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# Conformal Differential Geometry and Its Generalizations

(Parte **1** de 8)

Conformal Differential

Geometryand Its Generalizations

PURE AND APPLIED MATHEMATICS A Wiley-Interscience Series of Texts, Monographs, and Tracts

Founded by RICHARD COURANT

Editor Emeritus: PETER HILTON Editors: MYRON B. ALLEN Ill, DAVID A. COX, HARRY HOCHSTADT, PETER LAX, JOHN TOLAND

A complete list of the titles in this series appears at the end of this volume.

MAKS A. AKIVIS Ben-Gurion University of the Negev Beer-Shera, Israel

VLADISLAV V. GOLDBERG New Jersey Institute of Technology Newark; New Jersey

A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York Chichester Brisbane Toronto Singapore

This text is printed on acid-free paper. Copyright © 1996 by John Wiley & Sons, Inc.

Published simultaneously in Canada.

All rights reserved. This book is protected by copyright. No part of it. except brief excerpts for review. may be reproduced. stored in a retrieval system. or transmitted in any form or by any means, electronic, mechanical, photocopying. recording. or otherwise, without permission from the publisher. Requests for permission or further information should be addressed to the Permissions Department. John Wiley & Sons. Inc.. 605 Third Avenue, New York, NY 10158-0012.

Ubrary of Congress Catafogfag-in-Pubfleadon Data

Akivis, M. A (Maks Aizikovich) Conformal differential geometry and its generalizations / Maks A.

Akivis and Vladislav V. Goldberg.

p.cm. - (Pure and applied mathematics)

"A Wiley-Interscience publication." Includes bibliographical references (p.-) and indexes.

ISBN 0-471-14958-6 (cloth:alk. paper)

1. Geometry, Differential.1. Gol 'dberg, V. V. (Vladislav Viktorovich)I. Title.Ill. Series: Pure and applied mathematics

(John Wiley & Sons: Unnumbered) QA641.A587 1996 516.3'63-dc20 96-31348

Contents

Introduction ix

CHAPTER 1CONFORMAL AND PSEUDOCONFORMAL SPACES

CHAPTER 2HYPERSURFACES IN CONFORMAL SPACES 31

CHAPTER 3SUBMANIFOLDS IN CONFORMAL AND PSEUDOCONFORMAL SPACES73

CHAPTER 4CONFORMAL STRUCTURES ON A DIFFERENTIABLE MANIFOLD119

4.4A conformal structure on a hypersurface of a projective space 150

Notes 160

CHAPTER 5THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES 163

CHAPTER 6GEOMETRY OF THE GRASSMANN MANIFOLD 221

CONTENTS vii

CHAPTER 7MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES267

almost Grassmann manifold274 7.3The complete structure object of an almost Grassmann manifold 281 7.4Manifolds endowed with semiintegrable almost Grassmann structures 292 7.5Multidimensional (p + 1)-webs and almost Grassmann

Bibliography 323 Symbols Frequently Used355 Author Index359 Subject Index363

Introduction

This book presents the foundations and applications of local conformal differential geometry and the theory of conformal, Grassmann, and almost Grassmann structures.

Conformal differential geometry was developed within the framework of classical differential geometry at the end of the last and the beginning of this century. It included concepts such as surfaces with isothermic or spherical curvature lines, canal surfaces, congruence of circles, triply orthogonal systems of surfaces, and conformal differential invariants and conformally invariant differential quadratic forms of a surface (G. Darboux, G. Fubini, A. Ribaucour,

A. Voss, and others). L. Berwald's paper of 1927 contains a survey of works in this area.

However, in the 1920s affine and projective differential geometries became independent branches of differential geometry (E. tech, G. Fubini, E. J. Wilczynski, and others), while conformal differential geometry lagged behind in its development. This phenomenon can be explained by the fact that in works on affine and projective differential geometry, the coordinate systems natural for these geometries, namely the affine and projective systems, had been used, whereas in the works on the conformal differential geometry the investigations were conducted in the rectangular Cartesian coordinate system.

Only G. Thomsen in 1923-1925 and E. Vessiot in 1926-1927 started to use the pentaspherical coordinates (introduced long before by G. Darboux) and tensor analysis in their studies in conformal differential geometry.

A part of W. Blaschke's book of 1929 and T. Takasu's papers of 1928-1938, and his book of 1938 were devoted to the differential geometry of the conformal space C", the Laguerre space, and the space whose fundamental group is the group of spherical transformations of S. Lie. In all these works the differential geometry of submanifolds of spheres was considered.

The 1918 paper of H. Weyl was very important for the development of conformal differential geometry. In this paper H. Weyl studied conformal invariants of Riemannian metrics and their relation to general relativity, which was intensively developing at that time. Following Weyl's ideas, in the 1920s and 1930s E. Cartan, V. Hlavaty, S. Sasaki, J. A. Schouten, I. M. Thomas, T. Y. Thomas, K. Yano, and others intensively developed the theory of multidimensional conformally connected spaces. However, in most of these works the conformal differential geometry of submanifolds was constructed by means of Riemannian geometry. The authors of these works did not go beyond ob- taining the Frenet equations and finding their integrability conditions. The complete bibliography of these works can be found in the book Ricci Calculus by J. A. Schouten (1924).

Along with proper conformal geometry, pseudoconformal geometry is also of great importance. One of the reasons is that pseudo-Riemannian metrics are used in general relativity, and these metrics lead to the study of pseudoconfor- mal structures.

After World War I the geometry of submanifolds of the conformal space

C" was intensively developed. As apparatus, tensor methods and the method of exterior forms and moving frames were applied (M. A. Akivis, A. P. Norden,

V. I. Vedernikov, L. L. Verbitsky, and others). In addition the conformal theory of manifolds of spheres of different dimensions was investigated (R. M. Geidelman, B. A. Rosenfeld, V. I. Vedernikov, and others).

Although multidimensional conformal differential geometry is important for other parts of differential geometry and in other branches of mathematics, and there are numerous papers on the subject, there is as yet no book in which multidimensional conformal differential geometry has been presented system- atically.

The last book devoted to the theory of conformal structures was the book by S. Sasaki published in 1948 in which conformal connections on submanifolds were the subject of study. However, there is the need for a book on conformal and almost Grassmann structures, since these structures find applications in a number of branches of mathematics and physics. The present book will fill the indicated gap in the literature on differential geometry.

There exists a connection between conformal geometry and the geometry of

Grassmann and almost Grassmann structures. It was F. Klein who noted that geometry of the manifold of straight lines of a three-dimensional space is equivalent to the geometry of a four-dimensional pseudoconformal space. Grassmann and almost Grassmann structures on a manifold are close to conformal structures, since both kinds of structures are determined on a manifold by a field of cones. The difference is that for conformal structures these cones are cones of second order, while for Grassmann and almost Grassmann structures they are more complicated algebraic cones called Segre cones. This is the reason for studying the conformal, Grassmann, and almost Grassmann structures in the framework of a unified theory.

(Parte **1** de 8)