This book covers the topics of differential manifolds, Riemannian metrics, connections, geodesics and curvature, with special emphasis on the intrinsic features of the subject. It treats in detail classical results on the relations between curvature and topology. The book features numerous exercises with full solutions and a series of detailed examples are picked up repeatedly to illustrate each new definition or property introduced.

Inhalt

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.- Connexions.- Canonical connexion 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's theorem.- Geodesics 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 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 manifolds.- D. Riemannian Submersions and Curvature.- Riemannian submersions and connexions.- 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 Cartan theorems.- H. Curvature and Volume.- Densities on a differential 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.- Traditional point of view: pinched manifolds.- Almost flat pinching.- Coarse point of view: compactness theorems of Gromov and Cheeger.- K. Curvature and Representations of the Orthogonal Group.- Decomposition of the space of curvature tensors.- Conformally flat manifolds.- The second Bianchi identity.- Chapitre 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.- V. Riemannian Submanifolds.