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PERSPECTIVES OF COMPLEX ANALYSIS, DIFFERENTIAL GEOMETRY AND MATHEMATICAL PHYSICS Proceedings of the 5th International Wo...

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PERSPECTIVES OF COMPLEX ANALYSIS, DIFFERENTIAL GEOMETRY AND MATHEMATICAL PHYSICS Proceedings of the 5th International Workshop on Complex Structures and Vector Fields

Editors

Stancho Dimiev Kouei Sekigawa

World Scientific

PERSPECTIVES OF COMPLEX ANALYSIS, DIFFERENTIAL GEOMETRY AND MATHEMATICAL PHYSICS

PERSPECTIVES OF COMPLEX ANALYSIS, DIFFERENTIAL GEOMETRY AND MATHEMATICAL PHYSICS Proceedings of the 5th International Workshop on Complex Structures and Vector Fields St. Konstantin, Bulgaria

3-9 September 2000

Editors

Stancho Dimiev Bulgarian Academy of Sciences, Bulgaria

Kouei Sekigawa Niigata University, Japan

V f e World Scientific wb

Singapore Jersey L • London • Hong Kong Sinaaoore • New NewJersev

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

PERSPECTIVES OF COMPLEX ANALYSIS, DIFFERENTIAL GEOMETRY AND MATHEMATICAL PHYSICS Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-4597-1

Printed in Singapore.

V

FIFTH I N T E R N A T I O N A L W O R K S H O P ON COMPLEX S T R U C T U R E S A N D V E C T O R FIELDS S e p t e m b e r 3 - 9 , 2000, St. Konstantin, Bulgaria

CONTRIBUTED COMMUNICATIONS SEMINAR LECTURES:

1. Toshiaki ADACHI Length spectrum of geodesic spheres in non-flat complex and quaternionic space forms 2. Georgi GANCHEV Canal hypersurfaces of second type 3. Hideya HASHIMOTO Weierstrass formula for super minimal J-holomorphic curves of a 6dimensional sphere and its applications 4. Milen HRISTOV Real hypersurfaces of Kaehler manifold (sixteen classes) 5. Ognyan KASSABOV, Georgi GANCHEV Almost hermitian manifolds of poinwise constant antiholomorphic sectional curvature 6. Kazunori KIKUCHI The quotient space of the complex projective plane under conjugation is a 4-sphere 7. Masatoshi KOKUBU On a generalization of CMC — 1 surfaces theory 8. Vladimir KOSTOV The Deligne-Simpson problem 9. Julian LAWRINOWICZ Holomorphic chains in analysis, geometry and differential equations 10. Rumyan LAZOV Convex preorders, function algebras and applications 11. SawaMANOFF Lagrangian fluid mechanics

VI

12. Velichka MILUSHEVA Hypersurfaces of conullty 2 which are 1-parameter systems of torses 13. Tejinder NEELON CR structures on singular sets 14. Nikolai NIKOLOV Localizations of invariant functions 15. Kiyoshi OHBA The moduli space of dipoles 16. Kouei SEKIGAWA Integrability of almost Hermitian manifolds (mainly on the Goldberg conjecture) 17. Kahrlheinz SPALLEK Product decomposition of space germs P O S T E R SESSION:

Alexander KYTMANOV, Simona MYSLIVETS The teorem of analytic representation on hypersurface with singularities

vii Dedicated to the outstanding Dubna School in Theoretical Physics.

PREFACE This volume contains a part of the papers contributed by the participants of the Fifth International Workshop on topics in Differential Geometry, Mathematical Physics and Complex Analysis held at St. Constantin near Varna (Bulgaria). It contains investigations initiated in the previous Workshops during the period 1992-1998, mostly in classical trends, describing some perspectives. The Editors would like to express here their gratitude to Professor T. Oguro for his outstanding cooperation and efforts in the arrangement of this volume, and to Professors S. Manoff and R. Lazov for their help. Editors

CONTENTS

Preface The Deligne—Simpson Problem for Zero Index of Rigidity V. P. Rostov

vii 1

Theorems for Extension on Manifolds with Almost Complex Structures L. N. Apostolova, M. S. Marinov and K. P. Petrov

36

The Theorem on Analytic Representation on Hypersurface with Singularities A. M. Kytmanov and S. G. Myslivets

45

Pseudogroup Structures on Spencer Manifolds S. Dimiev

53

Type-Changing Transformations of Hurwitz Pairs, Quasiregular Functions, and Hyper Kahlerian Holomorphic Chains I J. Lawrynowicz and L. M. Tovar

58

Embedding of the Moduli Space of Riemann Surfaces with Igeta Structures into the Sato Grassmann Manifold Y. Hashimoto and K. Ohba

75

On the Quotient Spaces of S2 x S2 under the Natural Action of Subgroups of £>4 K. Kikuchi

80

Existence of Spin Structures on Cyclic Branched Covering Spaces over Four-Manifolds S. Nagami

86

Length Spectrum of Geodesic Spheres in Rank One Symmetric Spaces T. Adachi

93

Grassmann Geometry of 6-Dimensional Sphere, II H. Hashimoto and K. Mashimo Hypersurfaces in Euclidean Space which are One-Parameter Families of Spheres G. Ganchev and V. Mihova

113

125

Hypersurfaces of Conullity Two in Euclidean Space which are One-Parameter Systems of Torses G. Ganchev and V. Milousheva Real Hypersurfaces of a Kaehler Manifold (the Sixteen Classes) G. Ganchev and M. Hristov

135 147

Almost Contact B-Metric Hypersurfaces of Kaehlerian Manifolds with B-Metric M. Manev

159

Projective Formalism and Some Methods from Algebraic Geometry in the Theory of Gravitation B. G. Dimitrov

171

Geometry of Manifolds and Dark Matter /. B. Pestov

180

Lagrangian Fluid Mechanics S. Manoff

190

Transformation of Connectednesses G. Zlatanov

201

1

T H E DELIGNE-SIMPSON PROBLEM FOR ZERO I N D E X OF RIGIDITY* VLADIMIR PETROV ROSTOV To the memory of my mother We consider the Deligne-Simpson problem: Give necessary and sufficient conditions for the choice of the conjugacy classes CJ C gl(n,C) or Cj C GL(n,C), j = 1 , • • •, V + 1 . s o that there exist irreducible (p + 1) -tuples of matrices Aj g Cj whose sum is 0 or of matrices Mj £ Cj whose product is I. The matrices Aj (resp. Mj) are interepreted as matrices-residua of Fuchsian linear systems (resp. as monodromy operators of regular systems) on Riemann's sphere. We consider the case when the sum of the dimensions of the conjugacy classes Cj or Cj is In2 and we prove a theorem of non-existence of such irreducible (p+ l)-tuples.

1

Introduction

In the present paper we sonsider a particular case of the Deligne-Simpson problem (DSP): Give necessary and sufficient conditions for the choice of the conjugacy classes Cj c gl(n, C) or Cj c GL(n, C), j = 1, . . . , p + 1, so that there exist irreducible (p+l)-tuples of matrices Aj
=0

(1)

or M i . . . M p + i = /.

(2)

"Irreducible" means "not having a common proper invariant subspace", i.e. impossible to conjugate simultaneously the (p + 1) matrices to a block uppertriangular form. The problem is connected with the theory of linear regular systems of differential equations on Riemann's sphere: X = A{t)X

(3) 1

Here the n x n-matrix A(t) is merornorphic on C P , with poles at the points ai, . . . , a p +i; the unknown variables X form also a matrix n x u. Such a system is called regular at the pole aj if one has \\X(t - a,j)\\ = 0(\t - aj\N>) •RESEARCH PARTIALLY SUPPORTED BY INTAS GRANT 97-1644

2

for some Nj € R when the solution is restricted to a sector of sufficiently small radius and centered at aj. A particular case of a regular system is a Fuchsian one, i.e. with logarithmic poles: dX/dt=(^Aj/(t-aj))X

(4)

where Aj G gl(n,C) are its matrices-residua; in the absence of a pole at oo one has (1). As a result of the linear change of variables X

H->

W(t)X

(5)

the matrix A(t) of a regular system (3) undergoes the gauge transformation A(t)^-W~lW

+ W~1A(t)W 1

(6)

The n x n-matrix W is meromorphic on C P , its poles if any are usually among the points a,, and outside them det W i= 0. The only invariant of a regular system under the linear changes (5) is its monodromy group. This is the group generated by the monodromy operators. A monodromy operator is a linear operator mapping the solution space of a regular system onto itself. It is defined as follows: one fixes a base point a ^ aj for j = 1, ..., p + 1, the value at a of the solution X, i.e. a matrix B G GL(n, C) and a closed contour F passing through a. The monodromy operator M defined by the homotopy equivalence class of the contour T maps the solution X with X\t=a = B onto the value at a of its analytic continuation P along the contour (notation: X — i > XM). Fix (p + 1) contours whose homotopy equivalence classes generate 7Ti(CP 1 \{ai,..., ap+i}). One usually chooses the contours such that Tj consists of a segment [a, Xj] (XJ is close to aj), of a small circumference (centered at aj, passing through Xj, circumventing a.j counterclockwise and not containing inside any other pole a,) and of the segment [x_j,a]. We assume that for i ^ j one has Fi n Tj = {a} and that the index of the contour increases when one turns around a clockwise. For such a choice of the contours the monodromy operators Mj satisfy condition (2). This means that one can choose as generators of the monodromy group any p out of the p + 1 operators Mj. The monodromy group is an antirepresentation of 7Ti(CP 1 \{ai,..., r,r ap+i}) into GL(n,C) because one has X H-+J XMjMi (although we often write "representation" instead). The change of a and B changes the monodromy group to a conjugate one.

3

If the contours defining the operators Mj are chosen like above, then Mj is conjugate to the corresponding operator of local monodromy defined by a small lace circumventing the pole Qj counterclockwise. Therefore in the case of matrices Mj the DSP admits the interpretation: For which (p + 1)-tuples of local monodromies do there exist irreducible monodromy groups with such local monodromies ? Remark 1. The eigenvalues Xkj of the matrix-residuum Aj of a Fuchsian system are connected with cr^.j, the ones of the monodromy operator Mj by exp(27riAfcii) = crk,j.

