Riemannian Geometry

I. Differential Manifolds.- A. From Submanifolds to Abstract Manifolds.- Submanifolds of Rn+k.- Abstract manifolds.- Smooth maps.- B. Tangent Bundle.- Tangent space to a submanifold of Rn+k.- The manifold of tangent vectors.- Vector bundles.- Differential map.- C. Vector Fields.- Definitions.- Another definition for the tangent space.- Integral curves and flow of a vector field.- Image of a vector field under a diffeomorphism.- D. Baby Lie Groups.- Definitions.- Adjoint representation.- E. Covering Maps and Fibrations.- Covering maps and quotient by a discrete group.- Submersions and fibrations.- Homogeneous spaces.- F. Tensors.- Tensor product (digest).- Tensor bundles.- Operations on tensors.- Lie derivatives.- Local operators, differential operators.- A characterization for tensors.- G. Exterior Forms.- Definitions.- Exterior derivative.- Volume forms.- Integration on an oriented manifold.- Haar measure on a Lie group.- H. Appendix: Partitions of Unity.- II. Riemannian Metrics.- A. Existence Theorems and First Examples.- Definitions.- First examples.- Examples: Riemannian submanifolds, product Riemannian manifolds.- Riemannian covering maps, flat tori.- Riemannian submersions, complex projective space.- Homogeneous Riemannian spaces.- B. Covariant Derivative.- Connections.- Canonical connection of a Riemannian submanifold.- Extension of the covariant derivative to tensors.- Covariant derivative along a curve.- Parallel transport.- Examples.- C. Geodesics.- Definitions.- Local existence and uniqueness for geodesics, exponential map.- Riemannian manifolds as metric spaces.- Complete Riemannian manifolds, Hopf-Rinow theorem.- Geodesies and submersions, geodesies of PnC.- Cut locus.- III. Curvature.- A. The Curvature Tensor.- Second covariant derivative.- Algebraic properties of the curvature tensor.- Computation of curvature: some examples.- Ricci curvature, scalar curvature.- B. First and Second Variation of Arc-Length and Energy.- Technical preliminaries: vector fields along parameterized submanifolds.- First variation formula.- Second variation formula.- C. Jacobi Vector Fields.- Basic topics about second derivatives.- Index form.- Jacobi fields and exponential map.- Applications: Sn, Hn, PnR, 2-dimensional Riemannian manifolds.- D. Riemannian Submersions and Curvature.- Riemannian submersions and connections.- Jacobi fields of PnC.- O'Neill's formula.- Curvature and length of small circles. Application to Riemannian submersions.- E. The Behavior of Length and Energy in the Neighborhood of a Geodesic.- The Gauss lemma.- Conjugate points.- Some properties of the cut-locus.- F. Manifolds with Constant Sectional Curvature.- Spheres, Euclidean and hyperbolic spaces.- G. Topology and Curvature.- The Myers and Hadamard-Cartan theorems.- H. Curvature and Volume.- Densities on a differentiable manifold.- Canonical measure of a Riemannian manifold.- Examples: spheres, hyperbolic spaces, complex projective spaces.- Small balls and scalar curvature.- Volume estimates.- I. Curvature and Growth of the Fundamental Group.- Growth of finite type groups.- Growth of the fundamental group of compact manifolds with negative curvature.- J. Curvature and Topology: An Account of Some Old and Recent Results.- Traditional point of view: pinched manifolds.- Almost flat pinching.- Coarse point of view: compactness theorems of Cheeger and Gromov.- K. Curvature Tensors and Representations of the Orthogonal Group.- Decomposition of the space of curvature tensors.- Conformally flat manifolds.- The second Bianchi identity.- L. Hyperbolic Geometry.- Angles and distances in the hyperbolic plane.- Polygons with "many" right angles.- Compact surfaces.- Hyperbolic trigonometry.- Prescribing constant negative curvature.- M. Conformai Geometry.- The Moebius group.- Conformai, elliptic and hyperbolic geometry.- IV. Analysis on Manifolds and the Ricci Curvature.- A. Manifolds with Boundary.- Definition.- The Stokes theorem and integration by parts.- B. Bishop's Inequality Revisited.- Some commutations formulas.- Laplacian of the distance function.- Another proof of Bishop's inequality.- The Heintze-Karcher inequality.- C. Differential Forms and Cohomology.- The de Rham complex.- Differential operators and their formal adjoints.- The Hodge-de Rham theorem.- A second visit to the Bochner method.- D. Basic Spectral Geometry.- The Laplace operator and the wave equation.- Statement of the basic results on the spectrum.- E. Some Examples of Spectra.- The spectrum of flat tori.- Spectrum of (Sn, can).- F. The Minimax Principle.- The basic statements.- G. The Ricci Curvature and Eigenvalues Estimates.- Bishop's inequality and coarse estimates.- Some consequences of Bishop's theorem.- Lower bounds for the first eigenvalue.- H. Paul Levy's Isoperimetric Inequality.- The statement.- The proof.- V. Riemannian Submanifolds.- A. Curvature of Submanifolds.- Second fundamental form.- Curvature of hypersurfaces.- Application to explicit computations of curvatures.- B. Curvature and Convexity.- The Hadamard theorem.- C. Minimal Surfaces.- First results.- Some Extra Problems.- Solutions of Exercises.- I.- II.- III.- IV.- V.