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'

From the reviews of the third edition:

"This new edition maintains the clear written style of the original, including many illustrations ... examples and exercises (most with solutions)." (Joseph E. Borzellino, Mathematical Reviews, 2005)

"This book based on graduate course on Riemannian geometry ... covers the topics of differential manifolds, Riemannian metrics, connections, geodesics and curvature, with special emphasis on the intrinsic features of the subject. Classical results ... are treated in detail. ... contains numerous exercises with full solutions and a series of detailed examples which are picked up repeatedly to illustrate each new definition or property introduced. For this third edition, some topics ... have been added and worked out in the same spirit." (L'ENSEIGNEMENT MATHEMATIQUE, Vol. 50, (3-4), 2004)

"This book is based on a graduate course on Riemannian geometry and analysis on manifolds that was held in Paris. ... Classical results on the relations between curvature and topology are treated in detail. The book is almost self-contained, assuming in general only basic calculus. It contains nontrivial exercises with full solutions at the end. Properties are always illustrated by many detailed examples." (EMS Newsletter, December 2005)

"The guiding line of this by now classic introduction to Riemannian geometry is an in-depth study of each newly introduced concept on the basis of a number of reoccurring well-chosen examples ... . The book continues to be an excellent choice for an introduction to the central ideas of Riemannian geometry." (M. Kunzinger, Monatshefte für Mathematik, Vol. 147 (1), 2006)