Differential Geometry Proceedings of Symposia in Pure Mathematics

Home , Hopf conjecture, John Milnor, Robert Osserman

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

DIFFERENTIAL GEOMETRY 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, UE 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)

Printed in the United States of America

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 sem• inars, 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 Eliz• abeth 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 Bergery, L. Berard, 3,4 Ahlfors, L. V., 27, 31 Bernstein, S. N., 143, 149, 150, 173, 236, 250, Akin, E., 373, 381 268 Alexander, H., 212, 213, 214, 215 Bianchi, L., 248, 249,250,251 Allard, W. K., 146, 153, 155, 158, 159, 173, Bieberbach, L., 245, 251 211, 212,215,275 Blair, David E., 113 Allendoerfer, C. B., 102 Blanton, John D., 349, 349, 356 Almgren, F, J., Jr., 144, 146, 153, 156, 165, Bleeker, D., 186, 186 172,173, 211, 215, 215, 420, 427 Bochner, S., 254, 261, 262 Anderson, D. W., 362, 364, 364 Bombieri, E., 154, 156, 173 Anderson, P. G., 298, 300 Bonnesen, T., 245, 251 Arnol'd, V. I., 381, 382, 382 Bonnet, 102, 203, 247, 407, 408 Artin, M., 382, 382 Boothby, W. M., 175 Asada, A., 52 Bott, Raoul, 29, 30, 31, 31, 41, 52, 59, 67, 266, Asimov, Daniel, 316 268, 274, 274, 280, 281, 291, 292, 293, 293, Atiyah, M. F., 29, 30, 31, 31, 58, 67, 285, 293 303, 305, 306, 307, 308, 309, 313, 314, 314, Aubin, T., 69, 71 324,357, 357, 364, 364, 384,595 Avez, A., 106, 106 Bourbaki, N., 382 Bourguignon, Jean Pierre, 3, 33, 34, 35, 36, 37 Banchoff, Thomas F., 333, 339, 340, 342, 346, Bouzon, J., 62 347,371, 372, 375, 376, 380, 397,402 Brieskorn, E., 86,87, 382,382 Barbance, Ch., 67 Brown, E. H., Jr., 39,40,47,52 Barbosa, J. L., 161, 162, 173 Buoncristiano, S., 380, 380 Baum, P., 303, 306, 309 Beckenbach, E. F., 208, 212, 215 Cairns, S. S., 375, 380 Berger, M., 3, 13, 20, 23, 24, 26, 26, 34, 36, Calabi, E., 3,4, 7,12, 18, 19, 23, 26, 27, 31, 38, 37, 37, 98, 99,100, 409 161, 162,173, 204, 205, 251, 409

