PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS VOLUME XV

GLOBAL ANALYSIS

AMERICAN MATHEMATICAL SOCIETY

Providence, Rhode Island 1970

http://dx.doi.org/10.1090/pspum/015

Proceedings of the Symposium in Pure Mathematics of the American Mathematical Society

Held at the University of California Berkeley, California

July 1-26, 1968

Prepared by the American Mathematical Society

under National Science Foundation Grant GP-8410

SHIING-SHEN CHERN STEPHEN SMALE

Editors

Standard Book Number 8218-1415-X

Library of Congress Catalog Number 70-95271

Copyright ©1970 by the American Mathematical Society

AMS 1968 Primary Subject Classification 5750

Printed in the United States of America

All rights reserved except those granted to the United States Government May not be reproduced in any form without permission of the publishers

Reprinted 1988

The paper used in this book is acid-free and falis within the guidelines established to ensure permanence and durability. @

CONTENTS Preface . . . . . . . . . . Finsler Geometry on Sobolev Manifolds . . . .

BY J. DOWLING

The Manifold of Riemannian Metrics . . . . . BY DAVID G. EBIN

On the Differential Topology of Hilbertian Manifolds BY J. EELLS AND K. D. EL WORTHY

Differential Structures and Fredholm Maps on Banach Manifolds BY K. D. ELWORTHY AND A. J. TROMBA

Infinite Dimensional ^-Theory and Characteristic Classes of Fredholm Bundle Maps . . . . . . . .

BY ULRICH KOSCHORKE

Fredholm Maps and Gysin Homomorphisms . BY JACK JOHNSON MORAVA

Stability of Hilbert Manifolds BY NICOLE MOULIS

On the Group of Diffeamorphisms on a Compact Manifold BY HIDEKI OMORI

Critical Point Theory and the Minimax Principle BY RICHARD S. PALAIS

Transversal Approximation on Banach Manifolds . BY FRANK QUINN

Examples of Bernstein Problems for Some Nonlinear Equations BY EUGENIO CALABI

The Dirichlet Integral in Differential Geometry BY ROBERT B. GARDNER

Integral Distributions Determined by an Immersion BY ALFRED GRAY

Deformations of Complex Structure . . . . . BY PHILLIP A. GRIFFITHS

Compact Minimal Surfaces in S3 . BY H. BLAINE LAWSON, JR.

Some Properties of Solutions to the Minimal Surface System for Arbitrary Codimension . . . . . . . . . . .

BY ROBERT OSSERMAN

A Global Existence and Uniqueness Theorem for Generalized Parallelism . BY ALAN B. PORITZ

The Generic Conjugate Locus . . . . . . . . BY ALAN WEINSTEIN

Author Index . . . . . . . . . . . Subject Index . . . . . . . . . . .

283

293

299

303 305

PREFACE

The papers in these Proceedings grew out of lectures given at the fifteenth ISummer Mathematical Institute of the American Mathematical Society, whose topic was global analysis. The Institute was held at the University of California at Berkeley from July 1 to July 26,1968, and was partially financed by the National Science Foundation.

Notes of lectures were distributed at the time of the conference and some of the papers here are just as in those notes. These volumes, however, can be distinguished from the notes in the sense that in general the papers here are not just expositions of material that has or will appear elsewhere; most of the articles could just as well have appeared in Journals.

The unity given by the subject matter makes it desirable to collect them here. It is hoped that the volumes will provide an important start to the scientist who wishes to learn what is going on in that part of mathematics called global analysis.

The organizing committee for the institute consisted of: F. Browder, S.-S. Chern, L. Hormander, I. Singer, and S. Smale, with the co-editors serving as co-chairmen.

Seminar organizers were: F. Browder, E. Calabi, H. Goldschmidt, R. Hermann, C. Morrey, R. Palais, C. Pugh, I. Singer, and D. Spencer.

Finally the editors would like to thank the many people who made the institute and volumes possible. Of especially direct help to ourselves were Celeste Andrade, Ann Harrington, Gordon and Jacqueline Walker.

S.-S. Chern December 1968

S. Smale

AUTHOR INDEX

Roman numbers refer to pages on which a reference is made to an author or a work of an author. Italic numbers refer to pages on which a complete reference to a work by the author is given. Boldface numbers indicate the first page of the articles in the book.