2 2.1

Definitions and known facts The quantities dj, TJ and K; the construction \t; (poly)multiplicity vectors

Definition 2. A Jordan normal form (JNF) of size n is a collection of positive integers indexed by two indices — J " = {fr,,fc} — where k is the index of an eigenvalue, i is the index of the Jordan block of size bi.k with this eigenvalue; k = 1, . . . , p, i = 1, . . . , Sfc. We assume that all p eigenvalues are distinct and that for each k one has 6i,fc > . . . > bSktkConvention. All Jordan matrices and Jordan blocks are presumed to be upper-triangular. Definition 3. Denote by J(X) the JNF of the matrix X. We say that the DSP is solvable (resp. weakly solvable) for a given (p + l)-tuple of JNFs {J™} and given eigenvalues if there exists an irreducible (p + l)-tuple (resp. a (p + l)-tuple with trivial centralizer) of matrices Mj satisfying (2) or of matrices Aj satisfying (1), with J(Mj) = J™ or J(Aj) = J™ and with the given eigenvalues. By definition, the DSP is solvable for n — 1. For a given conjugacy class C (in gl(n, C) or GL(n, C)) we denote by d(C) its dimension (which is always even) and by r(C) the quantity m i n ^ c rk (X — XI) for X e C. The quantity ?i —r(C) is the greatest number of Jordan blocks with one and the same eigenvalue. We set dj = d(cj) (resp. dj = d{Cj)) and r = r c i ( i ) (resp. Tj = r(Cj)). The quantities r(C) and d(C) depend not on the conjugacy class C but only on the JNF defined by it. The following two conditions are necessary for the existence of irreducible (p + l)-tuples of matrices Mj satisfying (2) or of matrices Aj satisfying (1),

4

see [Si] and [Ko3], [Ko4]: di + ...+ dp+1 > 2n2 - 2 for all j ri H \-fj-\

(- r p + 1 > n

(an) (/3„)

Definition 4. The quantity K = 2n 2 — di — • • • — d p +i is called the index of rigidity. If condition (a n ) holds, then it takes the values 2, 0, —2, Call rigid the case K — 2 (i.e. for which condition ( a n ) is an equality). The rigid case has been studied in [Ka]. In the present paper we study the case K = 0. These two cases are of particular interest because they seem to contain all non-trivial examples when the DSP is not weakly solvable. (An example is called non-trivial if the JNFs J™ satisfy the conditions of Theorem 10 below.) Definition 5. Denote by JJ1 the JNF of size n defined by the class Cj or Cj and by {•/"} the (p + l)-tuple of these JNFs. For n > 1 define the map * : {JJ} •-> iJp} ^ the condition (pn) holds and the condition n + . . . + rp+i > 2n

(un)

does not hold. Namely, set n\ = (X)j=i ri) ~ n ' hence, rt\ < n. For each j the new JNF J" 1 is defined after J™ by choosing an eigenvalue with the maximal possible number n — r ; of Jordan blocks, by decreasing by 1 the sizes of the smallest n — m of them and by deleting the Jordan blocks of size 0. One has n — n\ < n — Tj because ((3n) holds. If there are several eigenvalues with maximal number of Jordan blocks, then we choose any of them. Definition 6. A multiplicity vector (MV) is a vector whose components are non-negative integers whose sum is n. Notation: A™ = (wi,j> • • • i m i,,j)> m i j > • • • > rrii, j , mij + • • • + mihj = n. The components have the meaning of the multiplicities of the eigenvalues of a matrix Aj or M 7 (for the sake of convenience we admit components equal to 0). A polymultiplicity vector (PMV) is the (p + l)-tuple of MVs defined by the eigenvalues of the matrices Aj or Mj. Remark 7. 1) In the case of diagonalizable matrices Aj or Mj the JNF J™ is completely defined by the MV A™ and the construction Vfr results in decreasing the biggest component of A™ by n — m to obtain A" 1 . 2) For a diagonal JNF defined by a MV A™ one has Tj=n — m\j and

5

3) If A n = (n) and if the matrix Aj or Mj is diagonalizable, then it is scalar. 2.2

Generic eigenvalues; non-genericity relations; the quantities I and £

We presume the necessary condition Y\det(Cj) = 1 (resp. J^Tr (CJ) = 0) to hold. This means that the eigenvalues ak,j (resp. \k,j) of the matrices from Cj (resp. Cj) repeated with their multiplicities, satisfy the condition n p+1

nn^

= i resp

-

fc=ij=i

n

P+l

EE^=O

(?)

^=1^=1

An equality of the form P+I

P+I

fe

x

n n °" .j= j=ifc€$,

resp

- s 5 i Afc<j=° j=ifce*j

is called a non-genericity relation; the sets $., contain one and the same number < n of indices for all j . Eigenvalues satisfying none of these relations are called generic. Reducible (p + l)-tuples exist only for non-generic eigenvalues (a reducible (p + l)-tuple of matrices can be conjugated to a block upper-triangular form, its restriction to each diagonal block is such a (p + 1)tuple of smaller size, and, hence, the eigenvalues of each diagonal block satisfy condition (2) or (1) which is a non-genericity relation). Remark 8. In the case of matrices Aj, if the greatest common divisor q of the multiplicities of all eigenvalues of all p + 1 matrices is > 1, then a non-genericity relation ( 7 B ) (called the basic non-genericity relation) results automatically from Yl^i0}) = 0 when one decreases q times the multiplicities of all eigenvalues. In the case of matrices Mj the equality fj a^.j = 1 implies that if one divides by q the multiplicities of all eigenvalues, then their product would equal £ = exp(27rifc/qr), 0 < k < q — 1, not necessarily 1. In this case a non-genericity relation holds exactly if £ is a non-primitive root of unity of order q. Indeed, denote by I the greatest common divisor of q and k. Then the product of all eigenvalues with multiplicities divided by I equals 1 which is the basic non-genericity relation (73) in the case of matrices Mj. Definition 9. In the case when the basic non-genericity relation (7s) holds, eigenvalues satisfying no non-genericity relation other than (JB) and its corollaries are called relatively generic. The following theorem is the basic result from [Ko3], [Ko4] and [Ko5]:

6

T h e o r e m 10. Let n > 1. The DSP is solvable for the conjugacy classes Cj or Cj (with generic eigenvalues, defining the JNFs J™ and satisfying conditions (an) and (pn)) if and only if either {./"} satisfies condition (u>n) or the construction \P : {J?} I~^ {Jj1} iterated as long as it is defined stops at a (p + 1)-tuple {J? } either with n' = 1 or satisfying condition ( u v ) . Proposition 11.

The construction

\I/ preserves

the index of

rigidity.

T h e proposition is proved in [Ko4]. R e m a r k 1 2 . 1) T h e result of the theorem does not depend on the choice one makes in \t of an eigenvalue with maximal number of J o r d a n blocks (if such (a) choice(s) is (are) possible). 2) Proposition 11 implies t h a t it suffices to check condition (an>) for the (p + l)-tuple of J N F s Jf without checking ( a „ ) for the J N F s JJ 1 . It does hold — if n' = 1, then ( o v ) is an equality (this is the rigid case, i.e. K = 2). If n' > 1 and condition (u>n>) holds for t h e J N F s J™ , t h e n ( o v ) holds and is a strict inequality, see [Ko3], Theorem 9. T h u s a posteriori one knows t h a t it is not necessary to check condition (an) in Theorem 10. 3 3.1

T h e basic result The case K = 0 for diagonalizable

matrices

L e m m a 1 3 . In the case K = 0 a monodromy group with trivial and with relatively generic eigenvalues is irreducible.

ccntralizer

T h e lemma is proved in [Ko5], see part 1) of Lemma 6 there. Making use of the lemma we shall not distinguish solvability from weak solvability of the D S P in the case K = 0. T h e o r e m 14. In the case of matrices Mj, the conditions of Theorem 10 upon the JNFs Jjl are necessary for the solvability of the DSP in the case K = 0. / / the conjugacy classes Cj defining the JNFs J" satisfy condition (Pn) and do not satisfy condition (u>n), then the solvability of the DSP for the conjugacy classes Cj implies the solvability of the DSP for the (p + 1)tuple of JNFs J " 1 = ^(J?) (see Subsection 2.1) for some relatively generic. eigenvalues with the same value, of S,. T h e theorem is proved in Section 4. In order to announce the basic result we need to introduce some technical notions (see Subsections 3.2 and 3.3).

7

Therefore we first announce the result for the case of diagonalizable matrices which does not need them. Theorem 15. 1) If' K = 0, if the JNFs defined by the classes Cj are diagonal, if q > 1, if £ is a non-primitive root of unity of order q and if the eigenvalues of the classes Cj are relatively generic, then the DSP is not weakly solvable for matrices Mj (hence, not solvable either). 2) If K = 0, if the JNFs defined by the classes Cj are diagonal, if q > 1 and if the eigenvalues of the classes c.j are relatively generic, then the DSP is not weakly solvable for matrices Aj (hence, not solvable either). A plan of the proof of the theorem is given at the end of this subsection. In the rigid case the construction ty stops at a (p + l)-tuple of onedimensional JNFs, see Theorem 10 and Remark 12, part 2). Lemma 16. In the case when n = 0 and the JNFs J™ are diagonal there are four possible (p + 1)-tuples of JNFs at which 9 stops. Their PMVs are: Case A) p = 3 (d, d)

(d, d)

(d, d)

Case B) p = 2 (d, d, d)

(d,d,d)

(d,d,d)

Case C) p = 2 (d,d,d,d)

(d,d,d,d)

{2d,2d)

(d, d)

Case D) p = 2 (d, d, d, d, d, d) {2d, 2d, 2d) (3d, 3d) In all cases d n) holds and is an equality. The lemma follows from Lemma 3 from [Kol] and from the notion of corresponding JNFs defined below in Subsection 3.3. Plan of the proof of Theorem 15: We prove part 1) first. We show that in each of the four cases A) - D ) from Lemma 16 (and when the conditions of 1) of the theorem are fulfilled) the DSP is not solvable; by Lemma 13 it is not weakly solvable either. This is done in Sections 5, 7, 6 and 8, one case per section. Section 5 is the longest and the most important of them because in the other three cases the proof is reduced to the one in Case A). Theorem 14 and Lemma 16 imply that in all possible cases covered by Theorem 15 the DSP is not weakly solvable. Part 2) of the theorem is proved in Section 9 using part 1). •

8

3.2

The basic technical tool

Definition 17. Call basic technical tool the way described below to deform analytically a (p + l)-tuple of matrices Aj satisfying (1) or of matrices Mj satisfying (2), with trivial centralizer. In the case of matrices Aj set Aj = Q~xGjQj, Gj being Jordan matrices. Look for matrices Aj of the form Aj = (I + Yli=i£iXj.i(e))~1Q7l(Gj + T.UerVj,{e))Q3{I + ELI^XJAS)) where e = (eu.'..,£„) e (C',0) and Vjj(e) are given matrices analytic in e. One chooses Vj^ such that tr (XT?=i J2i=i £iVj,i(£)) = 0 identically in e. One often has s = 1 and Vj,\ are such that the eigenvalues of the (p + l)-tuple of matrices Aj are generic for E / 0 . Often one has Vjj = 0 for all indices j but one, i.e. all matrices Aj but one remain within their conjugacy classes. In the case of (p + l)-tuples of matrices Mj with trivial centralizer look for Mj of the form

