Theory of Multicodimensional (n+1)-Webs

485 Pages·1988·41.091 MB·English

by  Vladislav V. Goldberg (auth.)

Theory of Multicodimensional (n+ I)-Webs
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
F. CALOGERO, Universita degli Studi di Roma, Italy
Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, U.S.S.R.
A. H. G. RINNOOY KAN, Erasmus University, Rotterdam, The Netherlands
G.-c. ROTA, M.l. T., Cambridge, Mass., U.S.A.
Vladislav V. Goldberg
Department of Mathematics,
New Jersey Institute of Technology, U.S.A.
Theory of
Multicodimensional
(n+l)-Webs
KLUWER ACADEMIC PUBLISHERS
DORDRECHT I BOSTON I LONDON
Library of Congress Cataloging in Publication Data
Goldberg, V. V. (Vladislav Viktorovich)
Theory of multicodimensional (n+1)-webs I Vladislav V. Goldberg.
p. cm. -- (Mathematics and its applications)
Bibliography: p.
Inc ludes index.
ISBN 90-277-2756-2
1. Webs (Differential geometry) I. Title. II. Series:
Mathematics and its applications (D. Reidel Publishing Company)
QA648.5.G65 1988
516.3'6--dcI9
88-158')5
CIP
ISBN-13: 978-94-010-7854-2 e-ISBN-13: 978-94-009-3013-1
DOl: 10.1007/978-94-009-3013-1
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© 1988 by Kluwer Academic Publishers
Softcover reprint of the hardcover 1s t edition 1988
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SERIES EDITOR'S PREFACE
Approach your problems from the right end It isn't that they can't see the solution. It is
and begin with the answers. Then one day, that they can't see the problem.
perhaps you will find the final question.
G.K. Chesterton. The Scandal of Father
'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'.
van Gulik's The Chinese Maze Murders.
Growing specialization and diversification have brought a host of monographs and textbooks on
increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields
does not grow only by putting forth new branches. It also happens, quite often in fact, that
branches which were thought to be completely disparate are suddenly seen to be related.
Further, the kind and level of sophistication of mathematics applied in various sciences has
changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical
economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the
structure of water meet one another in packing and covering theory; quantum fields, crystal defects
and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering;
and prediction and electrical engineering can use Stein spaces. And in addition to this there are
such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable
systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the
existing classification schemes. They draw upon widely different sections of mathematics. This pro
gramme, Mathematics and Its Applications, is devoted to new emerging (sub)disciplines and to such
(new) interrelations as exempla gratia:
- a central concept which plays an important role in several different mathematical and/or
scientific specialized areas;
- new applications of the results and ideas from ont area of scientific endeavour into another;
- influences which the results, problems and concepts of one field of enquiry have and have had on
the development of another.
The Mathematics and Its Applications programme tries to make available a careful selection of
books which fit the philosophy outlined above. With such books, which are stimulating rather than
definitive, intriguing rather than encyclopaedic, we hope to contribute something towards better
communication among the practitioners in diversified fields.
A web is a collection of d foliations in general position of the same codimension. For example
one has the case of 3-webs of curves in the plane (the first interesting case, and the subject of
numerous early studies). The French and German names for the concept are respectively "tissu"
and "Gewebe", which mean tissue, fabric, texture (in its original meaning of woven fabric), web;
and these words with these meanings convey a good initial intuitive picture of the kind of geometric
structure involved. An easy example of a 3-web of curves in the plane is given by the three systems
of lines x = const., y = const., x +y = const.. By definition a codimension foliation is locally like
a set of parallel hyperplanes. However, whether several foliations, i.e. a web, can be (locally)
'straightened out' simultaneously is a much tougher question. For example the trivial 3-web of lines
just mentioned has the following closure property. Take a point 0 and draw the three leaves of the
v
vi SERIES EDITOR'S PREFACE
three foliations through that point Take a neighboring point A on one of these leafs and walk
around along the leafs and the foliations as indicated in the figure below
2
3
2 3
In the case of the given trivial foliation one finishes back in the point A after traversing a hexagon,
This turns out to be a necessary and sufficient condition for a 3-web of curves in the plane to look
locally like our trivial example, i,e, to be linearizable,
There are, of course, a good many places in mathematics where multiple foliations come up
naturally, and where such results would thus be important For example in control theory, although,
as far as I know, this particular potential application of web theory remains unexplored,
It is an old theorem (1924) that a linearizable web in the plane consists of the tangents to a
curve of degree 3 in the projective plane, and thus the question arises whether every web arises in
some such algebraic manner.
This also provided a rather unexpected link with algebraic geometry, one of the first of many
interrelations of the theory of webs with various parts of the theories of symmetric spaces,
differential equations, algebraic geometry, integral geometry, singularities, and holomorphic map
pings, The author sums up some 11 such interrelations in his own preface,
After the initial flowering of the subject which resulted in the classics by Bol-Blaschke and
Blaschke there was a dormant period. More recently starting around 1970 there was a substantial
resurge of interest, resurge of interest, which still continues, led and/ or sparked by Akivis, the
author himself, Chern and Griffiths.