441 442 AUTHOR INDEX

Carleman, T., 208,215 Federer, H., 145, 146, 153, 154, 155, 157, 158, do Carmo, M., 127,130,134,162,163,164,173 165, 167, 168, 170, 172, 173, 214, 215, 420, 226,245,251 421,422,427 Cartan, Elie, 14, 130, 131, 134, 191, 195, 196, Fenchel, W., 245, 251 197, 199, 200, 200, 239, 285 Ferus, D., 120, 123, 245, 251 Cartan, H., 293 Flanders, H., 235, 236, 236 Cecil, Thomas E., 115 Fleming, W. H., 153, 155, 157, 158, 165, 173, Cenkl, B., 265, 268, 274 174, 212,214,215 deCesare, K., 277 Floyd, E. E., 298, 299, 300, 323, 324, 330 Chavel, I., 410 Freedman, M. H., 411 Cheeger, J., 35, 36, 37, 37, 73, 74, 87, 107,108, Freudenthal, H., 410 340,347,349,356,311,380 Frolicher, A., 41, 52 Chen, Bang-Yen, 119, 120, 121, 123, 182, Fuks, D. B., 267, 274, 314, 357, 358, 361, 362, 255, 258, 259, 261, 262 363, 364,364 Chen, Kuo-Tsai, 39, 40, 42, 44, 46, 48, 49, 50, Funk, 109, 409 52 Chern, S. S., 33, 37, 37, 41, 52, 72, 102, 127, Gardner, Robert B., 125, 133, 134, 240, 242, 128, 129, 130, 131, 134, 135, 137, 151, 152, 243 161, 164, 173, 226, 235, 236, 294, 306, 365, Gauduchon, P., 13, 23, 26,26, 37,37 367, 370, 383, 384, 395, 400, 402, 408, 431, Gauss, K. F., 102,125,130, 200, 203, 247,407 437 Gelfand, I. M., 267, 274, 314, 357, 358, 361, Chevalley, C, 53, 57, 61, 62, 64, 67 362,363,364, 364,421,427 Choquet-Bruhat, Y., 55, 67 deGiorgi, E., 153, 154, 156, 157, 173 Chow, 169, 231 Giusti, E., 154, 156,173 Civita, 195 Godbillon, C, 291, 294, 307, 308, 309 Codazzi, 200 Godement, 285 Cohn, 407 Goldberg, S. I., 135, 256, 262 Conlon, Lawrence, 275 Graev, M. I., 421, 427 Conner, P. E., 298,299,300, 323,324,330 Green, Leon W., 407, 409 Courant, R., 150, 173, 212, 215 Griffiths, P. A., 4, 7, 11, 12,12, 23, 26, 26 Crumeyrolle, A., 53, 53, 54, 55, 56, 57, 60, 61, Gromoll, D., 74, 86, 87, 107, 108, 407, 409, 62, 65, 67, 67 409 Gromov,311,321,410 Deschamps, A., 34, 35, 36,37 Grossman, Nathaniel, 137,413 Dieudonne, J., 14, 25, 26 Grove, K., 409, 409 Dombrowski, P., 76, 87, 94 Guillemin, V., 266,267,274,357,363,364 Douglas, J., 149, 150, 154, 173 Gulliver, R., 150,774,411 Gunning, R. C, 25, 26, 27, 28, 30, 31 Earle, C. J., 25, 26, 26, 27 Eberlein, P., 409 Haefliger, Andre, 265, 268, 269, 274, 274, 291, Ebin, D. G., 3, 4, 13, 20, 26, 35, 36, 37, 98, 293, 294, 303, 305, 309, 311, 314, 321, 321, 99, 100 357,364,411 Eells, J., Jr., 25, 26, 26, 27, 120, 189, 190, Halperin, S., 313, 340, 345, 347, 349, 356, 376, 200 380 Efimov, N. V., 183, 184, 185, 186, 408 Hamilton, R. S., 28, 31 Ehresmann, Charles, 370 Hardy, G. H., 243, 243 Ehrlich, Paul Ewing, 69, 69, 71 Hartman, P., 246, 251 Eliasson, H., 94 Harvey, R., 143, 169, 170, 171, 174 Epstein, D. B. A., 277, 281, 315, 319, 380 Heitsch, James L., 72 Erbacher, J. A., 121,123, 111, 182 Helgason, S., 410 Escobales, Richard H., Jr., 405, 405 Herbert, R., 437 van Est, W. T., 273, 292 Herman, M„ 315, 319 AUTHOR INDEX 443