Mj = ( ' + £

£

>XJA£))

[M]

+ £ £*NjA£)) (i + £ £*xjAe))

(8)

where the given matrices Nj^ are analytic in e 6 (C s , 0) and one looks for Xj^ analytic in e. Like in the case of matrices Aj one can set Mj = Q~1GjQj, Nj,i = QjlVjtiQj. For both cases the existence of the matrices Xj^ analytic in e is proved in [Ko4]. 3.3

Correspondence between Jordan normal forms

Definition 18. For a given JNF Jn = {6j,fc} define its corresponding diagonal JNF J'n. A diagonal JNF is a partition of n defined by the multiplicities of the eigenvalues. For each k fixed the collection {fr^fc} is a partition Vk of Yliei ^.fc- The diagonal JNF J'n is the disjoint sum over k of the partitions dual to VkExample 19. Consider the JNF J 1 7 = {{6,4,3}{3,1}}, i.e. with two eigenvalues, the first with three Jordan blocks of sizes 6,4,3 and the second with two blocks of sizes 3,1. The partition of 13 dual to (6,4,3) is (3,3,3,2,1,1), the one of 4 dual to (3,1) is (2,1,1). Hence, the diagonal JNF corresponding to J 1 7 is defined by the MV (3,3,3,2,2,1,1,1,1) (in decreasing order of the multiplicities).

9

Proposition 20. Consider a JNF Jn and its corresponding diagonal JNF J'n defined by a MV A = ( m i , . . . , mv), m\> . ..> mv. Choose an eigenvalue of Jn with maximal number n — r(Jn) of Jordan blocks and decrease the sizes of the k' smallest of these blocks by 1, k' < n — r(Jn) — this defines a new JNF Jn~k'. Set A* = (mi - k',m2, • • • ,m„). Then the MV A* defines a diagonal JNF corresponding to Jn~k . Corollary 21. The {p + 1)-tuples of JNFs J™ J j " where for each j J? corresponds to «/.'" satisfy or not the conditions of Theorem 10 simultaneously. The propositions and corollary from this subsection are proved in [Ko4]. Proposition 22. 1) If the JNF J'n corresponds to the JNF Jn, then r(Jn) = r(J'n) and d(Jn) = d(J,n). 2) To each diagonal JNF there corresponds a unique JNF with a single eigenvalue. Remark 23. Denote by G a Jordan matrix and by G" a diagonal matrix defined as follows: the diagonal entries of G' in the last but s positions of the Jordan blocks of G with given eigenvalue A are equal among themselves and different from the ones in the last but m positions for m ^ s, TO, S e N*. Then the matrix G + eG', 0 / £ £ (C,0) is diagonalizable and its JNF is the diagonal JNF corresponding to J(G) (the proof can be found in [Ko4]). Hence, if one applies the basic technical tool with s — 1 and Gj, Vjt\ playing the roles respectively of G, G', then one sees that the weak solvability of the DSP for matrices Aj or Mj with given JNFs J " implies the one for diagonal JNFs corresponding to J™ and for nearby eigenvalues. 3.4

The result in the general case

Definition 24. We say that the conjugacy class C is continuously deform.ed into the class C" if either the classes C, C" are like the ones of the matrices G, G + eG' from Remark 23 or C" is just another conjugacy class defining the same JNF as C. We say that the ( p + l)-tuple of conjugacy classes Cj is continuously deformed into the (p + l)-tuple of conjugacy classes C[- if each class Cj is continuously deformed into the corresponding class Cj and the eigenvalues of the first (p + 1 )-tuple are homotopic to the ones of the second (p + l)-tuple. Throughout the homotopy there holds condition (7) and the MVs remain the same. Example 25.

Consider the triple of conjugacy classes C\, C2, C3 of size 12

10

each with a single eigenvalue A^ and with Jordan blocks of equal size ly. (Ai,A2,A3) = (i, 1,1), (I1J2J3) = (2,3,6). For these eigenvalues one has q = 12, £ = i which is not a primitive root of unity of order 12. One has I = 3. The basic non-genericity relation (73) is obtained by dividing the multiplicities of all eigenvalues by 3. The eigenvalues are relatively generic. To the triple of JNFs defined by the conjugacy classes Cj there corresponds the triple of diagonal JNFs defined by the PMV (6,6), (4,4,4), (2, 2, 2, 2, 2, 2). For this PMV one has q = 2 and by continuous deformation of the conjugacy classes Cj into diagonal ones with the above PMV one obtains £ = — 1 which is a primitive root of unity of order 2. (Indeed, for the classes Cj the product of the eigenvalues repeated each with the half of its multiplicity equals —1 which remains unchanged throughout the continuous deformation.) Definition 26. Denote by d the greatest common divisor of all quantities £j,m(c) where T,j,m(a) is the number of Jordan blocks of size m of a given matrix Mj or Aj and with eigenvalue a. It is true that d divides q and that q divides n. Remark 27. The quantity q does not increase under continuous deformations like in the above example. If one deforms continuously the conjugacy classes so that the eigenvalues of C be "as generic as possible" (i.e. satisfying only these non-genericity relations which are not destroyed by continuous deformations like the above ones), then one has q = d. Theorem 28. Suppose that 1) the conjugacy classes of the matrices Aj or Mj verify the conditions of Theorem 10; 2) they are continuously deformed into a (p+1) -tuple of conjugacy classes defining diagonal JNFs with q = d > 1, with relatively generic eigenvalues and in the case of matrices Mj with £ being a non-primitive root of unity of order q; 3) one has n = 0. Then for such conjugacy classes the DSP is not weakly solvable. Proof. Suppose that there exists a (p + 1 )-tuple of matrices Mj with trivial centarlizer which satisfies conditions 1), 2) and 3). Applying the basic technical tool with / = 1 and Gj, V^i like in Remark 23, one obtains the existence of a (p+ l)-tuple of diagonalizable matrices Mj with trivial centralizer, with relatively generic eigenvalues, with n = 0 and with £ being a non-primitive

11 root of unity of order q which contradicts Theorem 15. 4 4-1

•

P r o o f of T h e o r e m 14 The proof itself

D e f i n i t i o n 2 9 . A regular singular point of a linear system of ordinary differential equations is called apparent if its local monodromy is trivial. L e m m a 3 0 . Any monodromy group can be realized by a Fuchsian system on CP1 with at most one additional apparent singularity at a point Op+2 which can be chosen arbitrarily; for the eigenvalues X^.j of the matrices-residua Aj, j = l, • • •, p + 1 one has ReXkj € [0,1); one has J{Aj) = J(Mj) for j = 1, . . . , p + 1, Mj being the m.onodrom.y operators. T h e lemmas from this subsection except Lemmas 32 and 38 are proved in t h e subsequent ones (one proof per subsection). In what follows the points a i , . . . , ap+2 are fixed. D e f i n i t i o n 3 1 . A Fuchsian system belongs t o the class N if it has poles at the points aj the one at a p +2 being an apparent singularity, if its monodromy group is irreducible, and if at ap+2 t h e Laurent series expansion of t h e system looks like this: X = (Ap+2/(t-ap+2) where Ap+2 Denote function u. has ordBij

+ B(t-ap+2))X

(9)

= diag (MI, . . . , /x„), fij e Z, p,Y > . . . > \in. by ord u t h e order of t h e zero at a p + 2 of the germ of holomorphic A class N Fuchsian system is called normalized if for i < j one > /Xj — Hj.

L e m m a 3 2 . / / one has Ap+2 = d i a g ( ^ i i , . . . ,/J.n), f-j G Z, f.ii > ... > /x„, and if one has for i < j ord Bij > //j — /ij for B defined by (9), then the singularity at ap+2 is apparent. Indeed, t h e following change of variables brings the system locally, at a.p+2, t o a system without a pole at a p +2 (hence, t h e local monodromy at ap+2 is trivial): X ~ ( t L e m m a 3 3 . For a normalized for i = 1, . . . , n — 1.

op+2)diag('il--'4")X class NFuchsian

(10)

system one has \.i\— f-i+i < p

12

Definition 34. Set a = (p,\ + ... + fin)/n (mean value) and 6 = ((p,i ~ a)2 + ... + (/.in — a)2)/n (dispersion of the numbers /x,). Lemma 35. The monodromy group of a non-normalized class N Fuchsian system can be realized by a normalized class N Fuchsian system with the same conjugacy classes of the matrices A\, ..., Ap+\, with the same mean value and with a smaller dispersion of the numbers IM • Suppose that for K = 0 and for given diagonal conjugacy classes with relatively generic eigenvalues and not satisfying condition (uin) there exists a monodromy group with trivial centralizer (hence, irreducible by Lemma 13). Then for almost all relatively generic eigenvalues with the same value of £ there exist irreducible monodromy groups with such JNFs. Indeed, applying the basic technical tool, one can deform the given monodromy group into one with any nearby relatively generic eigenvalues and the same JNFs of the matrices Mj. Moreover, the deformation can be chosen such that the new matrices Mj will be diagonalizable and defining the JNFs corresponding to the initial ones. The set M of such monodromy groups is constructive and such is its projection V on the set of eigenvalues W, i.e. V is an everywhere dense constructive subset of W. Lemmas 30, 33 and 35 imply that for given conjugacy classes Cj of Afi, ..., Afp+i there exist finitely many sets Tj of eigenvalues /zfc = ^k,p+2 such that the monodromy group can be realized by a normalized class N Fuchsian system with such eigenvalues of Ap+2\ for j < p + 1 the eigenvalues X^.j are uniquely defined by the classes Cj, see Lemma 30. Consider gl(n, C)p+1 as the space of (p + 2)-tuples of matrices Aj whose sum is 0. Denote by Qi its subsets such that Ap+2 is diagonal, with eigenvalues fik £ r i : and for i < j there holds the condition ordBij > [li — Hj for B defined by (9) (recall that the poles aj are fixed). Hence, the sets Qi are constructible. A point from Qi defines a Fuchsian system (S). Fix a base point a different from the points aj and define the monodromy operators of the system with initial data X\t=a = I- The map which maps the matrices-residua A\, ..., Ap+2 into the (p + l)-tuple of monodromy operators of system (S) is a map Xi • & - •

M.