This is a modern, up-to-date, comprehensive book on webs a topic rich in interrelations and
(potential) application, unique in its coverage of higher codirnensional webs, and, as such, a volume
I am more than glad to welcome in this series.
The unreasonable effectiveness of mathemat As long as algebra and geometry proceeded
ics in science ... along separate paths, their advance was slow
and their applications limited.
Eugene Wigner But when these sciences joined company
they drew from each other fresh vitality and
Well, if you know of a better 'ole, go to it. thenceforward marched on at a rapid pace
towards perfection.
Bruce Bairnsfather
Joseph Louis Lagrange.
What is now proved was once only ima-
gined.
William Blake
Bussum, May 1988 Michiel Hazewinkel
Table of Contents
Preface
xv
CHAPTER 1
+
Differential Geometry of Multicodimensional (n I)-Webs
1.1 Fibrations, Foliations, and d-Webs W( d, n, r) of
xnr .....................
Codimension r on a Differentiable Manifold 1
1.1.1 Definitions and Examples ...................................... 1
1.1.2 Closed Form Equations of a Web W(n + l,n,r) and
Further Examples ........................................ . . . . . . 6
1.2 The Structure Equations and Fundamental Tensors of
a Web W(n + l,n,r) ................................................ 9
+
1.2.1 Moving Frames Associated with a Web W(n l,n,r) .... ...... 9
1.2.2 The Structure Equations and Fundamental Tensors
of a Web W(n + l,n,r) ........................................ 11
1.2.3 The Structure Equations and Fundamental Tensors
of a Web W(3, 2, r) ............................................ 16
1.2.4 Special Classes of 3-Webs W(3, 2, r) ............................ 17
1.3 Invariant Affine Connections Associated with a Web
W(n+1,n,r) ....................................................... 20
w;
1.3.1 The Geometrical Meaning of the Forms (<5) ................... 20
1.3.2 Affine Connections Associated with an (n + I)-Web ............. 21
1.3.3 The Affine Connections Induced by the Connnection
I'n+l on Leaves ................................................. 23
1.3.4 Affine Connections Associated with 3-Subwebs of an
+
(n I)-Web................................................... 24
1.4 Webs W(n + 1, n, r) with Vanishing Curvature....................... 31
+
1.5 Parallelisable (n 1)-Webs .......................................... 34
1.6 (n + 1)-Webs with Paratactical 3-Subwebs ........................... 38
1. 7 (n + 1)-Webs with Integrable Diagonal Distributions of
4-Subwebs .......................................................... 39
1.8 (n + 1)-Webs with Integrable Diagonal Distributions ................. 42
+
1.9 Transversally Geodesic (n 1)-Webs ................................. 46
+
1.10 Hexagonal (n 1)-Webs ............................................. 56
+
1.11 Isoclinic (n 1)-Webs ............................................... 58
Notes...................................................................... 65
viii TABLE OF CONTENTS
CHAPTER 2
Almost Grassmann Structures Associated with Webs Wen + 1, n, r)
2.1 Almost Grassmann Structures on a Differentiable Manifold ........... 69
2.1.1 The Segre Variety and the Segre Cone.......................... 69
2.1.2 Grassmann and Almost Grassmann Structures.................. 72
2.2 Structure Equations and Torsion Tensor of an Almost
Grassmann Manifold ................................................ 74
2.3 An Almost Grassmann Structure Associated with a Web
W(n+l,n,r)· ....................................................... 81
2.4 Semiintegrable Almost Grassmann Structures and
Transversally Geodesic and Isoclinic (n + I)-Webs ................... 84
2.5 Double Webs ........................................................ 87
2.6 Problems of Grassmannisation and Algebraisation and
+
Their Solution for Webs W(d,n,r), d;::: n 1 ....... ................. 90
2.6.1 The Grassmannisation Problem for a
WebW(n+l,n,r) ............................................ 90
2.6.2 The Grassmannisation Problem for a
+
Web W(d,n,r), d> n 1 ..................................... 91
2.6.3 The Algebraisation Problem for a Web W(3, 2, r) ............... 92
+
2.6.4 The Algebraisation Problem for a Web Wen l,n,r) ........... 94
2.6.5 The Algebraisation Problem for
WebsW(d,n,r),d>n+l ..................................... 95
Notes.. . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
CHAPTER 3
Local Differentiable 7J.-Quasigroups
+
Associated with a Web Wen 1,n,r)
3.1 Local Differentiable n-Quasigroups of a Web Wen + l,n,r) .......... 102
3.2 Structure of a Web W( n + 1, n, r) and Its Coordinate
n-Quasigroups in a Neighbourhood of a Point ........................ 107
3.3 Computation of the Components of the Torsion and Curvature
+
Tensors of a Web W( n 1, n, r) in Terms of Its Closed Form
Equations ............................................................ 110
3.4 The Relations between the Torsion Tensors and Alternators
of Parastrophic Coordinate n-Quasigroups ........................... 114
3.5 Canonical Expansions of the Equations of a Local
Analytic n-Quasigroup .............................................. 116
3.6 The One-Parameter n-Subquasigroups of a Local
Differentiable n-Quasigroup ......................................... 125
TABLE OF CONTENTS ix
3.7 Comtrans Algebras.................................................. 131
3.7.1 Preliminaries .................................................. 132
3.7.2 Comtrans Structures........................................... 133
3.7.3 Masking....................................................... 135
3.7.4 Lie's Third Fundamental Theorem for Analytic 3-Loops ......... 137
3.7.5 General Case of Analytic n-Loops .............................. 138
Notes...................................................................... 139
CHAPTER 4
+
Special Classes of Multicodimensional (n I)-Webs
+
4.1 Reducible (n I)-Webs............................. .•............... 140
+
4.2 Multiple Reducible and Completely Reducible (n I)-Webs .......... 147
+
4.3 Group (n I)-Webs................................................. 151
4.4 (2n + 2)-Hedral (n + I)-Webs........................................ 158
4.5 Bol (n + i)-Webs.................................................... 162
4.5.1 Definition and Properties of Bol and Moufang
+
(n I)-Webs.................................................. 162
4.5.2 The Bol Closure Conditions .................................... 164
+
4.5.3 A Geometric Characteristic of Bol (n I)-Webs ................ 173
4.5.4 An Analytic Characteristic of the Bol Closure
..............................................
