D I F F E R E N T I A L G E O M E T R Y

http://dx.doi.org/10.1090/pspum/003

PROCEEDINGS OF THE

THIRD SYMPOSIUM IN PURE MATHEMATICS

OF THE AMERICAN MATHEMATICAL SOCIETY

Held at the University of Arizona Tucson, Arizona

February 18-19, 1960

With the Support of the NATIONAL SCIENCE FOUNDATION

CARL B. ALLENDOERFER EDITOR

PROCEEDINGS OF

SYMPOSIA IN PURE MATHEMATICS

VOLUME III

D I F F E R E N T I A L G E O M E T R Y

AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND

1961

Library of Congress Catalog Card Number 50-1183

Prepared by the American Mathematical Society under Grant No. NSF-G10809 with the National Science Foundation

Copyright © 1961 by the American Mathematical Society Printed in the United States of America

All rights reserved except those granted to the United States Government. Otherwise, this book, or parts thereof, may

not be reproduced in any form without permission of the publishers.

CONTENTS

PAGE

INTRODUCTION vii

A Report on the Unitary Group 1 By RAOUL BOTT

Vector Bundles and Homogeneous Spaces 7 By M. F. ATIYAH and F. HIRZEBRUCH

A Procedure for Killing Homotopy Groups of Differentiable Manifolds . 39 By JOHN MILNOR

Some Remarks on Homological Analysis and Structures . . . . . . . . 56 By D. C. SPENCER

Vector Form Methods and Deformations of Complex Structures . . . . 87 By ALBERT NIJENHUIS

Almost-Product Structures 94 By A. G. WALKER

Homology of Principal Bundles 101 By ELD ON DYER and R. K. LASH OF

Alexander-Pontrjagin Duality in Function Spaces 109 By JAMES EELLS, JR.

The Cohomology of Lie Rings 130 By RICHARD S. PALAIS

On the Theory of Solvmanifolds and Generalization with Applications to Differential Geometry 138

By Louis AUSLANDER

Homogeneous Complex Contact Manifolds 144 By WILLIAM M. BOOTHBY

On Compact, Riemannian Manifolds with Constant Curvature. I . . . 155

By EUGENIO CALABI

Elementary Remarks on Surfaces with Curvature of Fixed Sign . . . . 181 By L. NlRENBERG

Canonical Forms on Frame Bundles of Higher Order Contact 186 By SHOSHICHI KOBAYASHI

On Immersion of Manifolds 194 By HANS SAMELSON

Index . 197

INTRODUCTION

This Symposium on Differential Geometry was organized as a focal point for the discussion of new trends in research. As can be seen from a quick glance at the papers in this volume, modern differential geometry to a large degree has become differential topology, and the methods employed are a far cry from the tensor analysis of the differential geometry of the lOSO's.

This development, however, has not been as abrupt as might be imagined from a reading of these papers. I t has its roots in the movement toward differ­ential geometry in the large to which mathematicians such as Hopf and Rinow, Cohn-Vossen, de Rham, Hodge, and Myers gave impetus. The objectives of their work were to derive relationships between the topology of a manifold and its local differential geometry. Other sources of inspiration were E. Cartan (whose fundamental contributions were recognized by many only after his death) and M. Morse and his calculus of variations in the large. One of the major new ideas was that of a fiber bundle which gave a global structure to a differentiable manifold more general than that included in the older theories. Methods and results of differential geometry were applied with outstanding success to the theories of complex manifolds and algebraic varieties and these in turn have stimulated differential geometry. The discovery by Milnor of invariants of the differential structure of a manifold which are not topological invariants estab­lished differential topology as a discipline of major importance.

GAEL B. ALLENDOERFER

University of Washington, Seattle, Washington

Vll

I N D E X

Ct-adic topology, 24 Affine connections, 94,186 Affine spaces, locally, 142 Alexander-Pontrjagin duality, 109

theorem, 124 Almost complex, 22 Almost-product structure, 94 Artin-Rees lemma, 24 Atlas

Eulerian, 159 Lagrangian, 160

Axioms of a cohomology theory, 14

Bidifferentiable transformations, closed pseudogroups of, 61

Borsuk's Extension Theorem, 120 Bott

isomorphism, 13 periodicity, 7

Bundles complex vector, 8 homology of principal, 101 fc-trivial, 49 of r-frames, 188 orien table, 115 ring of complex vector, 7 transverse, 115