For each point from M there exists at least one i such that the point has a preimage in Qi under xt- This means that there exists a point from M such that some neighbourhood of his is covered by Xi(Gi) f° r some i; we set i = 1. Indeed, the constructible set M. cannot be locally covered by a finite number of analytic sets of lower dimension. This and the irreducibility of M implies

13

that the set xi(Gi)

ls

dense in M.

Lemma 36. Suppose that A) the matrices-residua A\, ..., Ap+\ of a normalized class N Fuchsian system are diagonalizable, with generic eigenvalues; B) their (p + 1)-tuple is irreducible; C) none of these matrices has eigenvalues differing by a non-zero integer and each of them has a single integer eigenvalue Xj whose multiplicity is a (the) greatest one (hence, each monodromy operator Mj has an eigenvalue o-j = 1); D) all non-genericity relations satisfied by the eigenvalues of the monodromy operators Mj result from two relations, the first of which is the basic one ( 7 B ) the second being CTI . . . C T p + i = 1

(70)

E) one has Xj > 0 and X\+.. .+A p + i > (n 2 + /i)/x with ft = max(|/ii|, | ^ n | ) . F) the monodromy group can be analytically deformed into an irreducible one for nearby relatively generic eigenvalues and with the same JNFs of the matrices Mj. G) Condition (u)n) does not hold for the matrices Mj. Then the monodromy group of the Fuchsian system is with trivial centralizer. The projection V\ of the set Q\ on the space C s of eigenvalues Xkj {s depends on their multiplicities) is a constructive set. If V\ does not contain a point satisfying conditions C), D) and E) of the lemma, then codimo'Pi > 0, hence, X\{Q\) cannot be dense in M. Lemma 37.

The monodromy group of system, (4) with eigenvalues defined

as in Lemma 36 can be conjugated to the form I

n

r ) where $ is n\ x n\.

The subrepresentation $ can be reducible. The following lemma is proved in [Ko4]. Lemma 38.

The centralizer Z{<&) of the subrepresentation $ is trivial.

Thus the existence of an irreducible representation of rank n for which condition (ojn) does not hold implies the existence of the representation $ of rank n\ and with trivial centralizer. The JNFs defined by the matrices from $ are obtained from the initial (p+ 1) JNFs by applying the map \P. One can

14

deform the eigenvalues of $ so that they become relatively generic. For such eigenvalues the deformed representation $ is irreducible, see Lemma 13. If $ satisfies condition (u>ni), then we are done. If not, then we continue iterating 'J. In the end we stop at a representation of rank n' satisfying condition (u>n>). It is impossible to obtain a representation of rank 1 because its index of rigidity is 2, see Proposition 11. The eigenvalues of the representation $ define the same value of £ as the ones of the initial representation. Indeed, the eigenvalues from the initial one which are not in $ equal 1. • 4-2

Proof of Lemma 30

It is shown in [P] that any monodromy group can be realized by a regular system on CP1 which is Fuchsian at all poles but one. So one can add a (p+2)-nd monodromy operator equal to / to the initial operators Mj assuming that the system realizing this monodromy group has not p + 1 but p + 2 poles. Applying the result from [P] (reproved in [Aril], p. 131) one obtains a regular system (S) with the given monodromy group which is Fuchsian at a\, ..., ap+\ and which has a regular apparent singularity at ap+2- The point ap+2 ^ «j, j < p+1, is chosen arbitrarily and the JNFs of the matrices Aj are the same as the ones of the corresponding monodromy operators Mj for j = 1, ..., p+1. Moreover, ReXkj 6 [0,1). Remark 39. In [P] an attempt is made to prove that every monodromy group can be realized by a Fuchsian system on CP1 (without apparent singularities). This is one of the versions of the Riemann-Hilbert problem and the answer to it is negative, see [Bol]. We are referring above to the correct part of the attempt from [P] to prove the Riemann-Hilbert problem. See [Aril] pp. 130-135 as well. Make the singularity at ap+2 Fuchsian. Fix a matrix solution to system (4) with det X ^ 0. Its regularity and the triviality of the monodromy at ap+2 imply that it is meromorphic at aP+2Lemma 40 (A. Souvage). A meromorphic mapping from C n to Cn with a pole at ap+2 and nondegenerate for t ^ ap+2 can be represented in the form PH(t — ap+2)D where D is a diagonal matrix with integer entries, H is holomorphic and holomorphically invertible at Op+2 and the entries of the matrix P are polynomials in l/(t — a p +2), det P = const ^ 0. Perform in system (S) the change X >-^ P lX.

This change leaves the

15

system Fuchsian at a j , ..., ap+i and regular at ap+2 without introducing new singular points. At ap+2 the new system is Fuchsian. Indeed, the matrix (t — ap+2)D is a solution to the system (Fuchsian at ap+2) X = (D/(t — ap+2))X. The change of variables X >-^ HX leaves the latter system Fuchsian at ap+2 (the system becomes X = {-H~lH + H'l{D/{t - ap+2))H)X). D 4.3

Proof of Lemma 33

1°. The matrix B defined by equation (9) admits the Taylor series expansion B = Bo + (t — ap+2)Bi + (t — ap+2)2B2 + • • • • A direct computation shows that Bv = — Y^j=\ Aj/(a,j — ap+2)v• Suppose that for some io (1 < io < n — 1) one has /ij„ — fJ-i„+i > p + 1. Then for i < io, k > IQ + 1 one has /z, — /x^ > p + 1. 2°. Hence, all matrix entries A^.A,- with j < p + 1 and i, k like in 1° must be 0. Indeed, for each such i, k fixed the system of linear equations -Bjy;i,fe = 0, v = 1, . . . , p + 1 with unknown variables the entries Aj-^^ implies Aj-,i,k = 0 because it is of rank p+1 (its determinant is the Vandermonde one W(l/(ai - ap+2),..., l/(a P +i - aP+2)) and for j1 ^ j 2 one has aJL ^ ah). This means that the matrices-residua A\, . . . , Ap+\ are block lowertriangular, with diagonal blocks of sizes io and n — io- Hence, so are the monodromy operators, i.e. the monodromy group is reducible and the system is not from the class N. • 4-4

Proof of Lemma 35

1°. Recall that the matrix B was defined by equation (9). Assume for simplicity that ap+2 = 0. For i < j find an entry Bij with smallest value of m := —ordBij — [ij + fii. Hence, in > 0. If there are several possible choices, then we choose among them one with minimal value of j — i. Set Bitj = bt9 + o(|£| s ), b^Q (hence, g = ordB^-). 2°. Consider the change of variables X 1—> WX with W = I + (fij — m + g)Ejti/btm. It is holomorphic for t ^ 0, with det W = 1, hence, it preserves the conjugacy classes of the residua A\, . . . , Ap+i the system remaining Fuchsian there. At ap+2 the new residuum is lower-triangular, with diagonal entries equal to /zi, . . . , m-i, fij + g, /x i + i, . . . , /ij-i, ^ - g, fij+i, ..., /i n . The singularity at ap+2, in general, is no longer Fuchsian, but the order of the pole at ap+2 is < m; equality is possible only in position (j,i). This follows from rule (6) (the reader is invited to check the claim). Except on the diagonal poles of order > 1 at 0 can appear only in the entries (j, 1), (j, 2), . . . , (j, i), (j + 1, i), (j + 2, i), . . . , (n, i), see the choice of Bu in 1°.

16

3°. One deletes the polar terms below the diagonal by a change X >—> VX, V = I + V where each entry Vk\u of V is a suitably chosen polynomial pk,„ of 1/t, the non-zero entries being in the positions cited at the end of 2°. The degree of the polynomial pk,v is equal to the order of the pole in position (k, v) which has to disappear. We leave for the reader the proof that such a choice of the polynomials pk,v is really possible. 4°. As a result of the changes from 2° and 3° the system remains Fuchsian at cij for j < p + 1 and the conjugacy classes of its residua do not change because the matrix V is holomorphic for t ^ 0 and detV = 1. The system remains Fuchsian at 0 as well and the eigenvalues of Ap+2 change as follows: Mi l—> fJ'i — 9i f-j *-* llj + 5> the rest of the eigenvalues remain the same. (One should rearrange after this the eigenvalues pi in decreasing order by conjugating with a constant permutation matrix.) One checks directly that as a result of the change of the eigenvalues p,j the mean value a remains the same whereas S decreases. • 4-5

Proof of Lemma 36

1°. Suppose that the centralizer Z is nontrivial. Hence, it contains either a diagonalizable matrix D with exactly two different eigenvalues or a nilpotent matrix N ^ 0 such that iV2 = 0. 2°. Suppose that D = I

...

€ 2 with diagonal blocks of sizes /' and

n — V and with a ^= /3. Then the matrices Mj are block-diagonal with the same sizes of the diagonal blocks and the monodromy group is a direct sum. This follows from [Mj,D] = 0. Denote the two diagonal blocks of Mj by Sj and Tj (Sj is I' x /'). Hence, there are two subspaces of the solution space (Xi and X-i) which are invariant for the monodromy group and whose direct sum is the solution space. Denote by C', C" the conjugacy classes of the matrices Sj and Tj. 3°. Use a result from [Bol] (see Lemma 3.6 there): Lemma 41. The sum. of the eigenvalues X^j of the matrices-residua Aj corresponding to an invariant subspace of the monodromy group is a nonpositive integer. Remark 42. 1) Condition C) and Remark 1 imply that the equality exp(27riAfc.j) = akj defines (for j < p + 1 fixed) a bijection between the eigenvalues akj and the eigenvalues Xkj modulo permutation of equal eigenvalues. For j = p + 2 this is false (recall that Aj,%p+2 = Hk G Z,

17

°~\,p+2 — . . . — <7n,p+2 — ! ) •

2) When defining the sets of eigenvalues X^j corresponding to the subspaces X\ and X-2 it is true only for j < p + 1 but not for j = p + 2 that these sets are complementary to one another, i.e. one and the same eigenvalue Afc,p+2 = Mfc might appear in both sums while another one might appear in none of them. Indeed, present the eigenvalues Xk.j in the form ipkj + pk,j with ; := aiE! + 6,6' + A,;, a,: G N, 6j 6 Z, 61 + 62 = 0 where Ai (resp. A 2 ) is the sum of some /' (resp. n — V) eigenvalues AfciP+2 = //& (see Remark 42); hence, |A,| < iifi. One has at < n (evident), and |S'| < n/x (because the sum of all eigenvalues Xk,j (which is 0) is of the form gE' + Y2'k=i llk w ^ ^ 9 £ N , 1 < j < n; hence, |H'| < nfi/g < npi). If bi > 0, then i > 6i(n 2 + ? i ) / / - a i | E ' | - | A i | > (n2 + n)p-n2fi-~np, > 0. This contradicts Lemma 41. Hence, bi < 0. In the same way 62 < 0. Hence, b\ = 62 = 0. This means that equal eigenvalues of the blocks Sj and Tj have proportional multiplicities.