Condition (B~tn 180
Notes...................................................................... 188
CHAPTER 5
+
Realisations of Multicodimensional (n I)-Webs
+
5.1 Grassmann (n I)-Webs ............................................ 189
5.1.1 Basic Definitions ............................................... 189
5.1.2 The Structure Equations of Projective Space ................... 191
5.1.3 Specialisation of Moving Frames.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
5.1.4 The Structure Equations and the Fundamental Tensors of
a Grassmann (n + I)-Web...................................... 195
5.1.5 Transversally Geodesic and Isoclinic Surfaces of a
+
Grassmann (n I)-Web ....................................... 196
5.1.6 The Hexagonality Tensor of a Grassmann (n + I)-Web and
the 2nd Fundamental Forms of Surfaces Ue ..................... 198
5.2 The Grassmannisation Theorem for Multicodimensional
+
(n I)-Webs........................................................ 200
+
5.3 Reducible Grassmann (n I)-Webs .................................. 203
x TABLE OF CONTENTS
5.4 Algebraic, Bol Algebraic, and Reducible Algebraic (n + 1)-Webs ...... 206
+
5.4.1 General Algebraic (n I)-Webs ................................ 206
+
5.4.2 Bol Algebraic (n I)-Webs .................................... 207
+
5.4.3 Reducible Algebraic (n I)-Webs .............................. 209
+
5.4.4 Multiple Reducible Algebraic (n I)-Webs ..................... 210
5.4.5 Reducible Algebraic Four-Webs ................................ 213
+
5.4.6 Completely Reducible Algebraic (n I)-Webs.................. 214
+
5.5 Moufang Algebraic (n I)-Webs.................................... 218
5.6 (2n + 2)-Hedral Grassmann (n + I)-Webs............................ 222
5.7 The Fundamental Equations of a Diagonal 4-Web Formed by
Four Pencils of (2r)-Planes in p3r ................................... 226
5.8 The Geometry of Diagonal 4-Webs in p3r ............................ 234
Notes...................................................................... 242
CHAPTER 6
Applications of the Theory of (n + I)-Webs
+
6.1 The Application of the Theory of (n I)-Webs to the
+
Theory of Point Correspondences of n 1 Projective Lines ........... 243
6.1.1 The Fundamental Equations ................................... 243
+
6.1.2 Correspondences among n 1 Projective Lines and
One-Codimensional (n + I)-Webs .............................. 248
6.1.3 Parallelisable Correpondences .................................. 248
6.1.4 Hexagonal Correspondences .................................... 250
6.1.5 The Godeaux Homography ..................................... 252
6.1.6 Parallelisable Godeaux Homographies .......................... 253
+
6.2 The Application of the Theory of (n I)-Webs to the Theory
of Point Correspondences of n + 1 Projective Spaces ................. 253
6.2.1 The Fundamental Equations ................................... 253
+
6.2.2 Correspondences among n 1 Projective Lines and
+
Multicodimensional (n I)-Webs .............................. 258
6.2.3 Parallelisable Correpondences .................................. 260
6.2.4 Godeaux Homographies ........................................ 260
6.2.5 Parallelisable Godeaux Homographies .......................... 262
6.2.6 Paratactical Correspondences .................................. 263
+
6.3 Application of the Theory of (n I)-Webs to the Theory of
Holomorphic Mappings between Polyhedral Domains ................. 264
6.3.1. Introductory Note ............................................ 264
6.3.2 Analytical Polyhedral Domains in en, n > 1 .................... 265
6.3.3 Meromorphic Webs in Domains of en, n > 1 .................... 268
6.3.4 Partition Webs Generated by Analytical Polyhedral Domains ... 275
6.3.5 Partition Webs with Parallelisable Foliations ................... 278