Riemannian Manifolds: An Introduction to Curvature
by John M. Lee · John M. Lee
Graduate text that develops Riemannian metrics, connections, geodesics and curvature with a clear focus on what curvature means geometrically, culminating in Gauss-Bonnet, Cartan-Hadamard and Bonnet-Myers. Readers learn to prove core comparison theorems from first principles.
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Graduate textbook covering Riemannian metrics, affine and Levi-Civita connections, geodesics, curvature, Jacobi fields, the Hopf-Rinow and Hadamard theorems, and comparison theorems. Readers who know basic smooth manifolds learn to compute with curvature and prove classical global results.