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Riemannian Geometry [Petersen] - mklabgr - 08-17-2026 Riemannian Geometry Author: Peter Petersen Publisher: Springer Peter Petersen's Riemannian Geometry is an advanced graduate textbook designed to provide both a rigorous introduction and a pathway into modern research-level Riemannian geometry. Beginning with Riemannian metrics, connections and curvature, Petersen develops the fundamental machinery needed to understand geodesics, distance, sectional and Ricci curvature, and comparison geometry. A distinctive feature is the emphasis on understanding how local curvature controls global geometric and topological properties. The book assumes prior familiarity with smooth manifolds, tensors, differential forms and Lie groups, making it better suited to graduate students than to complete beginners.  The text becomes particularly valuable as it moves into more advanced subjects. Petersen treats sectional-curvature comparison, Ricci-curvature comparison, the Bochner technique, symmetric spaces, holonomy, convergence of Riemannian manifolds, Lie groups and Riemannian submersions. The third edition considerably expands the exercises and coordinate calculations, incorporates variational calculus, adds general curvature formulas for Lie groups and submersions, and includes newer results concerning manifolds of positive curvature. Petersen's treatment of the Bochner method is especially notable: the third edition introduces a streamlined tensor approach that connects curvature bounds with topological information.  One of the book's strongest characteristics is its combination of geometric and analytic methods. Instead of presenting Riemannian geometry merely as a collection of definitions and classical theorems, Petersen develops techniques that show how curvature, differential equations, topology and analysis interact. This makes the book useful not only as a one-year graduate course but also as a reference for students intending to specialize in differential or global geometry. It is demanding, but its breadth, exercises and treatment of modern comparison geometry make it an excellent bridge from introductory manifold theory to research-oriented Riemannian geometry.  Key takeaways Curvature is the central organizing idea: sectional, Ricci and related curvature quantities are used to extract global information about manifolds. Comparison geometry plays a major role: Petersen develops techniques for comparing manifolds under curvature bounds and studying their topology and geometry. Geometry meets analysis: the Bochner technique and related analytic methods illustrate how differential equations and tensor analysis can solve geometric problems. It goes well beyond an introductory text: topics such as holonomy, symmetric spaces, convergence, positive curvature and Riemannian submersions make it particularly valuable for graduate students moving toward research.  Overall: ★★★★★ — A comprehensive and sophisticated graduate text, especially strong for readers interested in curvature, comparison geometry and the interaction between geometry, topology and analysis. Springer — Peter Petersen, Riemannian Geometry (3rd ed.)