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Differential Geometry, Volume http://dx.doi.org/10.1090/pspum/003 DIFFERENTIAL GEOMETRY 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 DIFFERENTIAL GEOMETRY 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. It has its roots in the movement toward differ• ential 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 estab• lished differential topology as a discipline of major importance. GAEL B. ALLENDOERFER University of Washington, Seattle, Washington Vll INDEX Ct-adic topology, 24 Cobordism, 40 Affine connections, 94,186 Cochains, invariant, 135 Affine spaces, locally, 142 Cohomology, 109 Alexander-Pontrjagin duality, 109 group of a Lie d-ring, 137 theorem, 124 of Lie rings, 130 Almost complex, 22 operations, 18 Almost-product structure, 94 with values in the sheaves of Lie algebras Artin-Rees lemma, 24 of infinitesimal groups, 56 Atlas with values in sheaves of nonabelian Eulerian, 159 groups, 56 Lagrangian, 160 Cohomology theory Axioms of a cohomology theory, 14 axioms of, 14 periodic, 7 Bidifferentiable transformations, closed Compact, Riemannian manifolds, 155 pseudogroups of, 61 Complete germ, 72 Borsuk's Extension Theorem, 120 Completed representation ring of a torus, 26 Bott Completions of modules, 24 isomorphism, 13 Complex, almost, 22 periodicity, 7 Complex analytic Bundles differentiable r-manifold, 64 complex vector, 8 family of complex structures, 92 homology of principal, 101 Complex contact fc-trivial, 49 manifold, 144 of r-frames, 188 manifold, homogeneous, 146 orien table, 115 structure, 144 ring of complex vector, 7 Complex structures transverse, 115 complex analytic family of, 92 C-space, 146 deformation of, 87 Ci-map, 20 equivalence of, 89 Canonical form family of, 91 differential, 189 obstructions to deformation of, 89-90 structure equation of, 191 stability of, 90 Cartan, E. variations of, 89 invariant forms' for a continuous Complex vector bundles, 8 pseudogroup of differentiate trans• ring of, 7 formations, 85 Connections structure equations of, 186 affine, 94 Category, model, 59 linear (affine), 186 Characteristic class, 102 Constant curvature, 155-156 relative, 102 Contact form, 145 Chern character, 15, 29 Continuous pseudogroup of differentiable x-equivalent, 40, 46, 49, 53, 55 transformations, 81 Classification theorem, 29 invariant Cartan forms for, 85 Classifying spaces, 7, 28 Coordinate transformation, 118 Clifford-Klein spaces, 156 Co-orienting, 111 differentiable family of, 159 Curvature, 186 Closed pseudogroups of bidiffenertiable trans• constant, 155-156 formations, 61 Gauss, 181 197 198 INDEX d-trivial Lie d-ring, 131 Gauss curvature, 181 Deformations, homological analysis of, 69 Germ, Deformations of complex structures, 87 complete, 72 obstructions to, 89-90 effective, 72 Deformation of the T-manifold, 70 of deformation of the r-manifold, 71 germ of, 71 stable, 75 Derivative Gradient mapping, 182 lie, 134 Grating, 112 torsional, 99 Graves-Hildebrandt differentiability, 115 Differentiability Groups Graves-Hildebrandt, 115 killing homotopy, 39, 50 Differentiable sheaf of, 65 complex analytic or real analytic T-mani- unitarv, 1, 8 fold, 64 Weyl,23 family of T-manifolds, 70 Gysin homomorphism, 20, 114 Differentiable transformations continuous pseudogroups of, 81 Hildebrandt-Graves differentiability, 115 invariant Cartan forms for a continuous Homogeneous pseudogroup of, 85 complex contact manifold, 146 Differential spaces, 7, 31 form, 189 Homological analysis of deformations, 69 manifolds, 39 Homology of principal bundles, 101 Distributions, 94 Homomorphism, Gysin, 20, 114 Double exterior forms, 166 Homotopy Duality theorem, 110 complements, 113 Alexander-Pontrjagin, 124 killing classes, 43 killing groups, 39, 50 Eilenberg and Steenrod axioms, 7 Hypersurfaces, 181 Equivalence of complex structures, 89 Eulerian atlas, 159 Immersion of manifolds, 194 Existence in homological analysis, problem Implicit function theorem, 116-117 of, 75 Infinitesimal pseudogroup, 66 Extension Theorem of Borsuk, 120 Infinitesimally surjective, 75 Exterior forms, double, 166 Interior product, 133 Invariant /-relatedness for vector forms, 90 Cartan forms for a continuous pseudogroup Family of complex structures, 91 of differentiable transformations, 85 complex analytic, 92 cochains, 135 Family of r-manifolds, differentiable, 70 cohomology group of a Lie coring, 137 Frames, r-, 188 Invariants, ring of, 27 bundle of, 188 Isomorphism Function spaces, 109 Bott, 13 Fundamental class of the oriented pair, 113 Theorem of Leray, 111 r-manifolds r. (/) deformation of, 70 source of, 187 differentiable family of, 70 target of, 187 differentiable, real analytic or complex Jacobi identities for vector forms, 88 analytic, 64 Jet, r-, 187 germ of deformation of, 71 T-structure, 64 A;-parallelizable manifold, 49 T-vector field, 67 fc-trivial bundle, 49 INDEX 199 Killing homotopy Morse theory, 2 classes, 43 Multifoliate, 81 groups, 39, 50 Multiplication, Pontrjagin, 125 Klien-Clifford spaces, 156 differentiable family of, 159 Nilmanifold, 138 Noetherian ring, 24 <£-ring, 132 Normal degree, 194-195 Lagrangian atlas, 160 Leray Isomorphism Theorem, 111 Lie d-ring Obstructions to deformation of a complex cohomology group of, 137 structure, 89-90 d-trivial, 131 Orient, 111, 115 invariant cohomology group of, 137 Orientability, 111 cC-module over, 132 Orientable over R, 131 bundle, 115 Lie derivatives, 134 pair, 111 Lie group, compact, 25 Orientation sheet of the manifold pair connected, 23, 29, 36 (X, F), 111 representation ring of, 25 Oriented pair, fundamental class of, 113 Lie rings, cohomology of, 130 Orienting, co-, 111 Linear (affine) connection, 186 Locally Parallel, 99 afiine spaces, 142 Periodic cohomology theory, 7 stable, 74 Periodicity, Bott, 7 trivial, 74 7r-manifold, 46 Pontrjagin Manifold pair, orientation sheet of, 111 classes, 20 Manifolds, 39 multiplication, 125 compact, Riemannian, 155 numbers, 41 complex contact, 144 Pontrjagin-Alexander duality, 109 deformation of the T-, 70 theorem, 124 differentiable family of T-, 70 Primitive differentiable, real analytic or complex left, 167 analytic T-, 64 right, 167 germ of deformation of the T-, 71 Principal bundles, homology of, 101 homogeneous complex contact, 146 Product immersion of, 194 interior, 133 fc-parallelizable, 49 triple, 105 Mapping Projective space, 3 gradient, 182 Projector, 94 monotone, 182 Pseudogroup, 59 spherical image, 181 closed of bidifferentiable
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