18

5°. The monodromy group of a Fuchsian system satisfying the condition b\ = b2 = 0, see 4°, cannot be analytically deformed into an irreducible one for nearby relatively generic eigenvalues and with the same Jordan normal forms of the matrices My, this contradicts condition F). Indeed, suppose that there exists such a deformation analytic in e £ (C,0) (i.e. for almost all values of s ^ 0 the (p + l)-tuple is irreducible). For the (p + l)-tuple before the deformation the multiplicities of the equal eigenvalues a> j of the two diagonal blocks Sj and Tj are proportional for all j . This means that for all j one has d(Cj) = {V2/n2)d{Cj), d(Cj') = ((n l')2/n2)d{Cj). Indeed, if a diagonal JNF is defined by the PMV ( m i , . . . , ms), then a conjugacy class defining such a JNF is of dimension n'1 — X]i=i( TO i) 2 Hence, d(C[') + ...+ d{C'^+1) = 2(n - I')2, d{C[) + ... + d(C'p+1) = 2/' 2 (this follows from the proportional multiplicities) and for the representations M', M." defined by the matrices Sj, Tj one has Ext1 (M',M") = Ext1 (M",M')=0

(11)

6°. When one deforms analytically a (p + l)-tuple into a nearby one (see the basic technical tool) one can express the deformation as a superposition of two deformations - of a change of the eigenvalues (see the matrices Nj^(e) in (8)) and of a conjugation (see the matrices Xj^{e) there). One can choose the matrices Nj^ to be polynomials of the matrices Mj, i.e. block-diagonal, with diagonal blocks of sizes /' and n ~ I'. Hence, the two non-diagonal blocks of the matrices change (in first approximation w.r.t. e) only as a result of the conjugation. Condition (11) shows that up to conjugacy the (p + l)-tuple remains block-diagonal in first approximation w.r.t. e. Hence, one can conjugate it by a matrix analytic in s to make the non-diagonal blocks zero in first approximation w.r.t. e. In the same way one shows that the (p + l)-tuple is block-diagonal up to conjugacy of any order w.r.t. e. The deformation being analytic, the (/; + l)-tuple is block-diagonal up to conjugacy for e small enough and non-zero - a contradiction. 7°. If there exists N G Z like in 1°, then one can conjugate the matrix N /0 0 / \ (PjUjVjX and the matrices Mj to the form TV = 0 0 0 , Mj = \ 0 Qj W3 where

\00 0j

\0

0

Pj

the middle row and column of blocks might be absent. If they are absent, then the moriodromy group is a direct sum. Indeed, for the conjugacy classes Cj of the matrices Pj one has d{C'j) = d{Cj)/4, hence, d(C[) + - • •+d(C'p+1) = n2/2, see 5°. One has (11) with M.' = M" being the representation defined by the matrices Pj. Hence, the monodromy group is indeed a direct sum.

19

8°. Suppose that the middle row and column of blocks are present. Lemma 41 and conditions D) and E) imply that the multiplicities of the eigenvalues of the matrices Mj for the diagonal blocks Pj and Qj are pro( P Uportional. Indeed, the blocks Sj = Pj (the upper Pj) and Tj = I .-' nJ define invariant subspaces X\ and X2 of the monodromy group. Like in the case when D s Z, see 1°, and using the same notation one shows that equal eigenvalues of the matrices Sj and Tj are of proportional multiplicities. This implies that there holds (11), hence, the monodromy group is a direct sum of the groups defined by the blocks Sj and Tj, i.e. after a simultaneous conjugation of the matrices Mj one has Vj = Wj = 0. Hence, there exists D G Z like in 1° which possibility is already rejected. 4-6

Proof of Lemma 37

1°. The monodromy group can be conjugated to a block upper-triangular form. The diagonal blocks define either irreducible or one-dimensional representations. The eigenvalues of each diagonal block l x l satisfy the nongenericity relation (70) from Lemma 36. 2°. The lowest diagonal block is of size 1. Indeed, set Mj = I

J f

T

J where Lj is the restriction of Mj to the

lowest diagonal block (say, of size h). Denote by E, © the sets of eigenvalues o~k,j, j < P+ 1 participating respectively in ( 7 B ) , (70) and by E', 0 ' the sums of their respective eigenvalues Xkj- Hence, the set of eigenvalues of the blocks Li, . . . , Lp+i is of the form aS + b&, a e N, 6 0, b > 0, then condition (/3/,) is not fulfilled by the blocks Lj (this condition is necessary because these blocks define an irreducible monodromy group of h x /i-matrices). Indeed, for b = 0 it is not fulfilled because it is not fulfilled by the matrices Mj and the multiplicities of equal eigenvalues of Mj and Lj are proportional. When increasing h, i.e. when increasing b g Z while keeping a fixed it is only the biggest multiplicity that increases and it is of an eigenvalue equal to 1. Hence, the sum of the quantities Tj computed for the matrices Lj remains the same while their size h increases. On the other hand, one cannot have b < 0 because in this case the sum of the eigenvalues Xk.j corresponding to the invariant solution subspace on which the monodromy group acts with the blocks Qj would be positive which contradicts Lemma 41. Indeed, the sum of these eigenvalues equals <j> := cE! — 6 0 ' + A where A is the sum of some n — h eigenvalues of the matrix Ap+2- We prove that <j> > 0 like we prove that 0 in 4° of the proof of

20

Lemma 36. Hence, a = 0 which leads to b = 1. 3°. Denote by El the left upper (7* — 1) x (n - l)-block. Conjugate it to make all non-zero rows of the restriction of the (p + l)-tuple M of matrices Mj — I to II linearly independent. After the conjugation some of the rows of the restriction of M to II might be 0. In this case conjugate the matrices Mj by one and the same permutation matrix which places the zero rows of Mj — I in the last (say, m) positions (recall that the last row of Mj - I is 0, see 2°, so m > 1). Notice that if the restriction to II of a row of Mj — I is zero, then its last (i.e. n-th) position is 0 as well, otherwise Mj is not diagonalizable. 4°. There remains to show that m > n — 111. One has Mj =

Gj Rj

0 I I G GL(m,C). Denote by G the representation defined by the matrices Gj. We regard the columns of the (p + l)-tuple of matrices Rj as elements of the space F(G) (or just T for short) defined as follows. Set U* = (U\,..., Up+\). Set V = { U* I Uj = {Gj - I)Vj, Vj e Cm, ^ + J Gx... G3-XUj = 0 }, £ = { U* J Uj = (Gj - I)V, V e Cm }, T = V/6. Remark 43. there holds

If Rj = {Gj - I)V with V e Cm or with V e A/ m ,„- TO , then P+i YJGi---Gj~iR]=0 i=i

(12)

One has £ C V. Equality (12) with V e M m , „ _ m is condition (2) restricted to the block R. 5°. Each column of the (p + l)-tuple of matrices Rj belongs to the linear space T>. The latter is of dimension 6 = r\ + ... + rp+\ — (n — m). Indeed, the image of the linear operator Tj : (.) 1—> (Gj — /)(.) acting on C r a _ m is of dimension rj (every column of Rj belongs to the image of this operator, otherwise Mj will not be diagonalizable). The n — m linear equations resulting from (12) with Rj = Uj = (Gj - I)V, V € C m are linearly independent. Indeed, if they are not, then the images of all linear operators Tj must be contained in a proper subspace of C r a ~ m (say, the one defined by the first n — ?7i — l vectors of its canonical basis). This means that all entries of the last rows of the matrices Gj — / are 0. The matrices Mj being diagonalizable, this implies that the entire (n — m)-th rows of Mj — I are 0. This contradicts the condition the first n — m rows of the restriction to II of the (p + l)-tuple

21

of matrices Mj - I to be linearly independent, see 3°. 6°. The space T is of codimension n - m i n V, i.e. of dimension 0 - 2(n - m). Indeed, each vector-column V belongs to C n _ m and the intersection X of the kernels of the operators Tj is {0}, otherwise the matrices Mj would have a non-trivial common centralizer — if I ^ {0}, then after a change of the basis of C n _ m one can assume that a non-zero vector from J equals *(1,0,..., 0). Hence, the matrices Gj are of the form I „ ~„ I, G* e GL{n — m — 1,C), and one checks directly that [Mj, -Ei,n] = 0 for J5i,„ = {(5i_i,ra_j}. 7°. The columns of the (p + l)-tuple of matrices Rj (regarded as elements of J7) must be linearly independent, otherwise the monodromy group can be conjugated by a matrix I

n p

J, P <E GL{m,Q),

to a block-diagonal form in

which the right lower blocks of Mj are equal to 1, the monodromy group is a direct sum and, hence, its centralizer is non-trivial - a contradiction. This means that dim J 7 = 9 — 2(n — m) = ri + ... + rp+i — 2(n — m) > m which is equivalent to m > n — n\; recall that n\ = r\ + ... + rp+\ — n. In the case of equality (and only in it) the columns of the (p + l)-tuple of matrices Rj are a basis of the space T. •

5

Case A)

In this section we prove Theorem 44. The DSP is not solvable (hence, not weakly solvable, see Lemma 13) for quadruples of diagonalizable matrices Mj each with MV equal to (n/2,n/2) where n > 4 is even, the eigenvalues are relatively generic and £ is a non-primitive root of unity of order n/2. Remark 45. In case A) for relatively generic eigenvalues there exist only block-diagonal quadruples of matrices Mj with diagonal blocks (n/l) x (n/l). Their existence follows from [Ko5], Theorem 3. The non-existence of others follows from Theorem 44. The proof of the theorem consists of three steps. We assume that irreducible quadruples as described in the theorem exist. The first step is a preliminary deformation and conjugation of the quadruple which brings in some technical simplifications, the quadruple remaining irreducible and satisfying the conditions of the theorem, see the next subsection. At the second

22

step we discuss the possible eigenvalues of the matrix M\M2 after the first step, see Subsection 5.2. At the third step we prove that the new quadruple must be reducible, see Subsection 5.3. 5.1

Preliminary conjugation and deformation

Set S = M\M2 = (M4)'1(M3y1.

Denote by gh hj the eigenvalues of Mj.

