John M. Lee Bücher

Geared towards graduate students with a background in topological and differentiable manifolds, this textbook offers a comprehensive exploration of Riemannian geometry. The second edition features significant adaptations and expansions, enhancing its clarity and depth. It serves as a robust resource for understanding the intricacies of Riemannian structures, geodesics, and curvature, making it suitable for both one and two-semester courses. The updated content aims to facilitate a deeper grasp of the subject for advanced learners.

This graduate-level textbook introduces the theory of smooth manifolds, covering essential topics like tangent vectors, vector bundles, tensors, and Lie groups. The revised second edition emphasizes concrete understanding with intuitive discussions and illustrations, introducing key analytic tools earlier. Prerequisites include general topology and basic linear algebra.

Focusing on the geometric meaning of curvature, this textbook for graduate students emphasizes essential tools for studying Riemannian geometry. It covers metrics, connections, and geodesics before introducing the Riemann curvature tensor and submanifold theory for practical interpretation. The text aims to prove four fundamental theorems linking curvature and topology, including the Gauss-Bonnet and Cartan-Hadamard theorems. Selected topics are structured for a concise ten to fifteen-week course, prioritizing depth over breadth in the subject matter.