Abraham, Ralph, 136, 147, 155, 213, 215, 222, 301

Adams, J. F., 155 Ahlfors, L. V., 256, 273 Almgren, F. J., Jr., 225, 275, 282, 284 Anderson, D., 132 Anderson, R. D., 41, 42, 43 Andreotti, A., 256, 273 Arlt, D., 92, 132 Arnold, V., 301 Atiyah, M. F., 45, 53, 92, 114, 132, 136, 139,

140, 755 Aubin, Thierry, 230

Baily, W. L., Jr., 256, 273 Berger, M., 40 Bernstein, I., 211, 284 Bers, L., 256, 284, 290 Bessaga, C , 64, 88, 92, 157, 165, 203, 211 Boardman, J. M., 93 Bombieri, E., 225, 284, 290 Bonic, R., 83, 85, 92, 222 Borel, A., 40, 132, 269, 272, 273 Bott, R., 108 Bourbaki, N., 132, 155 Bredon, G., 155 Browder, F., 83, 92 Brown, Edgar, 96 Burghelea, Dan, 41, 43, 43, 44, 93, 157, 165

Calabi, Eugenio, 223, 230, 275, 281, 282 Cartan, H., 254, 260, 273 Chern, S.-S., 231, 240, 249, 251, 273 Conforto, F., 257, 273 Conner, P. E., 755 Corson, H. H., 203

De Giorgi, E., 225, 284, 290, 291 Deligne, Pierre, 96, 122, 125, 132 Dieudonne, J., 175, 183, 217, 222 Dixmier, J., 132 do Carmo, M., 239 Dold, A., 92, 132, 155 Dos Santos, N. M., 299, 301 Douady, Adrien, 47, 92, 95, 96, 98, 107, 125,

132, 203 Dowling, J., 1 Dugundji, J., 132 Dyer, E., 755

Eberlein, 3 Ebin, David G., 11, 40 Eells, James, 41, 43, 44, 64, 93, 96, 132, 136,

755, 157, 765, 183, 213, 217, 218, 222 Eilenberg, S., 66, 92, 141 Eliasson, Halldor I., 2, 5, 9, 10 Elworthy, K. D., 41, 43, 45, 51, 64, 93, 95, 103,

127, 128, 132, 136, 149, 150, 755, 215, 217, 218,22 2

Feldman, E., 301 Flanders, H., 237 Fleming, 284 Floyd, E. E., 755 Fox, R. H., 755 Frampton, John, 83, 92, 218, 222 Frobenius, 167

Gabriel, P., 755 Ganea, T., 277 Gardner, Robert B., 231 Geba, K., 45, 51,95 Gray, Alfred, 239, 249 Griffiths, Phillip S., 251, 273 Grothendieck, A., 156, 267, 269, 273 Grotomeyer, 231, 237 Gunning, R. C , 254, 273 Gusti, 284

Hardy, G. H., 237 Henderson, D. W., 41, 42, 43 Hirsch, M. W., 41, 44 Hirzebruch, Friedrich, 96, 132 Hodge, W. D. V., 258, 259, 264, 265, 273 Hsiung, C. C , 231, 233, 235, 237 Hu, S., 132 Huff, Melvyn, 201

Illusie, L., 132

Janich, K., 45, 53, 93, IK, 752, 156

Karoubi, M., 108, 132 Kelley, J." L., 156 Kervaire, M.,93 Klee, V. L., 93 Kobayashi, S., 40, 266 Kodaira, K., 257, 261, 273 Kohn, J. J., 251,275 Koschorke, Ulrich, 53,93,95, 752, 136,156,301

303

304 AUTHOR INDEX

Kuiper, Nicolass H., 41, 43, 43, 44, 47, 93, 107, 132,156, 157, 765,240,249

Kuranishi, M., 261,275 Kwack, 266

Landweber, P. S., 143, 144, 150, 156 Lang, S., 40, 47, 60, 79, 93, 96, 132, 156, 213,

215,222 Langlands, R. P., 262, 273 Langwitz, D., 282 Lawson, H. Blaine, Jr., 275, 282 Leray, J., 83, 84, 85, 86, 93 Leslie, J., 161,183 Lewy, H., 287, 291 Littlewood, J. E., 237 Lusternik, L., 186,277

