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Fundamentals of differential geometry. (English) Zbl 0932.53001
Graduate Texts in Mathematics. 191. New York, NY: Springer. xvii, 535 p. (1999).
Differential geometry has a long tradition. It has undergone several periods of intensive development. These were determined by new ideas emerging from differential geometry itself or from innovating changes in related disciplines such as differential topology and algebraic topology and last yet not least the spectacular evolutions in theoretical physics (gauge theories, interaction theories). Consequently, differential geometry is a very rich field, comprising a great diversity of results. It is virtually impossible to depict it in one book only.
Nearly forty years back, two large volumes bearing the title “Foundations of differential geometry” [Vol. 1 (1963; Zbl 0119.37502) and Vol. 2 (1969; Zbl 0175.48504)] were published by S. Kobayashi and K. Nomizu in which the theory of differentiable fiber bundles is systematically employed for invariantly describing both local and global results in differential geometry. Soon afterwards, M. Spivak, elaborating a “Comprehensive introduction to differential geometry”, came to publish a work spanning 5 volumes [Vol. 1 (1970; Zbl 0202.52001), Vol. 2 (1970; Zbl 0202.52201), Vol. 3 (1975; Zbl 0306.53001), Vol. 4 (1975; Zbl 0306.53002) and Vol. 5 (1975; Zbl 0306.53003)] of impressive size.
There are also many other successful books that were more or less concerned with encompassing the essentials of differential geometry. However, though with recognized subjectivity, we would place this book by Professor S. Lang directly after the two mentioned above. It comprises, in a very readable form, the fundamentals of differential geometry. It is to be mentioned that the whole theory is presented in the general framework of Banach spaces. This allows the author to simplify many proofs and to provide very modern treatments for old and new subjects.
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.
This book is interesting, useful and deserves to be in any mathematical library.

53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
58-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to global analysis
53Bxx Local differential geometry
58Axx General theory of differentiable manifolds
53Cxx Global differential geometry
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds