DIFFERENTIAL GEOMETRY

http://dx.doi.org/10.1090/pspum/027.1

PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS

VOLUME XXVII, PART 1

DIFFERENTIAL GEOMETRY

AMERICAN MATHEMATICAL SOCIETY

PROVIDENCE, RHODE ISLAND

1975

PROCEEDINGS OF THE SYMPOSIUM IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY

HELD AT STANFORD UNIVERSITY

STANFORD, CALIFORNIA

JULY 30-AUGUST 17, 1973

EDITED BY

S. S. CHERN and R. OSSERMAN

Prepared by the American Mathematical Society with the partial support of National Science Foundation Grant GP-37243

Library of Congress Cataloging in Publication Data

Symposium in Pure Mathematics, Stanford University, 1973. Differential geometry.

(Proceedings of symposia in pure mathematics; v. 27, pt. 1-2)

"Final versions of talks given at the AMS Summer Research Institute on Differential Geometry."

Includes bibliographies and indexes. 1. Geometry, Differential-Congresses.

I. Chern, Shiing-Shen, 1911- II. Osserman, Robert. III. American Mathematical Society. IV. Series. QA641.S88 1973 516'.36 75-6593 ISBN0-8218-0247-X(v. 1)

Copyright © 1975 by the American Mathematical Society

Printed in the United States of America

UE

All rights reserved except those granted to the United States Government. This book may not be reproduced in any form without the permission

of the publishers.

CONTENTS

Preface ix

Riemannian Geometry

Deformations localement triviales des varietes Riemanniennes 3 By L. BERARD BERGERY, J. P. BOURGUIGNON AND J. LAFONTAINE

Some constructions related to H. Hopf's conjecture on product manifolds . . . 33 By JEAN PIERRE BOURGUIGNON

Connections, holonomy and path space homology 39 By KUO-TSAI CHEN

Spin fibrations over manifolds and generalised twistors 53 By A. CRUMEYROLLE

Local convex deformations of Ricci and sectional curvature on compact manifolds 69

By PAUL EWING EHRLICH

Transgressions, Chern-Simons invariants and the classical groups 72 By JAMES L. HEITSCH AND H. BLAINE LAWSON, JR.

A class of compact manifolds with positive Ricci curvature 73 By HORACIO HERNANDEZ-ANDRADE

Tangent bundles with Sasaki metric 89 By MU-CHOU LIU

Curvature and critical Riemannian metric 95 By Yosio MUTO

Axioms for the Euler characteristic 99 By HOWARD OSBORN

Riemannian manifolds without conjugate points 103 By JOHN J. O'SULLIVAN

Local and global properties of convex sets in Riemannian spaces 105 By ROLF WALTER

v

VI

On the volume of manifolds all of whose geodesies are closed 107 By ALAN WEINSTEIN

Submanifolds

On a generalization of the catenoid 113 By DAVID E. BLAIR

Geometric applications of critical point theory to submanifolds of complex projective space and hyperbolic space 115

By THOMAS E. CECIL

Mean curvature vector of a submanifold 119 By BANG-YEN CHEN

Geometry of submanifolds of Euclidean spaces 125 By ROBERT B. GARDNER

The Hopf conjecture concerning surfaces in E3 135 By S. I. GOLDBERG

Relative Chern-Lashof theorems 137 By NATHANIEL GROSSMAN

A Sobolev inequality for Riemannian submanifolds 139 By DAVID HOFFMAN AND JOEL SPRUCK

Minimal varieties* 143 By H. BLAINE LAWSON, JR.

On a holomorphic analogue of vanishing normal scalar curvature 177 By GERALD D. LUDDEN AND KOICHI OGIUE

Complete, open surfaces in E3 183 By TILLA KLOTZ MILNOR

Minimal varieties in tori 189 By TADASHI NAGANO AND BRIAN SMYTH

Elie Cartan's work on isoparametric families of hypersurfaces 191 By KATSUMI NOMIZU

On Kaehler submanifolds 201 By KOICHI OGIUE

Isoperimetric and related inequalities 207 By ROBERT OSSERMAN

Minimal submanifolds with w-index 2 217 By TOMINOSUKE OTSUKI

A problem of Ordnungsgeometrie 229 By WILLIAM F. POHL

On the Hessian of a function and the curvatures of its graph 235 By ROBERT C. REILLY

Pairs of metrics on parallel hypersurfaces and ovaloids 237 By DONALD H. SINGLEY

Some left-over problems from classical differential geometry 245 By MICHAEL SPIVAK

•General lecture given at the Institute.

vii

Manifolds and submanifolds with vanishing Weyl or Bochner curvature tensor 253

By KENTARO YANO

Foliations

On the de Rham complex of BT 265 By B. CENKL

Locally free Lie transformation groups of codimension two 275 By LAWRENCE CONLON

On compact foliations 277 By K. DECESARE AND T. NAGANO

Semisimplicial Weil algebras and characteristic classes for foliated bundles in Cech cohomology 283

By FRANZ W. KAMBER AND PHILIPPE TONDEUR

Line fields transversal to foliations 295 By ULRICH KOSCHORKE

Classifying spaces for Riemannian foliations 303 By JOEL PASTERNACK

Codimension one plane fields and foliations 311 By PAUL A. SCHWEITZER, S. J.

The double complex of rk 313 By HERBERT SHULMAN

A local construction of foliations for three-manifolds 315 By WILLIAM P. THURSTON

The theory of foliations of codimension greater than one 321 By WILLIAM P. THURSTON

Locally free actions and Stiefel-Whitney numbers 323 By H. B. WlNKELNKEMPER

Algebraic and Piecewise Linear Topology

Stiefel-Whitney homology classes and singularities of projections for polyhedral manifolds 333

By THOMAS F. BANCHOFF

Axioms for characteristic classes of manifolds 349 By JOHN D. BLANTON AND PAUL A. SCHWEITZER, S. J.

On the Gelfand-Fuks cohomology* 357 By RAOUL BOTT

On H* (GStq; R/Z), range group of some Chern-Simons invariants 365 By JOHN S. MAFFEI

Euler singularities and homology operations 371 By CLINT MCCRORY

Isolated critical points of complex functions 381 By JOHN MILNOR

•General lecture given at the Institute.

viii

The Chern-Weil construction 383 By HOWARD OSBORN

Characteristic classes and singularities of mappings 397 By RICHARD PORTER

Miscellaneous

Riemannian submersions from spheres 405 By RICHARD H. ESCOBALES, JR.

Some open problems in differential geometry 407 By LEON GREEN

Some geometrical aspects of geodesy 413 By NATHANIEL GROSSMAN

The index theorem for closed geodesies 415 By W. KLINGENBERG

The topology of the solutions of a linear homogeneous differential equation on Rn 417

By NICOLAAS H. KUIPER

Unique structure of solutions to a class of nonelliptic variational problems . .419 By JEAN E. TAYLOR

Twist invariants and the Pontryagin numbers of immersed manifolds 429 By JAMES H. WHITE

Indexes

Author Index 441 Subject Index 447

Preface

The papers in these PROCEEDINGS represent the final versions of talks given at the AMS Summer Research Institute on Differential Geometry, which took place at Stanford University, Stanford, California, from July 30 to August 17, 1973. This Institute was made possible by a grant from the National Science Foundation. The organizing committee consisted of Raoul H. Bott, Eugenio Calabi, S. S. Chern, Leon W. Green, Shoshichi Kobayashi, Tilla K. Milnor, Barrett O'Neill, Robert Osserman, James Simons, I. M. Singer, with the coeditors serving as cochairmen.

The activities were divided between general lectures and seminar talks. In these PROCEEDINGS the general lectures have been distributed among the various sem­inars, according to their subject matter. Each part of the PROCEEDINGS consists of a group of seminars, whose titles and chairmen are as follows:

Part 1: Riemannian geometry (J. Cheeger) Submanifolds (K. Nomizu) Foliations (B. L. Reinhart) Algebraic and piecewise-linear topology (T. F. Banchoff and H. R. Gluck) Miscellaneous (B. O'Neill and J. Simons)

Part 2: Complex differential geometry (S. Kobayashi) Partial differential equations (J. L. Kazdan and F. W. Warner) Homogeneous spaces (J. Wolf) Relativity (T. Frankel)

Generally papers are included in the seminars in which they were presented, although in certain cases the contents would make them more appropriate in another section. In cases where a complete version of the talk appears elsewhere, only an abstract is included here, together with a reference to the full paper.

ix

X

A list of open problems submitted by participants was compiled by Leon Green. These are included at the end of Volume I in the Miscellaneous Section.

We should like to thank the seminar chairmen, and also the secretarial staff: Dorothy Smith and Muriel Toupin of AMS as well as Catherine Lowe and Eliz­abeth Plowman of the Stanford Mathematics Department, all of whose tireless efforts were a large factor in the success of the Institute.

S. S. CHERN

ROBERT OSSERMAN

JANUARY 1975

INDEXES

AUTHOR INDEX

Italic numbers refer to pages on which a complete reference to a work by the author is given. Roman numbers refer to pages on which a reference is made to a work of the author.

For example, under Oppenheim would be the page on which a statement like the following occurs: "This result had been conjectured by Oppenheim [13] in 1929, ..."

Boldface numbers indicate the first page of the articles in this volume.

Abraham, R., 236 Ahlfors, L. V., 27, 31 Akin, E., 373, 381 Alexander, H., 212, 213, 214, 215 Allard, W. K., 146, 153, 155, 158, 159, 173,

211, 212,215,275 Allendoerfer, C. B., 102 Almgren, F, J., Jr., 144, 146, 153, 156, 165,

172,173, 211, 215, 215, 420, 427 Anderson, D. W., 362, 364, 364 Anderson, P. G., 298, 300 Arnol'd, V. I., 381, 382, 382 Artin, M., 382, 382 Asada, A., 52 Asimov, Daniel, 316 Atiyah, M. F., 29, 30, 31, 31, 58, 67, 285, 293 Aubin, T., 69, 71 Avez, A., 106, 106

Banchoff, Thomas F., 333, 339, 340, 342, 346, 347,371, 372, 375, 376, 380, 397,402

Barbance, Ch., 67 Barbosa, J. L., 161, 162, 173 Baum, P., 303, 306, 309 Beckenbach, E. F., 208, 212, 215 Berger, M., 3, 13, 20, 23, 24, 26, 26, 34, 36, 37, 37, 98, 99,100, 409

Bergery, L. Berard, 3,4 Bernstein, S. N., 143, 149, 150, 173, 236, 250,

268 Bianchi, L., 248, 249,250,251 Bieberbach, L., 245, 251 Blair, David E., 113 Blanton, John D., 349, 349, 356 Bleeker, D., 186, 186 Bochner, S., 254, 261, 262 Bombieri, E., 154, 156, 173 Bonnesen, T., 245, 251 Bonnet, 102, 203, 247, 407, 408 Boothby, W. M., 175 Bott, Raoul, 29, 30, 31, 31, 41, 52, 59, 67, 266,

268, 274, 274, 280, 281, 291, 292, 293, 293, 303, 305, 306, 307, 308, 309, 313, 314, 314, 324,357, 357, 364, 364, 384,595

Bourbaki, N., 382 Bourguignon, Jean Pierre, 3, 33, 34, 35, 36, 37 Bouzon, J., 62 Brieskorn, E., 86,87, 382,382 Brown, E. H., Jr., 39,40,47,52 Buoncristiano, S., 380, 380

Cairns, S. S., 375, 380 Calabi, E., 3,4, 7,12, 18, 19, 23, 26, 27, 31, 38,

161, 162,173, 204, 205, 251, 409

441

442 AUTHOR INDEX

Carleman, T., 208,215 do Carmo, M., 127,130,134,162,163,164,173

226,245,251 Cartan, Elie, 14, 130, 131, 134, 191, 195, 196,

197, 199, 200, 200, 239, 285 Cartan, H., 293 Cecil, Thomas E., 115 Cenkl, B., 265, 268, 274 deCesare, K., 277 Chavel, I., 410 Cheeger, J., 35, 36, 37, 37, 73, 74, 87, 107,108,

340,347,349,356,311,380 Chen, Bang-Yen, 119, 120, 121, 123, 182,

255, 258, 259, 261, 262 Chen, Kuo-Tsai, 39, 40, 42, 44, 46, 48, 49, 50,

52 Chern, S. S., 33, 37, 37, 41, 52, 72, 102, 127,

128, 129, 130, 131, 134, 135, 137, 151, 152, 161, 164, 173, 226, 235, 236, 294, 306, 365, 367, 370, 383, 384, 395, 400, 402, 408, 431, 437

Chevalley, C , 53, 57, 61, 62, 64, 67 Choquet-Bruhat, Y., 55, 67 Chow, 169, 231 Civita, 195 Codazzi, 200 Cohn, 407 Conlon, Lawrence, 275 Conner, P. E., 298,299,300, 323,324,330 Courant, R., 150, 173, 212, 215 Crumeyrolle, A., 53, 53, 54, 55, 56, 57, 60, 61,

62, 65, 67, 67

Deschamps, A., 34, 35, 36,37 Dieudonne, J., 14, 25, 26 Dombrowski, P., 76, 87, 94 Douglas, J., 149, 150, 154, 173

Earle, C. J., 25, 26, 26, 27 Eberlein, P., 409 Ebin, D. G., 3, 4, 13, 20, 26, 35, 36, 37, 98,

99, 100 Eells, J., Jr., 25, 26, 26, 27, 120, 189, 190,

200 Efimov, N. V., 183, 184, 185, 186, 408 Ehresmann, Charles, 370 Ehrlich, Paul Ewing, 69, 69, 71 Eliasson, H., 94 Epstein, D. B. A., 277, 281, 315, 319, 380 Erbacher, J. A., 121,123, 111, 182 Escobales, Richard H., Jr., 405, 405 van Est, W. T., 273, 292

Federer, H., 145, 146, 153, 154, 155, 157, 158, 165, 167, 168, 170, 172, 173, 214, 215, 420, 421,422,427

Fenchel, W., 245, 251 Ferus, D., 120, 123, 245, 251 Flanders, H., 235, 236, 236 Fleming, W. H., 153, 155, 157, 158, 165, 173,

174, 212,214,215 Floyd, E. E., 298, 299, 300, 323, 324, 330 Freedman, M. H., 411 Freudenthal, H., 410 Frolicher, A., 41, 52 Fuks, D. B., 267, 274, 314, 357, 358, 361, 362,

363, 364,364 Funk, 109, 409

Gardner, Robert B., 125, 133, 134, 240, 242, 243

Gauduchon, P., 13, 23, 26,26, 37,37 Gauss, K. F., 102,125,130, 200, 203, 247,407 Gelfand, I. M., 267, 274, 314, 357, 358, 361,

362,363,364, 364,421,427 deGiorgi, E., 153, 154, 156, 157, 173 Giusti, E., 154, 156,173 Godbillon, C , 291, 294, 307, 308, 309 Godement, 285 Goldberg, S. I., 135, 256, 262 Graev, M. I., 421, 427 Green, Leon W., 407, 409 Griffiths, P. A., 4, 7, 11, 12,12, 23, 26, 26 Gromoll, D., 74, 86, 87, 107, 108, 407, 409,

409 Gromov,311,321,410 Grossman, Nathaniel, 137,413 Grove, K., 409, 409 Guillemin, V., 266,267,274,357,363,364 Gulliver, R., 150,774,411 Gunning, R. C , 25, 26, 27, 28, 30, 31

Haefliger, Andre, 265, 268, 269, 274, 274, 291, 293, 294, 303, 305, 309, 311, 314, 321, 321, 357,364,411

Halperin, S., 313, 340, 345, 347, 349, 356, 376, 380

Hamilton, R. S., 28, 31 Hardy, G. H., 243, 243 Hartman, P., 246, 251 Harvey, R., 143, 169, 170, 171, 174 Heitsch, James L., 72 Helgason, S., 410 Herbert, R., 437 Herman, M„ 315, 319

AUTHOR INDEX 443

Hermann, R., 67 Hernandez-Andrade, Horacio, 73 Hicks, Noel J., 238, 239, 243 Hilbert, D., 183, 242, 245, 251, 408 Hildebrandt, S., 150,174 Hilton, P. J., 350, 356 Hirzebruch, F., 349, 356, 382, 382 Hitchen, N., 86, 87 Hochschild, G., 266,273,274, 292,294, 314 Hoffman, David A., 120, 123, 128, 134, 139,

212,213,214,275 Holmgren, E., 245, 246, 251 Hopf, E., 102, 106,106, 182 Hopf, H., 33, 37, 119,123 Hsiang, Wu-yi, 145, 160, 162,174 Hsiung, C. C , 208, 275, 242, 407 Huber, A., 212, 275

Ishihara, S., 121, 725, 177, 182, 256, 262 Itoh, T., 226

Jacobson, N., 384, 385, 388, 389, 395 Jorgens, K., 249, 250, 257

Kamber, Franz W., 40, 52, 283, 283, 284, 285, 286, 290, 291, 292, 294

Karras, U., 298, 300 Kato, T., 31 Kaul, H., 208, 211,275 Kazdan, J. L., 71, 407 King, J., 168,174 Kirby, 280 Klingenberg, W., 74, 87, 415, 475 Klotz, T. (see also Milnor, T. K.), 119, 723,

185,187 Knapp, A., 143, 170, 171,174 Knutson, 408 Kobayashi, S., 33, 37, 89, 90, 92, 93, 94, 95,

164,173, 202, 205, 226, 306, 309 Kodaira,K.,5,ll,12,72,202 Koschorke, Ulrich, 295, 295, 299, 300, 300,

411 Kosmann, Y., 59,67 Kostant, B., 60 Kowalski, O., 95 Kreck, M., 298, 300 Kuiper, Nicolaas H., 130,131,134,417,418 Kulkarni, R. S., 255, 262 Kumpera, A., 266, 274

Lafontaine, J., 3,3,12,27 Lashof, R. K., 129,134,137,436, 437, 437

Lawson, H. Blaine, Jr., 72,143, 145, 147, 148, 149, 150, 152, 154, 155, 159, 160, 161, 162, 164,166,167,168,169,174, 251, 300,411

Leung, D. S., 191 Levi, 195 Lichnerowicz, A., 54,59,61,67 Lickorish, W. B. R., 315,319 Liebmann, H., 119, 121, 245, 257 Lima, E., 127, 130,134, 275, 276 Littlewood, J. E., 243, 243 Liu, Mu-Chou, 89, 89, 92, 95 Liu, R. C , 254, 262 Lovelock, D., 236, 236 Ludden, Gerald D., 723,177,182

MacPherson, Robert, 334, 397 Maeda, Y., 257, 262 Maffei, John S., 365 Martens, H. H., 28, 37 Marussi, Antonio, 413 Massey, W. S., 246, 257 Mather, J., 315, 317, 379, 364, 364, 373, 376,

380 Matsushima, Y., 190,190 Mazet, E., 13,23,26,26, 37,37 McCrory, Clint, 334, 347,347,371, 397 Meusnier, 230 Meyer, W., 33, 74, 85, 86, 87, 408, 409, 409 Michael, J. H., 140,141, 211, 212, 214, 275 Millett, K., 277, 280 Milnor, John W., 33, 37, 74, 79, 87, 87, 130,

134, 183, 184, 185, 186, 190, 190, 251, 349, 356,313,380,3S1,382

Milnor, Tilla Klotz (see also Klotz, Tilla), 183,183,185,186,757

Minkowski, 245 Mizutani, T., 298, 300, 300 Montgomery, D., 279, 281 Moore, J. D., 132,134, 245, 257 Moser, J., 132,734,315 Mostert, P., 328 Mostow, G. D., 266,273,274,292,294 Munzer, H. F., 200,200 Muto, Yosio, 97,97,99,100

Nagano, Tadashi, 189, 277 Nakagawa, H., 204, 205 Narasimhan, R., 169,174 Nash, J., 132,134 Neumann, W. D., 298, 300, 328, 330 Nijenhuis, A., 41, 52 Nirenberg, L., 5, 12, 72, 246, 257

444 AUTHOR INDEX

Nishikawa, S., 257, 262 Nitsche, J. C. C , 150, 174, 211, 215, 249, 252,

411,411 Noether, G., 28, 31 Nomizu, Katsumi, 33, 37, 89, 90, 92, 94, 95,

115, 116, 777, 121, 123, 177, 752, 191, 191, 198,199,200, 203,205, 306,309

Novikov, S. P., 275, 276, 315, 319

Ochiai, T., 202,205 Ogiue, Koichi, 177, 179, 182, 182, 201, 202,

203, 204, 205,410 Omari, H., 174 O'Neill, B., 179,180,752 Orlik, P., 85, 87 Osborn, Howard, 101, 102, 103, 383, 390, 395 Ossa, E., 298,300 Osserman, Robert, 119,123,150,151,152,173,

174, 175, 185, 186, 757, 207, 212, 213, 214, 275,411

O'Sullivan, John J., 105,105,106 Otsuki, Tominosuke, 217, 217, 218, 219, 220,

221,223,22(5,227,409

Palais, R. S., 59,67,198,200 Pasternack, Joel S., 280,281,303,304,306,310,

324, 330, 330 Penrose, R., 64, 65, 66, 67 Phillips, A., 311,572, 321 Pitts, John, 146 Plateau, 149, 153,411 Pogorelov, A. V., 184,757,246,252 Pohl, William F., 229, 229, 232, 233, 429, 430,

434,436,437 Poincare, H., 4,10,16,20 Polya, G., 243,243 Pontrjagin, L. S., 339,347, 365, 366,370, 375 Poor, W. A., 108 Poritz,A.B.,411 Porter, Richard, 334, 339,347,397 Preissmann, A., 34,37

Rado, T., 149,150,775,208,212,275 Raghunathan, M. S., 11,72 Rauch,H.E.,27,30,31,37 Raymond, B., 312,372 Reeb, G., 129,277,257,315,317,319,319 Reid, W. T., 208,275 Reidemeister, K., 107,108 Reifenberg, E. R., 155, 775 Reilly, Robert C , 235, 236, 408 Reinhart, B. L., 102,103, 304,310,411,411

deRham,G., 155,775 Robbin, J.W., 475 Rodriguez, L., 115,116,777 Rosenberg, H., 275,276,318,330 Rossi, H., 773 Rourke, C. P., 379,380 Roussarie, R., 328,330 Rozenfeld, 268 Ruh, E. A., 120, 723,189,190 Rund, H., 236 Ryan, P. J., 256,262

Sacksteder, R., 275,276 Sampson, J. H., 120, 189,190 Sanderson, B. J., 379, 380 Sard, A., 129, 157 Sario, L., 27, 37 Sasaki, S., 62, 89, 95, 250 Satake, I., 277, 280, 257 Sato, S., 261, 262 Schneider, R., 241 Schouten, J. A., 247, 253, 257, 262 Schweitzer, Paul A., S. J., 311, 311, 372, 349,

349,356,410 Segal, G., 265,268,269,274,357 Segre, B., 195 Seifert, R., 346 Sentenac, P., 34, 35, 36, 37 Sergeraert, F„ 315, 379 Shapiro, A., 58, 67 Shiftman, B., 169,174,211,275 Shiohama, K., 122, 723 Shulman, Herbert, 274,274, 292,294, 306,310,

313, 314 Siebenmann, 280 Sim6es,P., 154,168,775 Simon, L. M , 140,141,211,212,214,275 Simon, U., 236 Simons, J., 72,147,153,156,157,163,164,166,

169,174,175,294 Singer, I. M., 4,14,26 Singley, Donald H., 237, 238, 239, 242, 243 Smale, S., 436,437,437 Smith, B., 723 Smith, P. A., 380 Smith, Ted, 200 Smyth, Brian, 116, 777,120,121, 723,177, 752,

