Lectures on differential geometry / Bennett Chow and Yutze Chow.

Includes bibliographical references and index.

Geometry of submanifolds of Euclidean space -- Differential calculus of submanifolds -- Tangent and tensor bundles -- Curvature and local geometry -- Smooth manifolds -- Riemannian manifolds -- Differential forms -- Gauss-Bonnet theorem -- Bundles and connections -- Elliptic and parabolic equations -- Minimal surfaces -- Geometric flows and Ricci flow.

This graduate-level textbook provides a comprehensive and vertically integrated introduction to differential geometry and geometric analysis. Beginning with the classical geometry of submanifolds in Euclidean space, the book develops the foundations of smooth manifolds, Riemannian geometry, differential forms, bundles, connections, and curvature. The authors systematically introduce global geometric methods including the Gauss-Bonnet and Chern-Gauss-Bonnet theorems, comparison geometry, harmonic analysis on manifolds, and geometric topology. A substantial portion of the text is devoted to geometric analysis, including elliptic and parabolic partial differential equations, minimal surfaces, curve shortening flow, and Ricci flow on surfaces. Emphasizing both intuition and rigorous mathematical development, the book connects differential geometry with topology, analysis, and mathematical physics through numerous examples, exercises, and applications. Designed for graduate students and researchers, it serves as both a textbook and a reference work for advanced study in modern differential geometry and geometric analysis.