zbMATH — the first resource for mathematics

Geometry of manifolds. Reprint of the 1964 original. (English) Zbl 0984.53001
Providence, RI: AMS Chelsea Publishing. xii, 273 p. (2001).
This excellent book is a new printing of the version [Geometry of manifolds (1964; Zbl 0132.16003)] which was reviewed by M. Stoka.
As stated in the new preface, this is a “faithful copy” with corrections of errors and insertions of historical nature.
The fundamentals of differential geometry are exposed and the emphasis is on Riemannian geometry.
The last two chapters are a peculiarity with respect to similar introductory texts. In chapter ten, conditions for isometric immersions in a Riemannian manifold are established. Chapter eleven deals with the first and second variation of arc length leading to Synge’s formula (1926). Then, using the index form, one proves Morse’s theorem on the number of focal points. After that, the minimum locus \(L(p)\) is introduced and one obtains a result of Klingenberg (1959) which is a necessary condition on a point \(m\in L(p)\) for the existence of a closed geodesic with ends at a point \(p\) passing through \(m\). Rauch’s comparison theorem is established (where here too the absence of conjugate points allows to obtain the result).

53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
53B21 Methods of local Riemannian geometry
53C05 Connections, general theory
53C22 Geodesics in global differential geometry