Page 7 - Curvature and Homology

PREFACE

The purpose of this book is to give a systematic and "self-contained"
account along modern lines of the subject with which the title deals,
as well as to discuss problems of current interest in the field. With this
statement the author wishes to recall another book, "Curvature and
Betti Numbers," by K. Yano and S. Bochner; this tract is aimed at
those already familiar with differential geometry, and has served
admirably as a useful reference during the nine years since its appearance.
In the present volume, a coordinate-free treatment is presented wherever
it is considered feasible and desirable. On the other hand, the index
notation for tensors is employed whenever it seems to be more adequate.
The book is intended for the reader who has taken the standard courses
in linear algebra, real and complex variables, differential equations, and
point-set topology. Should he lack an elementary knowledge of algebraic
topology, he may accept the results of Chapter I1 and proceed from
there. In Appendix C he will find that some knowledge of Hilbert space
methods is required. This book is also intended for the more seasoned
mathematician, who seeks familiarity with the developments in this
branch of differential geometry in the large. For him to feel at home
a knowledge of the elements of Riemannian geometry, Lie groups, and
algebraic topology is desirable.
The exercises are intended, for the most part, to supplement and to
clarify the material wherever necessary. This has the advantage of
maintaining emphasis on the subject under consideration. Several might
well have been explained in the main body of the text, but were omitted
in order to focus attention on the main ideas. The exercises are also
devoted to miscellaneous results on the homology properties of rather
special spaces, in particular, &pinched manifolds, locally convex hyper-
surfaces, and minimal varieties. The inexperienced reader should not be
discouraged if the exercises appear difficult. Rather, should he be
interested, he is referred to the literature for clarification.
References are enclosed in square brackets. Proper credit is almost
always given except where a reference to a later article is either more
informative or otherwise appropriate. Cross references appear as (6.8.2)
referring to Chapter VI, Section 8, Formula 2 and also as (VI.A.3)
referring to Chapter VI, Exercise A, Problem 3.
The author owes thanks to several colleagues who read various parts
of the manuscript. He is particularly indebted to Professor M. Obata,
whose advice and diligent care has led to many improvements. Professor
R, Bishop suggested some exercises and further additions. Gratitude is
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