Riemannian geometry and geometric analysis are flourishing fields with applications in physics, statistics, and machine learning. This textbook develops both fundamental concepts and more advanced topics shaped by recent progress. The 8th edition expands coverage with a systematic treatment of total scalar curvature, from which the Einstein equations, Ricci flow, and the Yamabe problem emerge, and includes new perspectives on generalized sectional and Ricci curvatures as well as Kirillov's coadjoint orbits. It introduces core notions such as geodesics, connections, and curvature, alongside key tools of geometric analysis, including harmonic functions, forms, eigenvalues, the Dirac operator, and heat flow, and highlights major variational principles like harmonic maps, YangMills, GinzburgLandau and Seiberg-Witten. The book offers a coherent geometric framework while equipping readers with practical methods for further study and research.

Provides a solid foundation for research Starts from the very basics and leads to advanced topics Features new material on scalar, sectional and Ricci curvatures

Autorentext

Jürgen Jost is a Scientific Member of the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany, an Honorary Professor at the Department of Mathematics and Computer Sciences at Leipzig University, a PI at ScaDS.AI Dresden/Leipzig, ****and an External Faculty Member of the Santa Fe Institute for the Sciences of Complexity, New Mexico, USA. He is also the author of numerous books and over 500 publications in scientific journals.

Inhalt

Chapter 1. Riemannian Manifolds.- Chapter 2. Lie Groups and Vector Bundles.- Chapter 3. The Laplace Operator and Harmonic Differential Forms.- Chapter 4. Connections and Curvature.- Chapter 5. Bochner Identities, Dirac Operators and Eigenvalues.- Chapter 6. Geometry of Submanifolds.- Chapter 7. Geodesics and Jacobi Fields.- Chapter 8. The Geometry of Ricci Curvature.- Chapter 9. Nonpositive Curvature.- Chapter 10. A Survey on Curvature and Topology.- Chapter 11. Symmetric Spaces and Kahler Manifolds.- Chapter 12. Morse Theory and Floer Homology.- Chapter 13. Harmonic Maps between Riemannian Manifolds.- Chapter 14. Harmonic Maps from Riemann Surfaces.- Chapter 15. Variational Problems from Quantum Field Theory.