Differential and Riemannian manifolds / Serge Lang

The present book is an introduction to differential geometry, starting from recalling differential calculus and going through all the basic topics such as manifolds, vector bundles, vector fields, the theorem of Frobenius, Riemannian metrics and curvature. The more specialized chapters of the book deal with integration over Riemannian manifolds, the theorem of Stokes, and some of its applications. The author's intention is to rewrite the proofs of classical results in a coordinate-free formalism, which is suitable for extending these results to infinite-dimensional manifolds (modeled on a Banach space, in general (on a Hilbert space for Riemannian geometry)). With some exceptions (e.g. the Hopf-Rinow theorem) most classical results are actually shown to hold in the infinite-dimensional setting. However, the book seems (to the reviewer) difficult to read without some previous acquaintance with classical differential geometry, and this is the major criticism one may bring (for an introductory book on differential geometry). A minor (by comparison) criticism regards the comments adopted by the author which seem (to this reviewer) sometimes naive ("Even if via the physicists with their Feynman integration one eventually develops'' (p. vi), or "For one among many nice applications of the indefinite case, cf. for instance, [He84] and [Gu91], dealing with Huygens' principle'' (p. 170)) and other times just too coloquial ("replete with the claws of a rather unpleasant prying insect such as Γijkl'' (p. vii)). This reviewer finds it difficult to believe in the successful adoption of the book under review as a textbook for an introductory class on differential geometry, yet it may be useful to the researcher wishing to learn about infinite-dimensional geometry. (MathSciNet)

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