McAlpin, J., 136, 755, 157, 165, 213, 217, 222

Mayer, A. L., 256, 273 Mazur, B., 93, 157 Michael, E., 203 Milnor, J., 93,132, 139, 756, 277 Minkowski, 231, 233, 235 Miranda, M., 290 Morava, Jack Johnson, 131, 135 Morrey, C. B., Jr., 10, 278, 282 Morse, Marston, 187, 188, 193, 277 Moser, J., 38, 40, 111, 183, 284 Moulis, Nicole, 41, 44, 93, 157, 765 Mukherjea, K. K„ 43, 44, 93, 136, 215

Nagamo, M., 83, 93 Narasimhan, R., 269, 273 Nash, J., 2, 8, 10, 188 Neubauer, G., 47, 93, 132 Newlander, A., 251,275 Nirenberg, L., 251,275 Nitsche, J. C. C , 288, 297 Nomizu, K., 40 Novikov, S. P., 44, 143, 144, 155, 756 Nussbaum, R., 83, 92

Omori, Hideki, 39, 40,167 O'Neill, B., 240, 247, 248, 249, 249 O'Neill, D., 756 Osserman, Robert, 227, 230, 283, 297

Palais, Richard S., 2, 5, 6, 10, 12, 29, 34, 40, 41, 45, 51, 93, 96, 103, 110, 752, 135, 756, 185, 277

Polya, G., 237 Poritz, Alan B., 293, 298 Porteous, I. R., 95, 125, 755 Puppe, D., 755

Quinn, Frank, 73, 136, 147, 756, 213

Renz, P., 43 Robbin, J., 755, 222 Rossi, H., 254, 273 Rothe, E. H., 45, 93, 211

Sampson, J. H., 183 Sard, 73 Schauder, J., 83, 84, 85, 86, 93 Scherrer, 231, 237 Schmid, W., 257, 262, 265, 273 Schnirelman, L., 186, 277 Schori, R., 41,45 Schwartz, J. T., 40, 209, 272 Segal, G., 135, 136, 756 Seminaire Henri Cartan, 132, 256 Serre,J.-P., 144,756, 188 Shahin, J. K., 231, 233, 237, 237 Shih, W., 755 Siegel, C. L., 254, 273 Simons, J., 225, 284, 297, 298 Smale, S., 41, 54, 73, 74, 86, 93, 95, 130, 755,

135,146,147,756,186,194,277,272,213,222 Solovay, 103 Spanier, E. H., 66, 93 Spencer, D. C , 257, 261, 273 Spivak, M., 756 Stallings, J., 756 Stampacchia, G., 284, 297 Steenrod, N. E., 66, 92, 93, 133, 141 Stiel, E., 239, 240, 242, 248, 249, 249 Stong, R., 756 Svarc, A. S., 45, 53, 93, 103, 755

Tate, J., 267, 273 Taylor, A. E., 84, 93 Thorn, R., 755, 139, 756 Tromba, A. J., 43, 45, 83, 92, 93, 95, 127, 755,

136, 149, 150, 755

Uhlenbeck, Karen, 2, 10, 10

Vainberg, M. M., 85, 93 Verdier, J. L., 756

Warner, F., 239, 299, 301 Weinstein, Alan, 37, 299, 301 Wells, J., 214, 222 West, J. E., 43, 43, 44 Weyl, H., 255, 273 Whitehead, G. W., 756 Whitehead, H.C. , 41,44 Whitney, H., 756 Wood, R., 108, 755 Wu, H., 266, 273

Yoshida, K., 178,183

Zisman, M., 755

SUBJECT INDEX

Adjoint, 26 Algebraic family of algebraic varieties, 262 Analytic space, 254 Atiyah-Palais-Svarc bundle, 104, 105 Automorphic form, 256

Banach bundle complex, 146 Banach space

Grassmannian manifold of, 96 Manifold of split maps, 98

From one Banach space into another, 96 Perturbation class, 46 Simpler, 106 Stable, 107 Uniformly C** smooth, 213

Banach manifold, 15 C-structure, 54

Fredholm maps, 53 Layer structures, 53

Orientations, 149 Bernstein problem, 224 Bernstein's theorem, 284 Bordism functor, 139 Bumpy metric theorem, 301 Bundle

