Foundations of differential geometry. Vol. II. (English) Zbl 0175.48504

Kobayashi, Shoshichi; Nomizu, Katsumi

Interscience Tracts in Pure and Applied Mathematics 15. New York-London-Sydney: Interscience Publishers a division of John Wiley and Sons (1969).

Volume 2 contains the applications of the heavy apparatus developed in Vol. I [New York etc.: Interscience Publishers (1963; Zbl 0119.37502)]. The book contains much material not otherwise available in book form and sometimes not easily accessible. The authors first study submanifolds of Riemannian manifolds with special emphasis on embedding and rigidity problems. These problems are also explored in more detail in 8 of the 17 notes at the end of the book. The next chapter deals with the variations of the length integral. The remaining chapters are Complex manifolds, Homogeneous Spaces, Symmetric Spaces and Characteristic Classes. While all these subjects are treated in recent and more detailed monographs, the present book is the only available general introduction and guide to the literature and seems to be most adequate for this purpose.
There are two pages of errata for vol. 1. The misprints in this volume, as far as discovered in a first reading, seem to be minor matters of indices and superscripts.

Reviewer: H. Guggenheimer

MSC:

53-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
53C20	Global Riemannian geometry, including pinching
53C24	Rigidity results
53C30	Differential geometry of homogeneous manifolds
53C35	Differential geometry of symmetric spaces
53C40	Global submanifolds
53C55	Global differential geometry of Hermitian and Kählerian manifolds

Keywords:

differential geometry; submanifolds of Riemannian manifolds; embedding and rigidity problems; length integral; complex manifolds; homogeneous spaces; symmetric spaces; characteristic classes

Citations:

Zbl 0119.37502