Riemannian Manifolds

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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.