C-space, 146 Ci-map, 20 Canonical form

differential, 189 structure equation of, 191

Cartan, E. invariant forms' for a continuous

pseudogroup of differentiate trans­formations, 85

structure equations of, 186 Category, model, 59 Characteristic class, 102

relative, 102 Chern character, 15, 29 x-equivalent, 40, 46, 49, 53, 55 Classification theorem, 29 Classifying spaces, 7, 28 Clifford-Klein spaces, 156

differentiable family of, 159 Closed pseudogroups of bidiffenertiable trans­

formations, 61

Cobordism, 40 Cochains, invariant, 135 Cohomology, 109

group of a Lie d-ring, 137 of Lie rings, 130 operations, 18 with values in the sheaves of Lie algebras

of infinitesimal groups, 56 with values in sheaves of nonabelian

groups, 56 Cohomology theory

axioms of, 14 periodic, 7

Compact, Riemannian manifolds, 155 Complete germ, 72 Completed representation ring of a torus, 26 Completions of modules, 24 Complex, almost, 22 Complex analytic

differentiable r-manifold, 64 family of complex structures, 92

Complex contact manifold, 144 manifold, homogeneous, 146 structure, 144

Complex structures complex analytic family of, 92 deformation of, 87 equivalence of, 89 family of, 91 obstructions to deformation of, 89-90 stability of, 90 variations of, 89

Complex vector bundles, 8 ring of, 7

Connections affine, 94 linear (affine), 186

Constant curvature, 155-156 Contact form, 145 Continuous pseudogroup of differentiable

transformations, 81 invariant Cartan forms for, 85

Coordinate transformation, 118 Co-orienting, 111 Curvature, 186

constant, 155-156 Gauss, 181

197

198 INDEX

d-trivial Lie d-ring, 131 Deformations, homological analysis of, 69 Deformations of complex structures, 87

obstructions to, 89-90 Deformation of the T-manifold, 70

germ of, 71 Derivative

lie, 134 torsional, 99

Differentiability Graves-Hildebrandt, 115

Differentiable complex analytic or real analytic T-mani-

fold, 64 family of T-manifolds, 70

Differentiable transformations continuous pseudogroups of, 81 invariant Cartan forms for a continuous

pseudogroup of, 85 Differential

form, 189 manifolds, 39

Distributions, 94 Double exterior forms, 166 Duality theorem, 110

Alexander-Pontrjagin, 124

Eilenberg and Steenrod axioms, 7 Equivalence of complex structures, 89 Eulerian atlas, 159 Existence in homological analysis, problem

of, 75 Extension Theorem of Borsuk, 120 Exterior forms, double, 166

/-relatedness for vector forms, 90 Family of complex structures, 91

complex analytic, 92 Family of r-manifolds, differentiable, 70 Frames, r-, 188

bundle of, 188 Function spaces, 109 Fundamental class of the oriented pair, 113

r-manifolds deformation of, 70 differentiable family of, 70 differentiable, real analytic or complex

analytic, 64 germ of deformation of, 71

T-structure, 64 T-vector field, 67

Gauss curvature, 181 Germ,

complete, 72 effective, 72 of deformation of the r-manifold, 71 stable, 75

Gradient mapping, 182 Grating, 112 Graves-Hildebrandt differentiability, 115 Groups

killing homotopy, 39, 50 sheaf of, 65 unitarv, 1, 8 Weyl,23

Gysin homomorphism, 20, 114

Hildebrandt-Graves differentiability, 115 Homogeneous

complex contact manifold, 146 spaces, 7, 31

Homological analysis of deformations, 69 Homology of principal bundles, 101 Homomorphism, Gysin, 20, 114 Homotopy

complements, 113 killing classes, 43 killing groups, 39, 50

Hypersurfaces, 181

Immersion of manifolds, 194 Implicit function theorem, 116-117 Infinitesimal pseudogroup, 66 Infinitesimally surjective, 75 Interior product, 133 Invariant

Cartan forms for a continuous pseudogroup of differentiable transformations, 85

cochains, 135 cohomology group of a Lie coring, 137

Invariants, ring of, 27 Isomorphism

Bott, 13 Theorem of Leray, 111

r. (/) source of, 187 target of, 187

Jacobi identities for vector forms, 88 Jet, r-, 187

A;-parallelizable manifold, 49 fc-trivial bundle, 49

INDEX 199

Killing homotopy classes, 43 groups, 39, 50

Klien-Clifford spaces, 156 differentiable family of, 159

<£-ring, 132 Lagrangian atlas, 160 Leray Isomorphism Theorem, 111 Lie d-ring

cohomology group of, 137 d-trivial, 131 invariant cohomology group of, 137 cC-module over, 132 over R, 131

