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) Copyright © 1975 by the American Mathematical Society Printed in the United States of America 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 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.