Lemma 46. The triple M\, AI2, S"1 admits a conjugation to a block uppertriangular form with diagonal blocks of sizes only 1 or 2. The restriction of the triple to each diagonal block of size 2 is irreducible. Indeed, suppose that the triple is in block upper-triangular form, its restrictions to each diagonal block being irreducible (in particular, the triple can be irreducible, i.e. with a single diagonal block). The restriction of Mj to each diagonal block (say, of size k) is diagonalizable and has eigenvalues gj and hj, of multiplicities 1° and k — 1°. Hence, the conjugacy class of the restriction of Mj to the block is of dimension 2l°(k — 1°) < k2/2. An irreducible triple with such blocks of M\ and AI2 of size k > 1 can exist only for k = 2, in all other cases condition (a/t) does not hold. Indeed, the conjugacy class of the restriction of 5 to the diagonal block is of dimension < k2 ~k. Hence, the sum of the three dimensions is < k2/2 + fc2/2 + k2 — k = 2fc2 - k which is < 2fc2 - 2 if k > 2. • Give a more detailed description of the diagonal blocks of the triple Mi,M2,S~l after the conjugation (in the form of lemmas; Lemmas 47, 50 and 51 are to be checked directly). Lemma 47. 1) There are four possible representations defined by diagonal blocks of size 1 of the triple; we list them by indicating the couples of diagonal entries respectively of Mi and Mi: P 9i,92\

Q hi,h-2;

R gi,h-2;

U hi,g2.

2) Denote by V and W any two of these couples. For a given V there exists a unique W (denoted by ~V) such that the corresponding diagonal entries of both Ali and M2 are different. One has P = —Q and R = —U. 3)One has dim Ext 1 (V, W) = 1 if and only ifV = —W. In the other cases one has dim Ext 1 (F, W) = 0. Lemma 48. of type -V.

There are equally many diagonal blocks of type V as there are

23

Indeed, consider first the case when there are no blocks of size 2. Denote by p\ q', r' and u' the number of blocks P, Q, R and U. The multiplicities of the eigenvalues imply that p' + r' = p' + u' = q' + u' — q' + r' = n/2. Hence, r' = u' and p' — q'. If there are blocks of size 2, then each of them contains once each of the eigenvalues <7i, 52, h\, hi and the proof is finished in the same way as in the particular case considered above. • Lemma 49. In an irreducible representation defined by a 2 x 2-block the eigenvalues of S can equal any couple (A,/z) (with A// = g\h\g2hi) which is different, from (g\gi,h\h-i) and (g%h-2, gihi)'. Indeed, one can show (the easy computation is omitted) that if the eigenvalues of S equal gig2, h\li2 or #1/12, 52^1, then the triple is triangular up to conjugacy. On the other hand, if one fixes Mi =diag(<Ji,/ii) and varies M2 within its conjugacy class, one can obtain any trace of the product Mi A/2 • The determinant of the product being fixed, this means that M1M2 can belong to any non-scalar conjugacy class the product of whose eigenvalues equals <7i/ii<72/i2- (The choice of the eigenvalues excludes the possibility S to be scalar.) • Lemma 50.

The semi-direct sums defined by two diagonal blocks of size 1

are up to conjugacy of one of the types: (Afi,M2) = or I (

\\mhij,

1 ,I

\m'g, 2JJ1 I

\\s hi )'U'/» 2 JJ

with either r = r' = 0 or s = s' = 0 but not both

(resp. with either u = u' = 0 or m = m' = 0 but not both). Such semi-direct sums exist only for couples (V, — V), see 2) and 3) from Lemma ^1. The centralizers of these semi-direct sums are trivial. Denote by $, respectively an irreducible representation of rank 2 defined by a diagonal block of the triple Mi, Af2, S 1 - 1 and a representation which is either irreducible and non-equivalent to $ or one-dimensional (i.e. of type P, Q, R or U, see Lemma 47) or a semi-direct sum of two one-dimensional ones (V, —V), see Lemmas 47 and 50. Lemma 51.

One has d i m E x t 1 ^ , * ) = d i m E x t 1 ^ , $) = 0.

Definition 52. We say that the triple Mi, M 2 , S'1 or M 3 , M4,S is in special form if it is block-diagonal, each diagonal block Z?M being itself block upper-triangular, its diagonal blocks being of equal size which is either 1 or 2.

24

In the case of size 2 all diagonal blocks of each block BM define equivalent representations. In the case of size 1 the block B^ is of size 2 and defines a semi-direct sum, see Lemma 50. Thus a triple in special form is block upper-triangular with diagonal blocks of size 2 defining either irreducible representations or semi-direct sums like in Lemma 50. Lemma 53. One can deform the matrices Mj within their conjugacy classes (without changing the matrix S) so that after the deformation each of the triples Mi, M2, S 1 - 1 and M3, M4, S after a suitable conjugation is in special form. The two conjugations are, in general, different. The lemma is proved in Subsection 5.4. 5.2

The possible eigenvalues of the matrix S

The eigenvalues of the matrix S (even when they are distinct) must satisfy certain equalities — for every diagonal block of size 2 (irreducible or not) of the triple Mj, Mi, S~l (resp. M3, M4, S) the eigenvalues A, fi of S must satisfy the condition fin/ii^^A-V-1 = 1 (resp. g^higihiX^ = 1). In what follows we denote the eigenvalues of S by s». Let the triple M\, M2, S"1 (resp. M3, M\, S) be in special form. For each eigenvalue s* denote by t(si) (resp. by u(si)) the eigenvalue of S in the same diagonal 2 x 2-block of the triple with s*. Note that t(t(si)) = s, = M(U(S,)). One has i(sj) = U(SJ) if and only if £ = 1 (and this holds for alii = 1, . . . , n). Set i\ = 1. For the eigenvalue s\ = six find Sj2 = £(si,), then find Sj3 = u(si2), then Sj4 = t(si3), then again s,5 = U(SJ 4 ) etc. Thus one has s W l = t(sl:,) for i/ odd (hence, t(slu+1) = *(t(si„)) = s , J and s ii/+1 = u(s,-J for J/ even (hence, U(SJ„ + 1 ) = u(u(siv)) = s,;,,). Denote by m the least value of a for which one has ia = 1. It is clear that m — 1 is even. Lemma 54.

For ;/ odd one has s^ + 1 = ^s,;^_ i; /or v even one has Sj„+1 =

Indeed, there holds gihig2hos~^(i(sj„))_1 = g-zhi,g\h\Sivu(.s,;iJ = 1 and l _1 s n t = i 3 i ' j = £• Hence, ^ <(sj, / ) = u( i„)- For v odd this yields £ _1 Sj„ +1 = £~1t(si„) = u(siu) = u(u(Siv_l)) = «*„_,, for J/ even in the same way it gives Lemma 55.

One has m — 1 < n/2 and m — 1 divides n/2.

25 Proof. Recall that £ = exp(2kni/(n/2)) = exp(ikiri/n) (see Subsection 2.2). If k = 0, i.e. £ = 1, then S3 = si, i.e. m — 1 = 1, and the statement holds. Let k ^ 0. Then Si+( m _i) = ( £ ) _ m + 1 s i = s i (Lemma 54). Hence, ( ^ - m + i = 1 ; [e 4 f c ( m _ 1) = 2?iZ (/ is defined in Subsection 2.2), i.e. k(m 1) = (n/2)l. The minimality of m (hence, of m — 1 as well) implies that m — 1 and I are relatively prime, i.e. m — 1 divides n/2. The non-primitivity of £ implies k > 1. Hence, m — 1 < n/2. • Remark 56. Lemma 55 implies that the set of eigenvalues of S can be partitioned into n/(2m —2) sets Mi, . . . , Afn/(2m-2) each consisting of (2m — 2) eigenvalues (denoted again by s,) with the properties S2fc+2 = £s2fc, S2fe+i ^ £ _1 s 2 fc_i, S2fc-iS2fe = gihig2h2 and s^s^fc+i = 53^354^4- If some of the sets Mi are identical, then we define their multiplicities in a natural way. Two nonidentical sets Mi have no eigenvalue in common. In what follows we change the indexation — equal (different) indices indicate identical (different) sets Mi. 5.3

End of the proof of Theorem 44

Case 1) The matrix S has at least two different sets MiThen the upper-triangular form of the triple M j , A/2, S is in addition block-diagonal, the restrictions of the matrix S to two different diagonal blocks having no eigenvalue in common. Indeed, it suffices to rearrange the blocks ,BM from the special form putting first all the blocks J5M with eigenvalues of S from Mi (repeated with its multiplicity — this defines the diagonal block i?i), then all blocks with eigenvalues of S from M2 (this defines the diagonal block R2) etc. The size of the block Ri equals U, times the number of eigenvalues from Mi, h e N*. The triple A/3, A/4, S admits a conjugation to the same block-diagonal form. Hence, if the triple A/i, A/2, S'1 is block-diagonal (with diagonal blocks Ri), to give the same form of the triple A/3, M4, S one has to use as conjugation matrix one commuting with S, hence, a block-diagonal one with diagonal blocks of the sizes of the blocks Rt. Hence, both triples are simultaneously block-diagonal, i.e. the quadruple A/i, A/2, A/3, A/4 is block-diagonal, i.e. reducible. Case 2) There is a single set Mi repeated n/(2m — 2) times. In this case one can deform the matrices Mj, j = 1, 2, so that the matrix S have at least two different sets Mi of eigenvalues. Definition 57.

