A Comprehensive Introduction to Differential Geometry Books in Order

The opening volume develops the mathematical foundations of differential geometry through manifolds, differential structures, tangent bundles, tensors, and vector fields. It introduces the language and tools needed to study geometric spaces beyond the familiar setting of Euclidean calculus.

This volume moves from plane and space curves into the development of surface geometry. Curvature, torsion, and Gauss's work connect classical geometric questions with the mathematical structures that underpin the series' treatment of modern differential geometry.

The third volume develops classical surface theory using the foundations established earlier in the series. It examines principal curvatures, minimal and ruled surfaces, geodesics, the fundamental theorem of surface theory, and connections between curvature and global geometry.

Extending surface theory into higher dimensions and codimensions, this volume studies submanifold equations and constant curvature spaces. Second variation, Jacobi fields, and variations of length, area, and volume connect local calculations with questions about geodesics and minimal surfaces.

The concluding volume brings partial differential equations into the study of isometric embeddings and rigidity. It then develops the generalized Gauss-Bonnet theorem and characteristic classes, linking geometric curvature with topology and providing a bibliography for further study.