This established reference work continues to lead its readers to some of the hottest topics of contemporary mathematical research. Besides several smaller additions, reorganizations, corrections, and a systematic bibliography, the main new features of the 4th edition are a systematic introduction to Kahler geometry and the presentation of additional techniques from geometric analysis. From the reviews: "This book provides a very readable introduction to Riemannian geometry and geometric analysis. The author focuses on using analytic methods in the study of some fundamental theorems in Riemannian geometry, e.g., the Hodge theorem, the Rauch comparison theorem, the Lyusternik and Fet theorem and the existence of harmonic mappings. With the vast development of the mathematical subject of geometric analysis, the present textbook is most welcome. [..] The book is made more interesting by the perspectives in various sections." - "Math Reviews".
Table of Contents
Fundamental Material * De Rham Cohomology and Harmonic Differential Forms * Parallel Transport, Connections, and Covariant Derivatives * Geodesics and Jacobi Fields * A Short Survey on Curvature and Topology: Symmetric Spaces and Kahler Manifolds * Morse Theory and Floer Homology * Variational Problems from Quantum Field Theory * Harmonic Maps * Appendix A: Linear Elliptic Partial Differential Equations * Appendix B: Fundamental Groups and Covering Spaces * Bibliography * Index.