189,203,205 Spanier, E. H., 400,402 Spencer, D. C , 5,11,12,72,266,274 Spivac, Michael, 14, 15, 21, 26, 245, 245, 246,

247,248,250,251,252

AUTHOR INDEX 445

Spruck, Joel, 139 Stasheff, J. D., 40, 52, 265, 270, 274, 274, 313,

314, 349,356, 373,380 Steenrod, N. E., 67,378,380 Sternberg, S., 4,14,26,266,267,274 Stiefel, E., 340, 347, 349, 356 Stoker, J. J., 246,252 Stokes, 241,242,436 Stong, R., 299,300, 324,330 Sullivan, D., 40,52, 371, 373, 379, 380 Synge, J. L., 33,143, 250

Tachibana, S., 254, 256, 262 Takagi,H., 254,262 Takagi, R., 122,123,191,196,200,200 Takahashi, T., 162,175, 191, 196, 200 Tamura, I., 301 Tani, M., 256,262 Tanno, S., 89,94,95,204,205 Taylor, Jean E., 419,419,420,427 Teichmuller, 26 Thorn, Rene, 229, 233, 339, 347, 371, 373, 375,

378,379, 380,380, 381,382, 397,402 Thorpe, J. A., 233, 233 Thurston, William P., 295, 296, 299, 300, 301,

311, 312, 312, 315, 315, 316, 317, 319, 321, 321,527,364,410

Tits, J., 410 Tjurina, G. N., 382,382 Toledo, Domingo, 340, 345, 347, 349, 356, 372,

376, 380 Tompkins, C , 131,754 Tondeur, Philippe, 40, 52, 283, 283, 284, 285,

286, 290, 291, 292, 294 Torelli, 4, 27, 28 Trauber, P., 357, 363, 364 Trudinger, N., 213, 215 Tsagas, G., 410 Tsukamoto, Y., 181,182

Varma, S., 419 Vey, J., 291, 294, 307, 308, 309 Vilenkin, N. Ja., 421,427

Vilms,J., 120,725,189,790 Vladimirova, S. M., 250, 252 Volkov, Ju. A., 250, 252 Vossen, 407

Wagreich, P., 382 Wallach, N. R., 162, 163, 772, 175 Walter, Rolf, 107, 107, 108,108 Wang, H. C , 410 Warner, F. W., 13, 20, 21, 27, 30, 71, 407 Watanabe, Y., 254, 262 Watts, C. E., 102,103 Weil, Andre, 4, 11, 72, 27, 28, 32, 40, 306, 328,

330, 383 Weiner, J., 251 Weinstein, Alan, 34, 37, 37, 71, 71, 109, 409,

410 Weiss, 175 Weyl, H., 253, 262 White, James H., 429, 429, 430, 435, 436, 437 Whitehead, J. H .C. , 51,52 Whitney, H., 340, 347, 349, 356, 376, 380 Whitney, W., 175 Winkelnkemper, H. E., 323 Willmore, T. J., 121, 122, 725 Wolf, J. A., 7, 72, 14, 23, 26, 410 Wood, John W., 312,572,315,579 Wood, R., 200 Wu, Wen-Tsun, 324,330,370, 380,380 WulfT, G., 427,427 Wunderlich,W., 251,252 Wylie, S., 350,356

Yamabe, 408 Yamaguchi, S., 261, 262 Yano, Kentaro, 89, 93, 94, 95, 121, 725, 177,

182,253,254, 256,258,259,261,262 Yau, S. T., 71, 71,120, 725,164,175

Zieschang, H., 315 Zippin, L., 279, 281 Zoll, 109

SUBJECT INDEX

Action of the diffeomorphism group, 35 slice, 35

Affine transformation groups, 89, 93 Algebraic curve, 229 Allendoerffer-Weil formula, 107 Almost-cocomplex structures, 62 Almost-complex structure, 59, 62 Amitsur complex, 284 Analytic quadratic differential, 120 Anchor ring, 122

Basic connection, 305 Bianchi identity, 387 Bi-invariant metric connection, 72 Bochner curvature tensor, 254 Bordism groups

of foliated manifolds, 296 of manifolds with g-plane fields, 296

Brieskorn variety, 79, 80

Calculus of variations, 419 Canonically associated spin fibrations, 65 Canonical Riemannian connection, 305 Catastrophes, 381 Category, 265 Category M, 350 Characteristic classes, 397 Characteristic number obstructions, 324 Chern class, 355 Chern homology class, 355 Chern-Simons invariants, 72 Chern-Weil construction, 283 Chern-Weil theorem, 102, 384 Classifying spaces, 303

