Riemannian Geometry

This text has been adopted at:University of Pennsylvania, PhiladelphiaUniversity of Connecticut, StorrsDuke University, Durham, NCCalifornia Institute of Technology, PasadenaUniversity of Washington, SeattleSwarthmore College, Swarthmore, PA University of Chicago, ILUniversity of Michigan, Ann Arbor"In the reviewer's opinion, this is a superb book which makes learning a real pleasure."Revue Romaine de Mathematiques Pures et Appliquees"This main-stream presentation of differential geometry serves well for a course on Riemannian geometry, and it is complemented by many annotated exercises." Monatshefte F. Mathematik"This is one of the best (if even not just the best) book for those who want to get a good, smooth and quick, but yet thorough introduction to modern Riemannian geometry."Publicationes MathematicaeContents: Differential Manifolds * Riemannian Metrics * Affine Connections; Riemannian Connections * Geodesics; Convex Neighborhoods * Curvature * Jacobi Fields * Isometric Immersions * Complete Manifolds; Hopf-Rinow and Hadamard Theorems * Spaces of Constant Curvature * Variations of Energy * The Rauch Comparison Theorem * The Morse Index Theorem * The Fundamental Group of Manifolds of Negative Curvature * The Sphere Theorem * Index Series: Mathematics: Theory and Applications

A translation (from the Portuguese) of the second edition of Geometria riemanniana (Instituto de Matematica Pura e Aplicada, Rio de Janeiro, Brazil, 1988), a text on the basic language of and some fundamental theorems in Riemannian geometry. Includes a substantial number of exercises.
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Riemannian Geometry is an expanded edition of a highly acclaimed and successful textbook (originally published in Portuguese)

Preface to the 1st edition * Preface to the 2nd edition * Preface to the English edition * How to use this book * 0. Differentiable Manifolds * 1. Riemannian Metrics * 2. Affine Connections; Riemannian Connections * 3. Geodesics; Convex Neighborhoods * 4. Curvature * 5. Jacobi Fields * 6. Isometric Immersions * 7. Complete Manifolds; Hopf-Rinow and Hadamard Theorems * 8. Spaces of Constant Curvature * 9. Variations of Energy * 10. The Rauch Comparison Theorem * 11. The Morse Index Theorem * 12. The Fundamental Group of Manifolds of Negative Curvature * 13. The Sphere Theorem * References * Index