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This established reference work continues to lead its readers to some of the hottest topics of contemporary mathematical research. The previous edition already introduced and explained the ideas of the parabolic methods that had found a spectacular success in the work of Perelman at the examples of closed geodesics and harmonic forms. It also discussed further examples of geometric variational problems from quantum field theory, another source of profound new ideas and methods in geometry.
The 6^{th} edition includes a systematic treatment of eigenvalues of Riemannian manifolds and several other additions. Also, the entire material has been reorganized in order to improve the coherence of the book.
From the reviews:
"This book provides a very readable introduction to Riemannian geometry and geometric analysis. ... With the vast development of the mathematical subject of geometric analysis, the present textbook is most welcome." Mathematical Reviews
"...the material ... is self-contained. Each chapter ends with a set of exercises. Most of the paragraphs have a section ‘Perspectives’, written with the aim to place the material in a broader context and explain further results and directions." Zentralblatt MATH
Content Level » Graduate
Keywords » 53B21, 53L20, 32C17, 35I60, 49-XX, 58E20, 57R15 - Dirac operator - Morse theory - Riemannian geometry - curvature - geodesic - harmonic map
Related subjects » Geometry & Topology - Theoretical, Mathematical & Computational Physics
1. Riemannian Manifolds.- 2. Lie Groups and Vector Bundles.- 3. The Laplace Operator and Harmonic Differential Forms.- 4. Connections and Curvature.- 5. Geodesics and Jacobi Fields.- 6. Symmetric Spaces and K¨ahler Manifolds.- 7. Morse Theory and Floer Homology.- 8. Harmonic Maps between Riemannian Manifolds.- 9. Harmonic Maps from Riemann Surfaces.- 10. Variational Problems from Quantum Field Theory.- A. Linear Elliptic Partial Differential Equations.- A.1 Sobolev Spaces.- A.2 Linear Elliptic Equations.- A.3 Linear Parabolic Equations.- B. Fundamental Groups and Covering Spaces.- Bibliography.- Index.
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