Br\ 265 for Rrq -structures, 305

Closed geodesic, 415 index of, 415

Closed leaf, 312 Cohn-Vossen inequality, 107 Complete intersection of hypersurfaces, 203 Complete manifolds, 107 Complex gradient, 78 Complex hessian, 79 Complex projective space, 115 Complex quadric, 117 Complex space form, 201 Cone, 154 Conformal Gauss equations, 259 Conformal groups, 67 Conformally flat Riemannian manifold, 253 Conjugate points, 105 Connection, 102, 384, 385, 387 Connection map, 89 Constant curvature, 3

complete surfaces of, 245-251 Constant mean curvature, 192 Constant principal curvatures, 193 Convex hull property, 213 Convex sets in Riemannian spaces, 107 Counter-orientation, 317 Critical point, 115, 381 Critical Riemannian metric, 98 Currents

representable by integration, 170 of type (p, p)t 168

Curvature, 102, 384, 385 Curvature tensor, 74

447

448 SUBJECT INDEX

Cut locus, 139

Deformations, 3 of metrics, 34

Density function, 155 Derivation, 384 Difference tensor, 133, 239

rank of, 240 Distance function, 107 Dual dimension, 129 Dual Pontrjagin classes, 354 Dual surface, 231

Ellipticity, 420 Enlarged group of spinoriality, 62, 63 Euler characteristic, 101 Euler class, 306, 390 Euler complex, 373 Euler singularity, 373 Euler space, 371, 373 Exotic C-characteristic classes, 306

f(z)=ZUzVlak(ak-l), 80 Fiber-preserving affine transformation, 94 File of M, 275 First derived characteristic homomorphism,

285 First osculating space of M, 127 Flat connection, 283 Focal point, 105, 116, 197 Focal variety, 198 Fold, 232 Foliated bundles, 283, Foliated manifolds

bordism groups of, 296 Foliations, 198, 277, 295, 311

parallel along 3, 275 P(3), 275

Full map, 372

*(*)=H;«i ****/(«*-1), 80 / \ , 313 Afoliation, 268 Gauss-Bonnet formula, 119 Gauss-Bonnet theorem, 102 Gauss curvature, 119

in the second fundamental form metric G', 240

Gauss curvature equation, 159 Gauss equations, 76,130, 257, 258 Gauss map, 128 Gelfand-Fuks cohomology, 314

Generalized characteristic homomorphism, 283

Generalized twistors, 64 Generalized Wirtinger inequality, 171 General variational problems, 420 Geodesies

closed, 109 Geometric homology operation, 379 Gradient flows, 129 Groups of spinoriality, 56, 57, 58, 62

Haefliger structure, 299 Hausdorff/7-measure, 145 Hessian, 126 Hessian tensor, 79 Hodge operator, 102, 389 Holomorphic analogue of vanishing normal

scalar curvature, 177 Holonomy groups of TM, 89 Homology operation, 378 Hyperbolic space, 116

metric spheres in, 116 Hypercohomology, 290 Hyperplane sections, 190 Hypersurfaces

complete intersection of, 203 isoparametric family of, 191 parallel family of, 191

Imbedding problem, 125 Immersion, 372 Index of closed geodesic, 415 Induced metric, 125 Infinitesimal affine transformation, 93, 94 Infinitesimal isometries, 93, 94 Integers localized away from 2, 354 Integral formulas, 241 Integrals-currents, 153 Integral varifold, 146 Intersection product, 351 Irreducible complete Riemannian manifold,

93 Irreducible, complete nonflat Riemannian

manifold, 89 Irreducible nonflat Riemannian manifold, 92,

94 Isometric immersions, 108 Isometry group, 93 Isoparametric family of hypersurfaces, 191 Isoperimetric inequality, 140, 207, 214 Isotropic twistors, 65

SUBJECT INDEX 449

Kahlerian Ricci flat manifold, 60 Kahler manifold, 166 Kahler submanifold, 201 Killing vector field, 3, 105

Lagrangian bundles, 415 Laplacian of mixed invariants, 242 Laplacian operators, 126, 241 Level hypersurfaces, 193 Lie algebras

reductive pair of, 288 Lie derivative, 4 Line field, 295 Linearly equivalent orbit system, 417 Lipschitz-Killing curvature, 129 Local connector, 89 Local convex deformations, 70 Locally free Lie transformation group of M

of dimension n — 2, 276 Locally free manifold, 324

Mainardi-Codazzi equations, 159 Manifolds

file of M, 275 locally free, 324 rank of A/, 275

Manifolds with #-plane fields bordism groups of, 296

Mass, 144 Mass-norm, 170 Massey triple product, 306 Mean curvature, 119, 139 Mean curvature vector, 120, 127 Mean curvature vector field, 148 Metrics, 102

mixed invariants, 238 pair of, 237

Metric projection, 107 Metric spheres in hyperbolic space, 116 Minimal immersion, 148 Minimal submanifold, 120, 148, 189, 199, 214 Minimal surfaces, 208 Minimal variety, 144 Mixed invariants of a pair of metrics, 238 Model deformation, 300 Morse function, 115