Lie derivatives, 134 Lie group, compact, 25

connected, 23, 29, 36 representation ring of, 25

Lie rings, cohomology of, 130 Linear (affine) connection, 186 Locally

afiine spaces, 142 stable, 74 trivial, 74

Manifold pair, orientation sheet of, 111 Manifolds, 39

compact, Riemannian, 155 complex contact, 144 deformation of the T-, 70 differentiable family of T-, 70 differentiable, real analytic or complex

analytic T-, 64 germ of deformation of the T-, 71 homogeneous complex contact, 146 immersion of, 194 fc-parallelizable, 49

Mapping gradient, 182 monotone, 182 spherical image, 181

Model category, 59 Module over a Lie d-ring <£, 132

basic, 132 cohomology of <£ with coefficients in, 136 invariant cohomology of £ with coefficients

in, 136 impairing of two, 132

Modules completions of, 24

Monotone mappings, 182

Morse theory, 2 Multifoliate, 81 Multiplication, Pontrjagin, 125

Nilmanifold, 138 Noetherian ring, 24 Normal degree, 194-195

Obstructions to deformation of a complex structure, 89-90

Orient, 111, 115 Orientability, 111 Orientable

bundle, 115 pair, 111

Orientation sheet of the manifold pair (X, F), 111

Oriented pair, fundamental class of, 113 Orienting, co-, 111

Parallel, 99 Periodic cohomology theory, 7 Periodicity, Bott, 7 7r-manifold, 46 Pontrjagin

classes, 20 multiplication, 125 numbers, 41

Pontrjagin-Alexander duality, 109 theorem, 124

Primitive left, 167 right, 167

Principal bundles, homology of, 101 Product

interior, 133 triple, 105

Projective space, 3 Projector, 94 Pseudogroup, 59

closed of bidifferentiable transfor­mations, 61

infinitesimal, 66 of bidifferentiable, bianalytic or biholo-

morphic transformations, 64 resolution of the sheaf of vector fields

associated with a continuous r (sheaf of T-vector fields), 85

Pseudogroup of differentiable transforma­tions, continuous, 81

invariant Cartan forms for, 85

200 INDEX

r-frames, 188 bundle of, 188

r-jet, 187 Real analytic, differentiable r-manifold, 64 Rees-Artin lemma, 24 Representation ring

completed, 27 completed of a torus, 26 of a compact Lie group, 25

Resolution of the sheaf of vector fields associated with a continuous pseudo-group r (sheaf of T-vector fields), 85

Riemann-Roch theorem, 7, 20 Rigid, 78 Ring

Noetherian, 24 of complex vector bundles, 7 of invariants, 27

Saddle surfaces, 182 Sequence, Wang, 102-103 Sheaf

of groups, 65 of vector fields associated with a continuous

pseudogroup T (sheaf of r-vector fields), resolution of, 85

Solvmanifold, 138 Source of £ (/), 187 Spaces

classifying, 7, 28 CUfford-Klein, 156 differentiable family of Clifford-Klein, 159 function, 109 homogeneous, 7, 31 locally affine, 142 projective, 3 structure on topological, 60

Spectral sequence, 7, 16 Spherical image mapping, 181 Spinor representation, 33 Stability of complex structures, 90 Stable

germ, 75 locally, 74

Steenrod and Eilenberg axioms, 7 Stiefel-Whitney

classes, 20 numbers, 41

Structure almost-product, 94 complex contact, 144 equation of the canonical form, 191

BCDEFGHIJ-AMS-89876543

equations of E. Cartan, 186 T-,64 on a topological space, 60 (See Complex)

Submanifold, closed relative, 113 Surfaces, 181

saddle, 182 Surgery, 39-42, 44, 46, 54 Surjective, infinitesimally, 75 Suspension, 9

Target of £ (/), 187 Tietze's Theorem, 119 Todd genus, 21 Topological space, structure on, 60 Topology, a-adic, 24 Torsion, 95, 186

for vector forms, 88 Torsional derivatives, 99 Torus, 26

completed representation ring of, 26 Transformation, coordinate, 118 Transregular, 116 Transverse bundle, 115 Triple product, 105 Trivial locally, 74

Unitary group, 1, 8 Universal Coefficient Theorem, 126

Variability, index of, 72 Variations of a complex structure, 89 Vector bundles, complex, 8

ring of, 7 Vector fields

associated with a continuous pseudogroup r (sheaf of r-vector fields), resolution of the sheaf of, 85

r-,67 Vector forms, 87

/-relatedness for, 90 Jacobi identities, 88 torsion for, 88 types of, 88 vertical, 91

Wang sequence, 102-103 Weyl group, 23 Whitney-Stiefel

classes, 20 numbers, 41