curvature and homology, revised ed. - s. goldberg

CURVATURE AND Revised Edition Samuel I. Goldberg CURVATURE AND HOMOLOGY Revised Edition Samuel I. Goldberg Mathematicians interested in the curvature properties of Riemannian mani- folds and their homologic structures, an increasingly important and specialized branch of differential geometry, will welcome this excellent teaching text. Revised and expanded by its well-known author, this volume offers a systematic and self-contained treatment of subjects such as the topol- ogy of differentiate manifolds, curvature and homology of Riemannian man- ifolds, compact Lie groups, complex manifolds, and the curvature and homology of Kaehler manifolds. In addition to a new preface, this edition includes five new appendices con- cerning holomorphic bisectional curvature, the Gauss-Bonnet theorem, some applications of the generalized Gauss-Bonnet theorem, an application of Bochners lemma, and the Kodaira vanishing theorems. Geared toward readers familiar with standard courses in linear algebra, real and complex variables, differential equations, and point-set topology, the book features helpful exer- cises at the end of each chapter that supplement and clarify the text. This lucid and thorough treatment—hailed by Nature magazine as " a valuable survey of recent work and of probable lines of future progress"— includes material unavailable elsewhere and provides an excellent resource for both students and teachers. Unabridged Dover (1998) corrected republication of the work originally pub- lished by Academic Press, New York, 1962. New Preface. Introduction. Five new appendixes. Bibliography. Indices. 416pp. 5 3/8 x 8 1/2 Paperbound. Free Dover Mathematics and Science Catalog (59065-8) available upon request. Curvature and Homology SAMUEL I. GOLDBERG Department of Mathematics University of Illinois, Urbana- Champaign DOVER PUBLICATIONS, INC. Mineola, New York To my parents and my wife Copyright Copyright 8 1962, 1982, 1998 by Samuel I. Goldberg All rights reserved under Pan American and International Copyright Conventions Published in Canada by General Publishing Company, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario. Published in the United Kingdom by Constable and Company, Ltd., 3 The Lanchesters, 162-164 Fulham Palace Road, London W6 9ER. Bibliographical Note This Dover enlarged edition, first published in 1998, is an unabridged, corrected, and enlarged republication of the second printing (1970) of the work first published in 1962 by Academic Press, N.Y., as Volume 11 in the series Pure and Applied Mathematics. Five new Appendices, a new Preface, and additional reference titles have been added to this edition. Library of Congress Cataloging-in-Publication Data Goldberg, Samuel I. Curvature and homology 1 Samuel I. Goldberg. - Enl. ed. p. cm. Includes bibliographical references and indexes ISBN 0-486-40207-X (pbk.) 1. Curvature. 2. Homology theory. 3. Geometry, Riemannian. I. Title. QA645.G6 1998 5 1 6.3'624~2 1 98-222 1 1 CIP Manufactured in the United States of America Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501 PREFACE TO THE ENLARGED EDITION Originally, in the first edition of this work, it was the author's purpose to provide a self-contained treatment of Curvature and Homology. Sub- sequently, it became apparent that the more important applications are to Kaehler manifolds, particularly the Kodaira vanishing theorems, which appear in Chapter VI. To make this chapter comprehensible, Appendices F and I have been added to this new edition. In these Appen- dices, the Chern classes are defined and the Euler characteristic is given by the Gauss-Bonnet formula-the latter being applied in Appendix G. Several important recent developments are presented in Appendices E and H. In Appendix E, the differential geometric technique due to Bochner gives rise to an important result that was established by Siu and Yau in 1980. The same method is applied in Appendix H to F-structures over negatively curved spaces. S. I. GOLDBERG Urbana, Illinois February, 1998 [...]