Natural sum, 101 it-Circuit, 373 Nonelliptic variational problems, 419 Nonflat Riemannian manifolds, 94 Nonnegative curvature, 107

Normal mappings, 129 Normal space, 75 Normal vector valued quadratic differential

form, 127 rank of, 127

Odd dimensions, 62 Orbit system, 417

linearly equivalent, 417 topologically equivalent, 417

Oriented /̂ -rectifiable set, 152 Orthogonal complex gradient, 78 Orthogonal intersection, 80 Orthonormal basis, 76 Outer parallel set, 107 Ovaloids, 237

Parallel family of hypersurfaces, 191 Parallel hypersurfaces, 237 Pencil of symmetric bilinear forms, 132 Pfaffian, 102, 390 Pin (2, 53 Pin g-spin structure, 58

in the broad meaning, 62 strict meaning, 53, 56, 57 trivial, 55

Pin Q'-spin structure, 61 Poincare map, 415 Pontrjagin class, 354, 384, 398 Pontrjagin ring, 306 Positional sectional curvature, 33 Positive current, 168 Positive (/?, p)-current, 171 Positive vector, 170 />-Rectifiable set, 145 Prescribed volume problem, 419,420

direct method of solution, 419 Principal curvatures, 192 Product manifolds, 33 Profile, 372, 376 Projective space bundle, 296 Pseudogroup, 266

Quasi-umbilical point, 258

RG' = the Gauss curvature in the second funda­mental form metric G\ 240

U/^-structures, 303 Real grassmannian, 365 "Real" Witt's frame, 54 Rectifiable current, 153 Rectifiable set, 145

450 SUBJECT INDEX

Rectifiable varifold, 146 Reeb foliation, 299 Resolution, 4 de Rham cohomology, 271 de Rham complex, 266, 271 de Rham decomposition, 92 Ricci curvature, 74, 190 Ricci flat, 61 Ricci tensor, 74 Riemannian foliation, 303 Riemannian manifold, 105

the center of * i(Af), 105 conformally flat, 253 irreducible, complete, nonflat, 89, 92, 93, 94 with no conjugate points, 105 with no focal points, 105

Riemannian product, 132 Riemannian spaces

convex sets in, 107 Riemannian submanifolds, 139 Riemannian submersion, 36 Riemann surface, 4 Rigidity problem, 125 r-Isotropic vector field, 56

Sasaki connection, 89 Scalar curvature, 75 Scalar normal curvature, 177 Schubert decomposition, 365 Secant lines, 232 Secant spaces, 429 Secondary characteristic classes, 286 Second fundamental form, 107,147 Second fundamental tensor, 75 Sectional curvature, 74 Semisimplicial g-Z>(7-algebras, 285 Semisimplicial groupoid, 268 Semisimplicial methods, 283 Semisimplicial space, 271 Simons characters, 72 Simple group, 315 Singularity, 375, 378, 382 Slice, 35 Smith operations, 380 Sobolev inequality, 139 Spin fibrations

canonically associated, 65 Spinor bundle, 54 Spinoriality

enlarged group of, 62,63 groups of, 56, 57, 58, 62

Stable variety, 145

Stably tangential map, 352 Steenrod cohomology operation, 378 Stiefel class, 350 Stiefel homology class, 351 Stiefel-Whitney class, 375, 378, 397 Stiefel-Whitney homology classes, 371 Stake's Theorem, 241 Strict meaning Pin G-spin structure, 53, 56,

57 Submanifolds,

of complex projective space, 116 of hyperbolic space, 116

Symmetic bilinear forms, pencil of, 132

Symplectic spinors, 60

Tangent cones, 155 Third product, 391 Topological groupoid, 266

r \ 265 Topological^ equivalent orbit system, 417 Torelli theorem, 4 Torsion, 430 Total curvature, 107 Total mean curvature, 121 Total twist, 429, 430, 432 Transgression coefficients, 389 Transgression form, 389 Trivial Pin Q-spin structure, 55 Twist, 430 Twist invariants, 429 Twistorial fibrations, 64 Typical spinor space, 54

Umbilical point, 258 Universal homotopy operator, 289 Universal normal bundle, 305 Universal /^-structure, 305

K'(fl,..., fl^HOJnrm 80 Vanishing normal scalar curvature

holomorphic analogue of, 177 Variational problems without the hypothesis

of ellipticity, 420 Varifold, 144, 420

minimal variety, 144 stable variety, 144

Vector of type (p, p), 170 Vector valued second fundamental form, 126 Veronese surface, 199 ^-manifold, 277 Volume preserving immersion, 132

SUBJECT INDEX 451

WOk, 314 Whitney duality, 380 Weil homomorphism, 283 Wirtinger inequality, 167 Weyl conformal curvature tensor, 253

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