We say that a matrix is in s-block-diagonal (resp. in s-block

26

upper-triangular) form if it is block-diagonal (resp. block upper-triangular) with diagonal blocks all of size s. Set fj, = n/(2m - 2). Conjugate the triple Mi, M2, S to a (2m - 2)-block upper-triangular form where the diagonal blocks of the matrix S are with eigenvalues from A/"i: 0 Mi

\ 0

A

M

i

0

H.r.i H i;2,M

(TQU2

0 T J = 1,2,

A/j /

6' =

\0

0

T J

We assume that the blocks A/' and T are 2-block-diagonal. Deform analytically the left upper blocks of size 2m - 2 of the matrices Mi, M'i and S so that they remain 2-block-diagonal and the eigenvalues of S change to new ones, forming again a set of 2m — 2 eigenvalues like in Remark 56 but different from A/i. To this end one can keep the matrix M\ the same and vary the left upper block of the matrix M-i\ see Lemma 49. This block will become M!2 + ell, e 6 (C, 0), U G gl(2m - 2, C), and the one of S will equal Mi(M^ + eU). The other blocks of Mi, A/2 and S do not change. One can deform in a similar way the triple of matrices A/3 M 4 _1 (requiring the deformation of S to be the same in both triples). For s ^ 0 small enough the quadruple of matrices remains irreducible. However, there are already two different sets Mi of eigenvalues of S, so we are in Case 1) and the quadruple is block-diagonal. Hence, the initial quadruple is also reducible. 5.4

Proof of Lemma 53

Notation 58. Assume that the triple M\,M'2,S satisfies the conclusion of Lemma 46. Block-decompose each matrix from gl(n,C) the sizes of the diagonal blocks being the same as the ones of the triple Mi, M2, S. Denote the block of this decomposition in the i-th row and /c-th column of blocks by ([i, k]). By (i, k) we denote the matrix entry in the i-th row and A;-th column. 1°. Up to conjugacy the triple Mi, M2, S is block-diagonal, with two diagonal blocks (T and Y) which are block upper-triangular, their diagonal blocks being respectively of size 1 and 2, the latter defining irreducible representations. Indeed, whenever a block ([i,i + 1]) of the triple Mi, A/2, S is of size 1 x 2 or 2 x 1, it can be made equal to 0 by a simultaneous conjugation of the triple with a matrix of the form I + R where only the block ([i, i + 1]) of

27

R is non-zero. This follows from Lemma 51. After this in the same way one annihilates all blocks ([i, i + 2]) of size 1 x 2 or 2 x 1, then all blocks ([i, i + 3]) of these sizes etc. Then one rearranges the diagonal blocks putting the ones of size 1 first and the ones of size 2 next. This gives the claimed form. 2°. The block Y after conjugation becomes block-diagonal, its diagonal blocks B^ being block upper-triangular, their diagonal blocks being of size 2. The diagonal blocks of one and the sam.e (resp. of different) diagonal blocks _BM define equivalent (resp. non-equivalent) representations. This is proved by analogy with 1°, making use of Lemma 51. 3°. Denote by Vi, . . . , Vn the diagonal blocks of T. One can conjugate the triple A/i, M-2, S by an upper-triangular matrix so that after the conjugation only these blocks ([i,j]), i < j , remain possibly non-zero for which Vi = —Vj. This is proved like 1° and 2°, making use of 2) and 3) of Lemma 47. 4°. After a conjugation and deformation the block T of the triple Mi, Mi, S becomes block-diagonal, with upper-triangular diagonal blocks of size 2 defining semi-direct sums, see Lemma 50. The proof of this statement occupies 4°-5°. It completes the proof of the lemma. A conjugation of the triple M\, A/2, S with a permutation matrix places the set of blocks P and Q first and the set of blocks R and U last on the diagonal; the triple remains block upper-triangular, in addition it is blockdiagonal, the sizes of the diagonal blocks equal respectively j)P + jjQ and $R + §U (one of these sizes can be 0). It suffices to consider the case when only, say, blocks P and Q are present, in the general case the reasoning is the same. Observe first that the blocks P and Q can be situated on the diagonal in any possible order. The eigenvalues of the restrictions of S to the blocks P and Q being different, one can conjugate the triple with an upper-triangular matrix to make S diagonal. Moreover, all blocks {[i,j]), i < j , with V,: = Vj are 0, otherwise at least one of the matrices A/i, A/2 will not be diagonal. 5°. Consider first the case when the triple after this conjugation becomes diagonal. Rearrange the blocks in alternating order — P, Q, P, Q, .... Make non-zero the entries (1,2), (3,4), (5,6) etc. of the matrices Mj without changing the matrix S. With the notation from Lemma 50 this amounts to choosing s = s' = 0, r ^ 0, r' = — r/i2#f (look at the first couple (A/i,A/2) from the lemma). This gives the necessary block-diagonal form of the block T. The representations P and Q being non-equivalent, the centralizers of the diagonal blocks are trivial. Suppose now that the triple is not diagonalizable and that V] — P (the

28

case Vi = Q is considered by analogy). Denote by zi < . . . < ih the indices i for which Vi = Q. Denote by m the smallest iv for which at least one of the entries (k,m) of M\ and M2 is non-zero, k < m; by 3°, k is not among the indices iv. Denote the greatest such value of k byfco-Hence, all entries (i, fc0) (i < ko) and (ko,p,) (/•* < m ) °f ^ i a n d -^2 are 0, otherwise these matrices will not be diagonalizable. One can annihilate all entries (k', m) oi Mj where k' < ko by consecutively conjugating the triple Mi,M2,S by matrices of the form / + gEk\k0- Note that the values of k' are not among the indices iv. In a similar way one annihilates all entries (ko,k") of Mj with k" > m by consecutive conjugations with matrices of the form / + gE,n,k" • Hence, it is possible to conjugate the triple by a permutation matrix putting the fco-th and m-th rows and columns first and preserving its uppertriangular form; in addition, the triple will be block-diagonal with first diagonal block of size 2 (which is upper-triangular non-diagonal and with trivial centralizer). After this one continues in the same way with the lower block. In the end the block T will become upper-triangular and block-diagonal, with diagonal blocks of size 2 each of which is triangular non-diagonal with trivial centralizer. 6

Case C)

L e m m a 59. If K = 0 and if the DSP is solvable for a (p + 1)-tuple of conjugacy classes Cj with relatively generic eigenvalues defining the diagonal JNFs J™, then the DSP is solvable for any (p + 1)-tuple of JNFs Jjn and for any relatively generic eigenvalues with the same value of £ where for each j the JNFs J " and J ' " correspond to one another or are the same. The lemma is proved at the end of the subsection. Assume that there exist irreducible triples of diagonalizable matrices Mj such that M1M2M3 = / , the PMV of the eigenvalues of the matrices being equal to (d, d, d, d), (d, d, d, d), (2d, 2d). Denote by Ok,j the eigenvalues of Mj where k = 1, 2,3,4 if j = 1 or 2 and k = 1, 2 if j = 3. One can choose the eigenvalues of A/i and M% such that a\j = —
= MXM2,

L2 = B = M2Mi,

29

U = (Mi)-\M2)-2MU

U = (M^-2.

One has L1L2L3L4 = I. The matrices Lj are diagonalizable, their MVs equal (2d, 2d) and by Case A) they define a block-diagonal algebra C with 2fc blocks 2s x 2s. Hence, dimC < 8fcs2. The algebra C contains also the matrices (Lj)~l. Hence, it contains the matrices (Mi)2 = (L4)" 1 , MXM2 = Lu Af2Mi = L2 and (M2)2 = (M2M1)(L3)-1(M2M1)-1. Every matrix from the algebra V generated by M\ and M2 is of the form K + MiL + M2S with K, L, S e C. Hence, dim2? < 3dimC < n 2 = dimgl(n, C). By the Burnside theorem, the matrix algebra V is reducible. Proof of Lemma 59: 1°. Suppose that the DSP is not solvable for the JNFs J'™ and for some relatively generic but not generic eigenvalues. Prove that then it is not solvable for the JNFs J™ and for any such eigenvalues. Note first that the JNFs J " and J'™ satisfy the conditions of Theorem 10, see Corollary 21. 2°. An irreducible (p + l)-tuple Ti of matrices M3 with JNFs J™ can be realized by a Fuchsian system with diagonalizable matrices-residua Aj such that J(Aj) = J(Mj) for j < p+l and with an additional apparent singularity, see Subsection 4.1 with the definition of the sets Qi, the maps Xi and M.. One can choose i such that Xi(Gi) is dense in M. 3°. Vary the eigenvalues of the matrices Aj within the set Qi without changing their JNFs. For suitable eigenvalues (in general, with integer differences between some of them; see 5°) one obtains as monodromy group 7i' of the Fuchsian system one in which either J(Mj) = J j n or J(Mj) is subordinate to Jj", i.e. the multiplicities of the eigenvalues are the same and for each eigenvalue A and for each s e N, rk(Afj — A/) s is the same or smaller than should be, see the details in [Ko4]. One can assume that the eigenvalues of the matrices Mj are relatively generic. Such a monodromy group cannot be irreducible (otherwise one could deform it using the basic technical tool into a nearby one with the same eigenvalues and with J(Mj) = J j n for all j ; such irreducible monodromy groups do not exist by assumption). 4°. The monodromy group Ti' can be analytically deformed into the monodromy group Ji because both are obtained from the Fuchsian system for different eigenvalues of the matrices-residua. However, Ti' cannot be analytically deformed into a nearby irreducible monodromy group with JNFs as in Ti which is a contradiction. Indeed, if for all j one has J(Mj) = Jj™ in Ti', then the monodromy group Ti' must be block-diagonal with diagonal blocks of equal size and for the representations $ 1 , $2 defined by two diagonal blocks one has E x t ^ ^ i , $2) <

30

0 with equality if and only if $ 1 , $2 are not equivalent. The last inequality holds also if for some j , J(Mj) is subordinate to J'™. After this one applies the reasoning from 5°-8° of the proof of Lemma 36. 5°. It is explained in [Ko4] how to choose the eigenvalues from 3° to obtain the monodromy group H' with J(Mj) equal or subordinate to J j n . Their possible choice is not unique — if one adds to equal eigenvalues of the matrices Aj equal integers the sum of all added integers (taking into account the multiplicities) being 0, then one obtains a new possible such set of eigenvalues; different eigenvalues of a given matrix Aj must remain such and if two eigenvalues of a given matrix Aj differ by a non-zero integer, then the order of their real parts must be preserved. From all these a priori possible choices there is at least one which is really possible, i.e. for which there exists such a point from £,;. Indeed, Qi is constructible and its projection on the set of eigenvalues W must be dense in W, see Subsection 4.1. • 7

Case B)

Definition 60. A special triple is an irreducible triple of matrices Mj such that M1M2M3 = / , Mi — / and M2 — I being conjugate to nilpotent Jordan matrices consisting each of n / 3 Jordan blocks of size 3, Af 3 being diagonalizable, with three eigenvalues each of multiplicity n / 3 . The eigenvalues are presumed to be relatively generic but not generic. In the present subsection we prove that special triples do not exist. By Lemma 59, there exist no irreducible triples from Case B). Lemma 6 1 . Suppose that there exist special triples. Then there exist special triples satisfying the conditions i) Im (Mj - I) n Ker (M2-j - I) = {0}, 3 = 1,2 ii) C n = Ker (Mi -I)® Ker (Af2 - /) e (Im (Mi - /) n Im (M2 - I)). Corollary 62.

If there exist special triples, then there exist special triples / 0 0 0\ in which the matrices M\ - I, M% - I are of the form Mi - I = P O O ,

\Q ROj (0IV\ M-i - I = l 0 0 I \ i n which all blocks are (n/3) x (n/3), the matrices P arid \00 0/ R being non-degenerate.