Hermann, R., 67 Lawson, H. Blaine, Jr., 72,143, 145, 147, 148, Hernandez-Andrade, Horacio, 73 149, 150, 152, 154, 155, 159, 160, 161, 162, Hicks, Noel J., 238, 239, 243 164,166,167,168,169,174, 251, 300,411 Hilbert, D., 183, 242, 245, 251, 408 Leung, D. S., 191 Hildebrandt, S., 150,174 Levi, 195 Hilton, P. J., 350, 356 Lichnerowicz, A., 54,59,61,67 Hirzebruch, F., 349, 356, 382, 382 Lickorish, W. B. R., 315,319 Hitchen, N., 86, 87 Liebmann, H., 119, 121, 245, 257 Hochschild, G., 266,273,274, 292,294, 314 Lima, E., 127, 130,134, 275, 276 Hoffman, David A., 120, 123, 128, 134, 139, Littlewood, J. E., 243, 243 212,213,214,275 Liu, Mu-Chou, 89, 89, 92, 95 Holmgren, E., 245, 246, 251 Liu, R. C, 254, 262 Hopf, E., 102, 106,106, 182 Lovelock, D., 236, 236 Hopf, H., 33, 37, 119,123 Ludden, Gerald D., 723,177,182 Hsiang, Wu-yi, 145, 160, 162,174 Hsiung, C. C, 208, 275, 242, 407 MacPherson, Robert, 334, 397 Huber, A., 212, 275 Maeda, Y., 257, 262 Maffei, John S., 365 Ishihara, S., 121, 725, 177, 182, 256, 262 Martens, H. H., 28, 37 Itoh, T., 226 Marussi, Antonio, 413 Massey, W. S., 246, 257 Jacobson, N., 384, 385, 388, 389, 395 Mather, J., 315, 317, 379, 364, 364, 373, 376, Jorgens, K., 249, 250, 257 380 Matsushima, Y., 190,190 Kamber, Franz W., 40, 52, 283, 283, 284, Mazet, E., 13,23,26,26, 37,37 285, 286, 290, 291, 292, 294 McCrory, Clint, 334, 347,347,371, 397 Karras, U., 298, 300 Meusnier, 230 Kato, T., 31 Meyer, W., 33, 74, 85, 86, 87, 408, 409, 409 Kaul, H., 208, 211,275 Michael, J. H., 140,141, 211, 212, 214, 275 Kazdan, J. L., 71, 407 Millett, K., 277, 280 King, J., 168,174 Milnor, John W., 33, 37, 74, 79, 87, 87, 130, Kirby, 280 134, 183, 184, 185, 186, 190, 190, 251, 349, Klingenberg, W., 74, 87, 415, 475 356,313,380,3S1,382 Klotz, T. (see also Milnor, T. K.), 119, 723, Milnor, Tilla Klotz (see also Klotz, Tilla), 185,187 183,183,185,186,757 Knapp, A., 143, 170, 171,174 Minkowski, 245 Knutson, 408 Mizutani, T., 298, 300, 300 Kobayashi, S., 33, 37, 89, 90, 92, 93, 94, 95, Montgomery, D., 279, 281 164,173, 202, 205, 226, 306, 309 Moore, J. D., 132,134, 245, 257 Kodaira,K.,5,ll,12,72,202 Moser, J., 132,734,315 Koschorke, Ulrich, 295, 295, 299, 300, 300, Mostert, P., 328 411 Mostow, G. D., 266,273,274,292,294 Kosmann, Y., 59,67 Munzer, H. F., 200,200 Kostant, B., 60 Muto, Yosio, 97,97,99,100 Kowalski, O., 95 Kreck, M., 298, 300 Nagano, Tadashi, 189, 277 Kuiper, Nicolaas H., 130,131,134,417,418 Nakagawa, H., 204, 205 Kulkarni, R. S., 255, 262 Narasimhan, R., 169,174 Kumpera, A., 266, 274 Nash, J., 132,134 Neumann, W. D., 298, 300, 328, 330 Lafontaine, J., 3,3,12,27 Nijenhuis, A., 41, 52 Lashof, R. K., 129,134,137,436, 437, 437 Nirenberg, L., 5, 12, 72, 246, 257 444 AUTHOR INDEX

Nishikawa, S., 257, 262 deRham,G., 155,775 Nitsche, J. C. C, 150, 174, 211, 215, 249, 252, Robbin, J.W., 475 411,411 Rodriguez, L., 115,116,777 Noether, G., 28, 31 Rosenberg, H., 275,276,318,330 Nomizu, Katsumi, 33, 37, 89, 90, 92, 94, 95, Rossi, H., 773 115, 116, 777, 121, 123, 177, 752, 191, 191, Rourke, C. P., 379,380 198,199,200, 203,205, 306,309 Roussarie, R., 328,330 Novikov, S. P., 275, 276, 315, 319 Rozenfeld, 268 Ruh, E. A., 120, 723,189,190 Ochiai, T., 202,205 Rund, H., 236 Ogiue, Koichi, 177, 179, 182, 182, 201, 202, Ryan, P. J., 256,262 203, 204, 205,410 Omari, H., 174 Sacksteder, R., 275,276 O'Neill, B., 179,180,752 Sampson, J. H., 120, 189,190 Orlik, P., 85, 87 Sanderson, B. J., 379, 380 Osborn, Howard, 101, 102, 103, 383, 390, 395 Sard, A., 129, 157 Ossa, E., 298,300 Sario, L., 27, 37 Osserman, Robert, 119,123,150,151,152,173, Sasaki, S., 62, 89, 95, 250 174, 175, 185, 186, 757, 207, 212, 213, 214, Satake, I., 277, 280, 257 275,411 Sato, S., 261, 262 O'Sullivan, John J., 105,105,106 Schneider, R., 241 Otsuki, Tominosuke, 217, 217, 218, 219, 220, Schouten, J. A., 247, 253, 257, 262 221,223,22(5,227,409 Schweitzer, Paul A., S. J., 311, 311, 372, 349, 349,356,410 Palais, R. S., 59,67,198,200 Segal, G., 265,268,269,274,357 Pasternack, Joel S., 280,281,303,304,306,310, Segre, B., 195 324, 330, 330 Seifert, R., 346 Penrose, R., 64, 65, 66, 67 Sentenac, P., 34, 35, 36, 37 Phillips, A., 311,572, 321 Sergeraert, F„ 315, 379 Pitts, John, 146 Shapiro, A., 58, 67 Plateau, 149, 153,411 Shiftman, B., 169,174,211,275 Pogorelov, A. V., 184,757,246,252 Shiohama, K., 122, 723 Pohl, William F., 229, 229, 232, 233, 429, 430, Shulman, Herbert, 274,274, 292,294, 306,310, 434,436,437 313, 314 Poincare, H., 4,10,16,20 Siebenmann, 280 Polya, G., 243,243 Sim6es,P., 154,168,775 Pontrjagin, L. S., 339,347, 365, 366,370, 375 Simon, L. M, 140,141,211,212,214,275 Poor, W. A., 108 Simon, U., 236 Poritz,A.B.,411 Simons, J., 72,147,153,156,157,163,164,166, Porter, Richard, 334, 339,347,397 169,174,175,294 Preissmann, A., 34,37 Singer, I. M., 4,14,26 Singley, Donald H., 237, 238, 239, 242, 243 Rado, T., 149,150,775,208,212,275 Smale, S., 436,437,437 Raghunathan, M. S., 11,72 Smith, B., 723 Rauch,H.E.,27,30,31,37 Smith, P. A., 380 Raymond, B., 312,372 Smith, Ted, 200 Reeb, G., 129,277,257,315,317,319,319 Smyth, Brian, 116, 777,120,121, 723,177, 752, Reid, W. T., 208,275 189,203,205 Reidemeister, K., 107,108 Spanier, E. H., 400,402 Reifenberg, E. R., 155, 775 Spencer, D. C, 5,11,12,72,266,274 Reilly, Robert C, 235, 236, 408 Spivac, Michael, 14, 15, 21, 26, 245, 245, 246, Reinhart, B. L., 102,103, 304,310,411,411 247,248,250,251,252 AUTHOR INDEX 445

Spruck, Joel, 139 Vilms,J., 120,725,189,790 Stasheff, J. D., 40, 52, 265, 270, 274, 274, 313, Vladimirova, S. M., 250, 252 314, 349,356, 373,380 Volkov, Ju. A., 250, 252 Steenrod, N. E., 67,378,380 Vossen, 407 Sternberg, S., 4,14,26,266,267,274 Stiefel, E., 340, 347, 349, 356 Wagreich, P., 382 Stoker, J. J., 246,252 Wallach, N. R., 162, 163, 772, 175 Stokes, 241,242,436 Walter, Rolf, 107, 107, 108,108 Stong, R., 299,300, 324,330 Wang, H. C, 410 Sullivan, D., 40,52, 371, 373, 379, 380 Warner, F. W., 13, 20, 21, 27, 30, 71, 407 Synge, J. L., 33,143, 250 Watanabe, Y., 254, 262 Watts, C. E., 102,103 Tachibana, S., 254, 256, 262 Weil, Andre, 4, 11, 72, 27, 28, 32, 40, 306, 328, Takagi,H., 254,262 330, 383 Takagi, R., 122,123,191,196,200,200 Weiner, J., 251 Takahashi, T., 162,175, 191, 196, 200 Weinstein, Alan, 34, 37, 37, 71, 71, 109, 409, Tamura, I., 301 410 Tani, M., 256,262 Weiss, 175 Tanno, S., 89,94,95,204,205 Weyl, H., 253, 262 Taylor, Jean E., 419,419,420,427 White, James H., 429, 429, 430, 435, 436, 437 Teichmuller, 26 Whitehead, J. H.C., 51,52 Thorn, Rene, 229, 233, 339, 347, 371, 373, 375, Whitney, H., 340, 347, 349, 356, 376, 380 378,379, 380,380, 381,382, 397,402 Whitney, W., 175 Thorpe, J. A., 233, 233 Winkelnkemper, H. E., 323 Thurston, William P., 295, 296, 299, 300, 301, Willmore, T. J., 121, 122, 725 311, 312, 312, 315, 315, 316, 317, 319, 321, Wolf, J. A., 7, 72, 14, 23, 26, 410 321,527,364,410 Wood, John W., 312,572,315,579 Tits, J., 410 Wood, R., 200 Tjurina, G. N., 382,382 Wu, Wen-Tsun, 324,330,370, 380,380 Toledo, Domingo, 340, 345, 347, 349, 356, 372, WulfT, G., 427,427 376, 380 Wunderlich,W., 251,252 Tompkins, C, 131,754 Wylie, S., 350,356 Tondeur, Philippe, 40, 52, 283, 283, 284, 285, 286, 290, 291, 292, 294 Yamabe, 408 Torelli, 4, 27, 28 Yamaguchi, S., 261, 262 Trauber, P., 357, 363, 364 Yano, Kentaro, 89, 93, 94, 95, 121, 725, 177, Trudinger, N., 213, 215 182,253,254, 256,258,259,261,262 Tsagas, G., 410 Yau, S. T., 71, 71,120, 725,164,175 Tsukamoto, Y., 181,182 Zieschang, H., 315 Varma, S., 419 Zippin, L., 279, 281 Vey, J., 291, 294, 307, 308, 309 Zoll, 109 Vilenkin, N. Ja., 421,427 SUBJECT INDEX

Action of the diffeomorphism group, 35 Br\ 265 slice, 35 for Rrq -structures, 305 Affine transformation groups, 89, 93 Closed geodesic, 415 Algebraic curve, 229 index of, 415 Allendoerffer-Weil formula, 107 Closed leaf, 312 Almost-cocomplex structures, 62 Cohn-Vossen inequality, 107 Almost-complex structure, 59, 62 Complete intersection of hypersurfaces, 203 Amitsur complex, 284 Complete manifolds, 107 Analytic quadratic differential, 120 Complex gradient, 78 Anchor ring, 122 Complex hessian, 79 Complex projective space, 115 Basic connection, 305 Complex quadric, 117 Bianchi identity, 387 Complex space form, 201 Bi-invariant metric connection, 72 Cone, 154 Bochner curvature tensor, 254 Conformal Gauss equations, 259 Bordism groups Conformal groups, 67 of foliated manifolds, 296 Conformally flat Riemannian manifold, 253 of manifolds with g-plane fields, 296 Conjugate points, 105 Brieskorn variety, 79, 80 Connection, 102, 384, 385, 387 Connection map, 89 Calculus of variations, 419 Constant curvature, 3 Canonically associated spin fibrations, 65 complete surfaces of, 245-251 Canonical Riemannian connection, 305 Constant mean curvature, 192 Catastrophes, 381 Constant principal curvatures, 193 Category, 265 Convex hull property, 213 Category M, 350 Convex sets in Riemannian spaces, 107 Characteristic classes, 397 Counter-orientation, 317 Characteristic number obstructions, 324 Critical point, 115, 381 Chern class, 355 Critical Riemannian metric, 98 Chern homology class, 355 Currents Chern-Simons invariants, 72 representable by integration, 170 Chern-Weil construction, 283 of type (p, p)t 168 Chern-Weil theorem, 102, 384 Curvature, 102, 384, 385 Classifying spaces, 303 Curvature tensor, 74 447 448 SUBJECT INDEX