... holomorphic curvature and curvature 338 G.7 Holomorphic curvature and Euler-PoincarC characteristic 343 G.8 Curvature and volume 346 G.9 T h e curvature transformation 352 G.IO Holomorphic pinching and Euler-PoincarC characteristic 357 Appendix H A N APPLICATION OF BOCHNER'S LEMMA 361 H 1 A pure F-structure ... GAUSS-BONNET THEOREM 327 G.I Preliminary notions 329 G.2 Normalization of curvature 330 G.3 Mean curvature and Euler-Poincard characteristic 331 G.4 Curvature and holomorphic curvature 334 G.5 Curvature as an average 337 G.6 Inequalities between holomorphic curvature. .. in V, and this subspace in turn defines a 'unique' (n - p)-vector Note that for eachp, the spaces Ap(V) and An-p(V) have the same dimensions Any p-vector 5 and any (n 2 p)-vector q determine an n-vector 5 A q which in terms of the basis e = el A e, of An(V) may be expressed as f A7 =( f,~) e (1.4.3) where (5, q) E R It can be shown that this 'pairing' defines an isomorphism of AP(V) with ( A ~ - P (... definitions of contravariant and covariant vector Consider the triples (P, U,, gl-irjl j,) and (P, Up, ~l.-i~jl j8) They are said to be equivalent if P = P and if the nr+ constants ,$'1 .-' rjl ,, are related to the nr+ constants @- irjl j, the formulae by An equivalence class of triples (P, U,, @ irjl ,j.) is called a tensor o type f f f (r, S) over T p contravariant o order r and covariant o order s A... then da = dp on S The elements A,P(T*) of the kernel of d: AP(T*) -+ AP+l(T*) are called closed p-forms and the images A,P(T*) of AP-'(T*) under d are called exact p-forms They are clearly linear spaces (over R) The quotient space of the closed forms of degree p by the subspace of exact p-forms will be denoted by D ( M ) and called the p - d i d 1 cohomology group o M obtained ust'ng dzjbvntial forms... 'Z.+, ~- 8 I RIEMANNIAN MANIFOLDS I t is also possible to form new tensors from a given tensor I n fact, let (P, U,, ~l-irjl- j.) be a tensor of type (r, The triple (P, U,, I s i a.j) where the indices i, and j, are equal (recall that repeated , ,.j: indices ~ c d ~ c asummation from 1 to n) is a rLpesentative of a tensor te of type (r - 1, s - 1) For, eel since This operation is known as contraction and. .. denote the local coordinates of P in V They determine a base {dv"P)} in T,* and a dual base Cfi(P)} If in Tp we set f = gcrv(p)t, (1.3.3) it follows that c~v(P,i) ~v(P,t) = (1.3.4) Since V~(P,F) & * - - i r jl ja eil ir = il ia (p) (1.3.5) and ~ v ( P , t= &.-ir ) jl jH where {eil ,,jl Ja(P)} and {fil ," jl(P)) w'), ftl, irjl-.j~ p) (1J.6) are the induced bases in These are the equations defining gUv(P)... (modulo the fundamental differentiability lemma C.l of Appendix C), and the essential material and informati09 necessary for the treatment and presentation of the subject of curvature and homology is presented The idea of the proof of the existence theorem is to show that A-'-the inverse of the closure of A-is a completely continuous operator The reader is referred to de Rham's book ''VariCth Diffkrentiables"... written as g f and at other times the dot is not present The dot is also used to denote the (local) scalar product of vectors ) (relative to g However, no confusion should arise Symbol Page n-dimensional Euclidean space n-dimensional affine space An with a distinguished point complex n-dimensional vector space n-sphere n-dimensional complex torus n-dimensional complex... MANIFOLDS where p, q, and r are functions of class 2 (at least) of x, y, and s h h for its differential the 2-form Moreover, the 2-form has the differential In more familiar language, da is the curl of a and dp its divergence That dda = 0 is expressed by the identity div curl a = 0 We now show that the coefficients ails.,, of a differential form u can be considered as the components of a skew-symmetric tensor . CURVATURE AND Revised Edition Samuel I. Goldberg CURVATURE AND HOMOLOGY Revised Edition Samuel I. Goldberg Mathematicians interested in the curvature properties of Riemannian mani- folds and. 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. space, So is to be a separable, Hausdorff space with the further propetties: (i) So is compact (that is X(So) is closed and bounded); (ii) So is connected (a topological space is said to