Atiyah-Palais-Svarc bundle, 104, 105 Canonical, 97

y„(£),97,102 Vector, 109

Complex, Banach, 146 GLC-, 111

Stiefel-Whitney-(Pontrjagin, Chern) class of, 127

Stably equivalent, 111, 112 GLC(£)-, 103, 105

Orientable, 128, 130 Orientation, 128

ILH-tangent—, of X, 169 Index, 148 Jet, 15 Layer, 47 Normal, 14, 294

Bundle map Smooth, 31 Vector, 31

C1-diffeomorphism, 29 C*-ILH-manifold, modeled on E, 169

Strong, 169 Calculus of variations, 5 Canonical basis, 253 Cartan-Kahler Theorem, 295 Clifford torus, 275

Codazzi equation, 240 Condition (C), 208 Cone, convex positive, 18 Configuration tensor, 239 Conjugate locus, 299 Conjugate points, 299 Connections, 5

Affine, 19 Critical point theory

Critical values, 189 Method of steepest descent, 188 Regular values, 189

Curvatures, ith elementary symmetric function of the principal, 232

Degenerate Gauss map, 288 Degree theory, 73 Diffeomorphism

C1-, 29 Group of, 11 Orientation preserving, 37 Subgroup of volume preserving, 37

Diffeomorphism group, 15 Differential equations

Bernstein problem, 224 Existence theorem for, 22

Differential operator, 25 Elliptic, 26 ith elementary symmetric function of the

principal curvatures, 232 kth order, 26 kth order elliptic, 26 Laplace Beltrami, 232 Linear, 14 Nonlinear elliptic, 150 Selfadjoint elliptic, 27 With injective symbol, 26

Differentials, holomorphic, 253 Dirichlet integral formula, 231 Domain, fundamental, 271 Dual curve, 263

Elliptic differential operator, 26 kth order, 26 Nonlinear, 150 Selfadjoint, 27

Existence theorem for differential equations, 22 Exponential, 14 Exponential maps, 4, 13, 299

Smooth, 20

Ft(E,Efl 100 Family of supports, 136

305

306 SUBJECT INDEX

Fiber products, 144 Finsler manifold, 200

Well imbedded, 1, 2 Finsler metric, 1, 2 Finsler structures, 1, 200 Flag, 102 Frechet Lie group, 167 Frechet space, 13

Implicit function theorem for, 34 Fredholm complexes, 115 Fredholm maps, 45, 53,130, 146 Fredholm morphism, 104, 126

Characteristic class for, 115 XPt€, 120,121, 122, 123, 125,126

Pontrjagin class off, 131 Stiefel-Whitney-(Pontrjagin, Chern) class,

123,131 Fredholm operators, 45, 101, 146

Space of, 103,107 F0(E% 107

Frobenius theorem, 23 Functor, bordism, 139

Gauss equation, 240 Gauss map, generalized, 285 G-cobordism characteristic class, 143 G-cobordism group, 139 Geodesic reflection, 279 Geodesies, minimizing, 1 GLc-bundle, 111

Stably equivalent, 111, 112 Stiefel-Whitney-(Pontrjagin, Chern) class of,

127 GLC(£), 103, 130 GLc(£)-bundle, 103,105 Global solution, 284 Grassmannian, 109,129 Grassmannian manifolds, 101

%(E), 96 Group

Canonical basis, 253 Diffeomorphism, 15 G-cobordism, 139

Characteristic class, 143 GLC(£), 103, 130 Isometry—, of the metric, 12 Isotropy, 12, 14 kF(X), 113 Monodromy, 264 Primitive part, 258 (Strong) ILH-Lie, 169

Gysin homomorphism, 131,150

H° norm, 21 H° topology, 18, 21 Hs maps, 14, 15 Hilbertian lens spaces, 42 Hilbert manifold, 15, 19

m-function, 157 Hilbert torus, 129

Hodge index theorem, 265 Holomorphic differentials, 253 Hopf-Rinow Theorem, 1, 2 Hull, convex, 277

ILH-Lie group, (strong), 169 ILH-tangent bundle, of X, 169 Implicit function theorem for Frechet spaces, 34 Index bundle, 148 Infinite dimensional manifold, 11, 14 Infinitesimal bilinear relation, 261 Injective immersion, 14 Inner product, strong, 32 Intersection matrix, 253 Isometry

Tangent bundle, 293 Vector bundle, 293

Isotropy group, 12,14 Of the metric, 12

Jet bundle, 15

kF(X% 113 X-theoretical index of/, 131 JC-theory, 110,114 Kuiper space, 107,110,114,126,128

