zbMATH — the first resource for mathematics

Differential and Riemannian manifolds. 3rd ed.; Repr. of the orig. 1972. (English) Zbl 0824.58003
Graduate Texts in Mathematics. 160. Heidelberg: Springer-Verlag. xiii, 364 p. (1995).
This volume – the third edition of the book on differential manifolds [1st ed. (1972; Zbl 0239.58001), 2nd ed. (1985; Zbl 0551.58001)] – represents an introduction to differential topology, differential geometry and differential equations. It provides the basic concepts and some of the fundamental theorems in all three areas, with emphasis on the differential aspects of differential manifolds rather than the topological ones. The topics are mainly covered for infinite-dimensional manifolds, modeled on Banach and Hilbert spaces; in the finite- dimensional case there are treated the differential forms of top degree, Stokes’ theorem and several of its applications to the differential or Riemannian case.
In addition to the older versions, the text includes three chapters on Riemannian and pseudo-Riemannian geometry, calculus of variations, and applications to volume forms. The sections on sprays are rewritten and more examples illustrating Stokes’ theorem are present in this new version; also, the list of references is substantially enlarged.
The book can be successfully used as a text for a general graduate course leading into many directions and giving a broad perspective of the considered domains.

58-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to global analysis
53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
57-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
58A40 Differential spaces
53C20 Global Riemannian geometry, including pinching