Riemannian Geometry

• 1 Differential manifolds.- 1.A From submanifolds to abstract manifolds.- 1.B The tangent bundle.- 1.C Vector fields.- 1.D Baby Lie groups.- 1.E Covering maps and fibrations.- 1.F Tensors.- 1.G. Differential forms.- 1.H Partitions of unity.- 2 Riemannian metrics.- 2.A Existence theorems and first examples.- 2.B Covariant derivative.- 2.C Geodesies.- 2.D A glance at pseudo-Riemannian manifolds.- 3 Curvature.- 3.A. The curvature tensor.- 3.B. First and second variation.- 3.C. Jacobi vector fields.- 3.D. Riemannian submersions and curvature.- 3.E. The behavior of length and energy in the neighborhood of a geodesic.- 3.F Manifolds with constant sectional curvature.- 3.G Topology and curvature: two basic results.- 3.H. Curvature and volume.- 3.I. Curvature and growth of the fundamental group.- 3.J. Curvature and topology: some important results.- 3.K. Curvature tensors and representations of the orthogonal group.- 3.L. Hyperbolic geometry.- 3.M. Conformai geometry.- 4 Analysis on manifolds.-4.A. Manifolds with boundary.- 4.B. Bishop inequality.- 4.C. Differential forms and cohomology.- 4.D. Basic spectral geometry.- 4.E. Some examples of spectra.- 4.F The minimax principle.- 4.G Eigenvalues estimates.- 4.H. Paul Levy’s isoperimetric inequality.- 5 Riemannian submanifolds.- 5.A. Curvature of submanifolds.- 5.B Curvature and convexity.- 5.C Minimal surfaces.- A Some extra problems.- B Solutions of exercises.- List of figures.

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 fur Mathematik, Vol. 147 (1), 2006)