Periodicity, 108

Langlands conjecture, 262 Laplace Beltrami differential operator, 232 Layer bundle, 47 Layer spray, 60 Layer structure, 54 Lebesgue number, 29 Lens spaces, Hilbertian, 42 Lie group, Frechet, 167 Local cross section, 30

Smooth, 24 Lusternik-Schnirelman category, 186

Manifold \,qm 108 ApJE,E% 100,117,129

Cohomology class apq, 117,118, 121 Algebraic, 257 Ck-ILH—, modeled on £, 169

Strong, 169 Complex structures, 251

Integrability condition, 251 Convex positive cone, 18 Ff(£, F ) , 100 Hilbert, 15, 19

m-function, 157 Homogeneous complex, 260 Infinite dimensional, 11, 14 Layer spray, 60 Layer structure, 54 Orientable P-structures, 63 Polarized algebraic, 257 Sobolev, 1 Support function, 232

SUBJECT INDEX 307

Map CMLH-differentiable, 169 Condition (C), 208 Degenerate Gauss, 288 Exponential mapping, 299 Fredholm,45, 130, 146 Generalized Gauss, 285 Geodesic reflection, 279 Hs maps, 14, 15 Injective immersion, 14 Local cross section, 30

Smooth, 24 Parallel immersion, 234 Period mapping, 254, 261, 264 Residue mapping, 258 Sigmaproper, 213 Smooth bundle, 31 Transversal, 145 Tube mapping, 258 Vector bundle, 31

Parallel, 294 Second fundamental form, 294

Matrix Equivalent, 253 Infinitesimal bilinear relation, 261 Intersection, 253 Period, 253 Period mapping, 254, 261, 264

Inversion of the periods, 256 Negatively curved, 266

Riemann bilinear relations, 254 Generalized, 259

Space, 265 Mean curvature vector field, 294 Metrics, space of, 15

Minimax principle, 190, 210 Monodromy group, 264

Normal bundle, 14, 294 Normal deformation, 233 Nullity, spaces of, 240

Orientable P-structures, 63 Orientation preserving diffeomorphism, 37

Parallel immersion, 234 Parallel vector field, 293 Period mapping, 254, 261, 264

Inversion of the periods, 256 Negatively curved, 266

Period matrix, 253 Perturbation class, 46 Picard-Fuchs equations, 262 Picard-Lefschetz transformation, 269 Polarized algebraic manifold, 257 Pontrjagin class of/, 131 Pontrjagin-Thom theory, 86 Primitive cycles, 264 Pseudogradient vector, 205

Rational boundary component, 270 Residue mapping, 258 Ricci curvature, 35

Tensor, 35 Riemann bilinear relations, 254

Generalized, 259 Riemannian geometry, 1

Hopf-Rinow Theorem, 1, 2 Riemannian manifolds, 1, 2, 239

Conjugate locus, 299 Conjugate points, 299 Second fundamental form, 239

Riemannian metrics, set of, 12 Riemannian structure, 11

Invariant, 20 Strong, 14 Weak, 14, 18, 37

Riemann surfaces, algebraic family of, 255

Sard class, 218 Schubert varieties, 125 Siegel generalized upper-half-plane, 254 Sigmaproper, 213 Slice, 32, 33 Smale degree, 131, 132,135 Sobolev lemma, 15 Sobolev manifolds, 1 Sobolev space, 14 Spaces of nullity, 240 Stable equivalence, 114 Stiefel-Whitney-(Pontrjagin, Chern) class, 123,

131 OfGLc-bundle, 127

Subbundle, smooth involutive, 23 Submanifold

Minimal, 275 Second fundamental form, 276

Support function, 232 Surface

Constant mean curvature in R3, 281 Minimal, 275

Isolated singularities, 286

Tangent bundle isometry, 293 Tangent bundle, of AT, ILH-, 169 Tensor, configuration, 239 Topology of uniform C* convergence, 22 Torelli theorem, 256 Torus, Clifford, 275 Tube mapping, 258

Vanishing cycles, 268 Variational problem, 293 Varieties, minimal, 293 Vector bundle isometry, 293 Vector bundle neighborhoods, 15 Vector field

Mean curvature, 294 Parallel, 293

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