Cut locus, 139 Generalized characteristic homomorphism, 283 Deformations, 3 Generalized twistors, 64 of metrics, 34 Generalized Wirtinger inequality, 171 Density function, 155 General variational problems, 420 Derivation, 384 Geodesies Difference tensor, 133, 239 closed, 109 rank of, 240 Geometric homology operation, 379 Distance function, 107 Gradient flows, 129 Dual dimension, 129 Groups of spinoriality, 56, 57, 58, 62 Dual Pontrjagin classes, 354 Dual surface, 231 Haefliger structure, 299 Hausdorff/7-measure, 145 Ellipticity, 420 Hessian, 126 Enlarged group of spinoriality, 62, 63 Hessian tensor, 79 Euler characteristic, 101 Hodge operator, 102, 389 Euler class, 306, 390 Holomorphic analogue of vanishing normal Euler complex, 373 scalar curvature, 177 Euler singularity, 373 Holonomy groups of TM, 89 Euler space, 371, 373 Homology operation, 378 Exotic C-characteristic classes, 306 Hyperbolic space, 116 metric spheres in, 116 f(z)=ZUzVla (a -l), 80 k k Hypercohomology, 290 Fiber-preserving affine transformation, 94 Hyperplane sections, 190 File of M, 275 Hypersurfaces First derived characteristic homomorphism, complete intersection of, 203 285 isoparametric family of, 191 First osculating space of M, 127 parallel family of, 191 Flat connection, 283 Focal point, 105, 116, 197 Focal variety, 198 Imbedding problem, 125 Fold, 232 Immersion, 372 Foliated bundles, 283, Index of closed geodesic, 415 Foliated manifolds Induced metric, 125 bordism groups of, 296 Infinitesimal affine transformation, 93, 94 Foliations, 198, 277, 295, 311 Infinitesimal isometries, 93, 94 parallel along 3, 275 Integers localized away from 2, 354 P(3), 275 Integral formulas, 241 Full map, 372 Integrals-currents, 153 Integral varifold, 146 *(*)=H;«i ****/(«*-1), 80 Intersection product, 351 /\, 313 Irreducible complete Riemannian manifold, Afoliation, 268 93 Gauss-Bonnet formula, 119 Irreducible, complete nonflat Riemannian Gauss-Bonnet theorem, 102 manifold, 89 Gauss curvature, 119 Irreducible nonflat Riemannian manifold, 92, in the second fundamental form metric G', 94 240 Isometric immersions, 108 Gauss curvature equation, 159 Isometry group, 93 Gauss equations, 76,130, 257, 258 Isoparametric family of hypersurfaces, 191 Gauss map, 128 Isoperimetric inequality, 140, 207, 214 Gelfand-Fuks cohomology, 314 Isotropic twistors, 65 SUBJECT INDEX 449

Kahlerian Ricci flat manifold, 60 Normal mappings, 129 Kahler manifold, 166 Normal space, 75 Kahler submanifold, 201 Normal vector valued quadratic differential Killing vector field, 3, 105 form, 127 rank of, 127 Lagrangian bundles, 415 Laplacian of mixed invariants, 242 Odd dimensions, 62 Laplacian operators, 126, 241 Orbit system, 417 Level hypersurfaces, 193 linearly equivalent, 417 Lie algebras topologically equivalent, 417 reductive pair of, 288 Oriented /^-rectifiable set, 152 Lie derivative, 4 Orthogonal complex gradient, 78 Line field, 295 Orthogonal intersection, 80 Linearly equivalent orbit system, 417 Orthonormal basis, 76 Lipschitz-Killing curvature, 129 Outer parallel set, 107 Local connector, 89 Ovaloids, 237 Local convex deformations, 70 Locally free Lie transformation group of M Parallel family of hypersurfaces, 191 of dimension n — 2, 276 Parallel hypersurfaces, 237 Locally free manifold, 324 Pencil of symmetric bilinear forms, 132 Pfaffian, 102, 390 Mainardi-Codazzi equations, 159 Pin (2, 53 Manifolds Pin g-spin structure, 58 file of M, 275 in the broad meaning, 62 locally free, 324 strict meaning, 53, 56, 57 rank of A/, 275 trivial, 55 Manifolds with #-plane fields Pin Q'-spin structure, 61 bordism groups of, 296 Poincare map, 415 Mass, 144 Pontrjagin class, 354, 384, 398 Mass-norm, 170 Pontrjagin ring, 306 Massey triple product, 306 Positional sectional curvature, 33 Mean curvature, 119, 139 Positive current, 168 Mean curvature vector, 120, 127 Positive (/?, p)-current, 171 Mean curvature vector field, 148 Positive vector, 170 Metrics, 102 />-Rectifiable set, 145 mixed invariants, 238 Prescribed volume problem, 419,420 pair of, 237 direct method of solution, 419 Metric projection, 107 Principal curvatures, 192 Metric spheres in hyperbolic space, 116 Product manifolds, 33 Minimal immersion, 148 Profile, 372, 376 Minimal submanifold, 120, 148, 189, 199, 214 Projective space bundle, 296 Minimal surfaces, 208 Pseudogroup, 266 Minimal variety, 144 Mixed invariants of a pair of metrics, 238 Quasi-umbilical point, 258 Model deformation, 300 Morse function, 115 RG' = the Gauss curvature in the second funda• mental form metric G\ 240 Natural sum, 101 U/^-structures, 303 it-Circuit, 373 Real grassmannian, 365 Nonelliptic variational problems, 419 "Real" Witt's frame, 54 Nonflat Riemannian manifolds, 94 Rectifiable current, 153 Nonnegative curvature, 107 Rectifiable set, 145 450 SUBJECT INDEX

