Differential geometry is a subject related to many fields in mathematics and the sciences. The authors of this book provide a vertically integrated introduction to differential geometry and geometric analysis. The material is presented in three distinct parts: an introduction to geometry via submanifolds of Euclidean space, a first course in Riemannian geometry, and a graduate special topics course in geometric analysis, and it contains more than enough content to serve as a good textbook for a course in any of these three topics.

The reader will learn about the classical theory of submanifolds, smooth manifolds, Riemannian comparison geometry, bundles, connections, and curvature, the Chern–Gauss–Bonnet formula, harmonic functions, eigenfunctions, and eigenvalues on Riemannian manifolds, minimal surfaces, the curve shortening flow, and the Ricci flow on surfaces. This will provide a pathway to further topics in geometric analysis such as Ricci flow, used by Hamilton and Perelman to solve the Poincaré and Thurston geometrization conjectures, mean curvature flow, and minimal submanifolds.

The book is primarily aimed at graduate students in geometric analysis, but it will also be of interest to postdoctoral researchers and established mathematicians looking for a refresher or deeper exploration of the topic.