut-locus.- 3.F Manifolds with constant sectional curvature.- 3.G Topology and curvature: two basic results.- 3.G.1 Myers’ theorem.- 3.G.2 Cartan-Hadamard’s theorem.- 3.H. Curvature and volume.- 3.H.1 Densities on a differentiable manifold.- 3.H.2 Canonical measure of a Riemannian manifold.- 3.H.3 Examples: spheres, hyperbolic spaces, complex projective spaces.- 3.H.4 Small balls and scalar curvature.- 3.H.5 Volume estimates.- 3.I. Curvature and growth of the fundamental group.- 3.I.1 Growth of finite type groups.- 3.I.2 Growth of the fundamental group of compact manifolds with negative curvature.- 3.J. Curvature and topology: some important results.- 3.J.1 Integral formulas.- 3.J.2 (Geo)metric methods.- 3.J.3 Analytic methods.- 3.J.4 Coarse point of view: compactness theorems.- 3.K. Curvature tensors and representations of the orthogonal group.- 3.K.1 Decomposition of the space of curvature tensors.- 3.K.2 Conformally flat manifolds.- 3.K.3 The Second Bianchi identity.- 3.L. Hyperbolic geometry.- 3.L.1 Introduction.- 3.L.2 Angles and distances in the hyperbolic plane.- 3.L.3 Polygons with “many” right angles.- 3.L.4 Compact surfaces.- 3.L.5 Hyperbolic trigonometry.- 3.L.6 Prescribing constant negative curvature.- 3.L.7 A few words about higher dimension.- 3.M. Conformai geometry.- 3.M.2 Introduction.- 3.M.3 The Möbius group.- 3.M.4 Conformai, elliptic and hyperbolic geometry.- 4 Analysis on manifolds.- 4.A. Manifolds with boundary.- 4.A.1 Introduction.- 4.A.2 Stokes theorem and integration by parts.- 4.B. Bishop inequality.- 4.B.1 Some commutation formulas.- 4.B.2 Laplacian of the distance function.- 4.B.3 Another proof of Bishop’s inequality.- 4.B.4 Heintze-Karcher inequality.- 4.C. Differential forms and cohomology.- 4.C.1 The de Rham complex.- 4.C.2 Differential operators and their formal adjoints.- 4.C.3 The Hodge-de Rham theorem.- 4.C.4 A second visit to the Bochner method.- 4.D. Basic spectral geometry.- 4.D.1 The Laplace operator and the wave equation.- 4.D.2 Statement of basic results on the spectrum.- 4.E. Some examples of spectra.- 4.E.1 Introduction.- 4.E.2 The spectrum of flat tori.- 4.E.3 Spectrum of (Sn, can).- 4.F The minimax principle.- 4.G Eigenvalues estimates.- 4.G.1 Introduction.- 4.G.2 Bishop’s inequality and coarse estimates.- 4.G.3 Some consequences of Bishop’s theorem.- 4.G.4 Lower bounds for the first eigenvalue.- 4.H. Paul Levy’s isoperimetric inequality.- 4.H.1 The statement.- 4.H.2 The proof.- 5 Riemannian submanifolds.- 5.A. Curvature of submanifolds.- 5.A.1 Second fundamental form.- 5.A.2 Curvature of hypersurfaces.- 5.A.3 Application to explicit computations of curvatures.- 5.B Curvature and convexity.- 5.C Minimal surfaces.- 5.C.1 First results.- 5.C.2 Surfaces with constant mean curvature.- A Some extra problems.- B Solutions of exercises.- List of figures.