Rectifiable varifold, 146 Stably tangential map, 352 Reeb foliation, 299 Steenrod cohomology operation, 378 Resolution, 4 Stiefel class, 350 de Rham cohomology, 271 Stiefel homology class, 351 de Rham complex, 266, 271 Stiefel-Whitney class, 375, 378, 397 de Rham decomposition, 92 Stiefel-Whitney homology classes, 371 Ricci curvature, 74, 190 Stake's Theorem, 241 Ricci flat, 61 Strict meaning Pin G-spin structure, 53, 56, Ricci tensor, 74 57 Riemannian foliation, 303 Submanifolds, Riemannian manifold, 105 of complex projective space, 116 the center of * i(Af), 105 of hyperbolic space, 116 conformally flat, 253 Symmetic bilinear forms, irreducible, complete, nonflat, 89, 92, 93, 94 pencil of, 132 with no conjugate points, 105 Symplectic spinors, 60 with no focal points, 105 Riemannian product, 132 Tangent cones, 155 Riemannian spaces Third product, 391 convex sets in, 107 Topological groupoid, 266 Riemannian submanifolds, 139 r\ 265 Riemannian submersion, 36 Topological^ equivalent orbit system, 417 Riemann surface, 4 Torelli theorem, 4 Rigidity problem, 125 Torsion, 430 r-Isotropic vector field, 56 Total curvature, 107 Total mean curvature, 121 Sasaki connection, 89 Total twist, 429, 430, 432 Scalar curvature, 75 Transgression coefficients, 389 Scalar normal curvature, 177 Transgression form, 389 Schubert decomposition, 365 Trivial Pin Q-spin structure, 55 Secant lines, 232 Twist, 430 Secant spaces, 429 Twist invariants, 429 Secondary characteristic classes, 286 Twistorial fibrations, 64 Second fundamental form, 107,147 Typical spinor space, 54 Second fundamental tensor, 75 Sectional curvature, 74 Umbilical point, 258 Semisimplicial g-Z>(7-algebras, 285 Universal homotopy operator, 289 Semisimplicial groupoid, 268 Universal normal bundle, 305 Semisimplicial methods, 283 Universal /^-structure, 305 Semisimplicial space, 271 Simons characters, 72 K'(fl,..., fl^HOJnrm 80 Simple group, 315 Vanishing normal scalar curvature Singularity, 375, 378, 382 holomorphic analogue of, 177 Slice, 35 Variational problems without the hypothesis Smith operations, 380 of ellipticity, 420 Sobolev inequality, 139 Varifold, 144, 420 Spin fibrations minimal variety, 144 canonically associated, 65 stable variety, 144 Spinor bundle, 54 Vector of type (p, p), 170 Spinoriality Vector valued second fundamental form, 126 enlarged group of, 62,63 Veronese surface, 199 groups of, 56, 57, 58, 62 ^-manifold, 277 Stable variety, 145 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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