Fundamentals of differential geometry / Serge Lang

The content of the book is divided into three parts. The first part is devoted to “general differential theory” and contains chapters covering differential calculus, manifolds, vector bundles, vector fields and differential equations, operations on vector fields and differential forms, the theorem of Frobenius. The second part with the title “metrics, covariant derivatives, and Riemannian geometry” treats metrics, covariant derivatives and geodesics, curvature, Jacobi lifts and tensorial splitting of the double tangent bundle, curvature and the variation formula, an example of seminegative curvature, automorphisms and symmetries, immersion and submersions. The third part entitled “volume forms and integration” includes chapters on volume forms, integration of differential forms, Stokes' theorem (for a rectangular simplex, on a manifold, with singularities), applications of Stokes' theorem. In an Appendix, a spectral theorem for operators on Hilbert spaces is presented. The bibliography contains mainly books as well as very influential papers. A large index is also included. The graphical aspects were excellent solved. (Zentralblatt)

Bibliogr. p. 523-530. Index