Curvature and homology. Revised reprint of the 1970 ed. (English) Zbl 0962.53001

Goldberg, Samuel I.

Mineola, NY: Dover Publications, Inc. xviii, 395 p. (1998).

This is a revised reprint of the 1970 edition. The first edition was published in 1962 and reviewed in Zbl 0105.15601. The book still can serve as a good introduction into the complex differential geometry. This edition is supplied by appendices which, in particular, contain expositions of the Kodaira vanishing theorems and of the Chern classes. Two of the appendices deal with the topology of Kähler manifolds with positive holomorphic or holomorphic bisectional curvature. This relates to the famous Frankel conjecture proved by Y.-T. Siu and S.-T. Yau [Invent. Math. 59, 189-204 (1980; Zbl 0442.53056)] and by S. Mori [Ann. Math., II. Ser. 110, 593-606 (1979; Zbl 0423.14006)]. The book also contains an appendix which explains an extension of the Bochner theorem obtained by X. Rong who proved that a compact Ricci negatively-curved manifold has no nontrivial invariant pure \(F\)-structures [Duke Math. J. 91, No. 2, 381-392 (1998; Zbl 0962.53026)].

Reviewer: Iskander A.Taimanov (Novosibirsk)

MSC:

53-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
53C55	Global differential geometry of Hermitian and Kählerian manifolds
53C20	Global Riemannian geometry, including pinching

Keywords:

Riemannian manifold; differential topology; curvature; compact Lie group; complex manifold; Kähler manifold