31

The lemmas and the corollary from this section are proved at its end. Let the matrices Mj be like in Corollary 62. Consider the matrices

fI00\

fiov-i\

f0I0\

/ooo\

Ni =

0/ / , N2 = \ P I 0 , G = 0 0 0 , H = 0 0 0 . \00 / / \Q0IJ \000j \0R0j Hence, each of the matrices Ny—I,N2 — I,G + H,G and H is nilpotent and conjugate to a Jordan matrix consisting of n/'i blocks of size 2 and of n / 3 blocks of size 1. One has (to be checked directly) N2(I + H)=M1, i.e. (I + G)N1=M2,

M1-I=(N2-I){I i.e. M2-I=(I

GH = G2 =H'2 = HG = 0,

(Ni-I)G

+ H) + (G + H)-G + G)(N1-I) +G = 0,

(N2-I)H

=0

(13) (14) (15)

Hence, N2(I + G + H)Ni = M\M2. Denote by A the matrix algebra generated by the matrices N\ — I, G + H and ./V2 — / , by B the one generated by A/i and A/2. Lemma 63.

The matrix algebra A is reducible and dim*4 < n 2 /2.

One has B = A + GA + AG + GAG + GAGA + AG AG + ••• (*). Indeed, every product of the matrices M\ —I and M2 — I (in any order and quantity) is representable as a linear combination of such products of the matrices JVi — / , N2-I,G + H and G, see (13), (14) and (15). On the other hand, one has AG C A. Indeed, denote by Y a product of the matrices N\ — I, N2 — I, G + H (in any quantity and order). If its right most factor is Ni - I or G + H, then by (15) one has YG = 0. If it is N2 - I, then YG = Y(G + H)-YH = Y(G + H) e A. This together with (*) implies that B = A + GA (**). Suppose that the couple of matrices A/i, M2 is irreducible. Then by the Burnside theorem the algebra B equals gl(n, C), i.e. d i m S = n 2 . The restriction of each matrix from B to the last 2n/3 rows is the restriction to them of a matrix from A, see (**). This means that dim ,4 > 2n 2 /3 which contradicts Lemma 63. Hence, special triples do not exist. Proof of Lemma 61: 1°. Recall that the three conjugacy classes Cj of the matrices Mj belong to SL(n,C). Denote by U the variety of irreducible representations (i.e. triples (A/i, A/2, A/3) defined up to conjugacy) where Mj e Cj c SL(n,C), MiM2M3 = / . Find dimZ//. One has to consider the cartesian product C\ x C2 C (SL(n,C) x SL(n,C)). The algebraic variety V C (SL(n,C))2 of irreducible couples of matrices My, M2 such that A/i € C\, M2 6 C2 and

32 (MIMJ)-1

G C3 is the projection in C\ x C2 of the intersection of the two varieties in C\ x C2 x SL(n, C): the cartesian product Ci x C2 x C3 and the graph of the mapping (Cx x C 2 ) 9 (Mi, M 2 ) >-> A/3 = M 2 ~ 1 Mf 1 G SX(n, C). This intersection is transversal which implies the smoothness of the variety V (this can be proved by analogy with 1) of Theorem 2.2 from [Ko2]). Thus dimV = (£5=1 dim Cj) - [( n 2 - 1) - dimC 3 ] (here (n 2 - 1) - dimC 3 = codim SL („ C )C73). Hence, dimV = dimCi + dimC 2 + dimC3 — n 2 + 1. 2°. In order to obtain dimW from dimV one has to factor out the possibility to conjugate the triple Mi, M2, M3 with matrices from SL(n,C). No non-scalar such matrix commutes with all the matrices (M\,M2,M-z) due to the irreducibility of the triple and to Schur's lemma. Thus dimW = dim V - dim SL(n, C) = Ylj=i d i m Cj " 2 ' ft2 + 2 = 2. 3°. The subvariety It' c U on which one has dim(Ker (Mj — I) n Im (M2-J — /)) > 0 for j = 1, 2 is of positive codimension in U. Indeed, its dimension is computed like the one of U, by replacing the cartesian product C\ x C2 by its subvariety on which one has dim(Ker (Mj —7)nlm (M2-J — I)) > 0 for j — 1, 2. This subvariety is of positive codimension. Hence, the condition dim(Ker (Mj - I) C\ Im (M2-j - I)) > 0 for j = 1, 2 cannot hold for all points from U. Condition ii) follows from statement i). D Proof of Corollary 62: 1°. One has dimKer(A/i - I) = dim Ker (M 2 - I) = n / 3 . Condition ii) of Lemma61 implies that dim(Im (Mi —7)nIm(M2 —/)) = n/3; recall that Ker (Mj - I) c Im (Mj - I), j = 1, 2. Choose a basis of Cn such that the first n / 3 vectors are a basis of Ker (M2 — I), the next n / 3 vectors are a basis of Im (Mi — I) fl Im (M2 — / ) and the last n / 3 vectors are a basis of Ker(A/i — I). Hence, in this basis the matrices of Mi — I, M2 — I look /0WV'\ like this: M 1 - I = \ P ' T 0 \ , M2 - I = 0 U Y ) (all blocks are 0 0 0

(n/3) x (n/3)). 0 WU2 WUY\ 2°. One has (M 2 - 1 ) = 0 U3 U2Y = 0. The rank of the matrix \0 0 0 / 3

j j , equals n / 3 because rk (M 2 — I) = 2n/3. Therefore the equalities ( U3 ) = ( 0 ) a n d ( U2Y ) = ( 0 ) i m p l y r e s P e c t i v e l y U2 = 0 an d UY = 0. It follows from rk(M 2 - I) = 2n/3 that rk(U Y) = n / 3 . Hence, the equality (U2 UY) = (0 0) implies U = 0.

33

3°. In the same way one proves that T = 0. A simultaneous conjugation IWY 0 0 \ of Mi - I and M2 - I with the matrix 0 Y 0 brings them to the

V o

oi)

desired form. Note that det W ^ 0 ^ det Y and det P' ^ 0 ^ det i?' due to rk(Mi - I) =rk(M 2 - J) = 2n/3. Hence, det P ^ 0 ^ det fi. D Proo/ of Lemma 63: Recall that one has N2(I + G + H)Ni = MiAf2 and that the matrix M\Mi is diagonalizable with three eigenvalues each of multiplicity n / 3 . Hence, the quadruple of matrices N2, I + G + H, Ni and (MiAh)"1 (their product is / ) is reducible — if the map * is applied to the quadruple, then one obtains a quadruple of conjugacy classes of size 2n/3 the first three of which are each with a single eigenvalue and with n/3 Jordan blocks of size 2 and the fourth of which is diagonalizable, with two eigenvalues each of multiplicity n / 3 . One can apply the basic technical tool to such a quadruple and deform it into one with relatively generic but not generic eigenvalues and in which all four matrices are diagonalizable and have two eigenvalues of multiplicity n / 3 . This is a quadruple from Case A) (recall that the value of £ is preserved), hence, block-diagonal up to conjugacy with diagonal blocks of one and the same size (Remark 45). Hence, there exist only block-diagonal up to conjugacy quadruples of matrices Ar2, I + G + H, Ni and (AfiAf 2 ) _1 and all their diagonal blocks are of the same size. The dimension of such a matrix algebra is < n 2 / 2 with equality if and only if there two diagonal blocks. • 8

Case D)

Set s = n/l (I was defined in Subsection 2.2). Hence, n = 6/cs, A: > 1 and the MVs of Mi, M2, M3 equal respectively (sk, sk, sk, sk, sk, sk), (2sk, 2sk, 2sk), (3sk, 3sk). Case D) can be reduced to Case B) like this: if the DSP is solvable in case D), then using Lemma 59 one can choose the eigenvalues of M$ to be ± 1 , i.e. (Mi)2 = / , and the ones of Mi to form three couples of opposite eigenvalues; hence, the MV of (Mi)2 is (2sk,2sk:2sk) and one has (A/i)~ 2 = M2(M3M2M3). Hence, the three matrices (Mi) 2 , M 2 and M3M2M3 == (M 3 ) - 1 M 2 M3 are from Case B). By assumption, they define a block diagonal matrix algebra A with 2k diagonal blocks 3sx3s (Remark 45). Hence, dim„4 < 18A;s2. The algebra A contains the matrices (Mi) 2 , (M 3 ) 2 , ( M i ) - I M 3 = M 2 and A/ 3 (Afi) _1 = M3M2M3. Every matrix from the algebra B generated by (A-fi) -1 and M3 (this is also the algebra generated by M i , M2 and M3) is representable as

34

K + MXL + M3N, K, L, N e A. Hence, d i m S < 54ks2 < n2 = 36k2s2 and this cannot be gl(n,C). By the Burnside theorem, B is reducible. 9

Proof of Theorem 15 in the case of matrices Aj

Suppose that the Deligne-Simpson problem is weakly solvable in one of cases A)- D) for matrices Aj with relatively generic but not generic eigenvalues. By Lemma 13 it is solvable as well. Construct a Fuchsian system with matrices-residua from an irreducible triple or quadruple corresponding to one of the four cases and with relatively generic eigenvalues. One can multiply the matrices-residua by c* € C so that no two eigenvalues differ by a non-zero integer and the eigenvalues of the rnonodromy operators become relatively generic. Hence, the rnonodromy group of the system is irreducible. Indeed, if it were reducible, then the eigenvalues of the diagonal blocks would satisfy only the basic non-genericity relation and its corollaries. The sum of the corresponding eigenvalues of the matrices-residua is 0 and, hence, one can conjugate simultaneously the matrices-residua to a block upper-triangular form, see [Bo2], Theorem 5.1.2. The irreducibility of the rnonodromy group contradicts part 1) of Theorem 15. References [Ar] V.I. Arnold, Chapitres supplernentaires de la theorie des equations differentielles ordinaires, Edition Mir, Moscou, 1980. [Aril] V.I. Arnold, V.I. Ilyashenko, Ordinary differential equations, in Dynamical Systems I, Encyclopaedia of Mathematical Sciences, t. 1, Springer 1988. [Bol] A.A. Bolibrukh, The Riemann-Hilbert problem, Russian Mathematical Surveys 45(1990), no. 2, pp. 1-49. [Bo2] , 21-ya problema Gil'berta dlya lineynykh Fuksovykh sistem, Trudy Matematicheskogo Instituta imeni V.A. Steklova. No. 206 The 21-st Hilbert problem for Fuchsian linear systems (in Russian). [Ka] N.M. Katz, Rigid local systems, Annals of Mathematics, Studies Series, Study 139, Princeton University Press, 1995. [Kol] V.P. Kostov, On the existence of rnonodromy groups of fuchsian systems on Riemann's sphere with unipotent generators, Journal of Dynamical and Control Systems, vol. 2, JV° 1, 1996, p. 125-155. [Ko2] , Regular linear systems on C P 1 and their rnonodromy groups, in Complex Analytic Methods in Dynamical Systems (IMPA, January

35

1992), Asterisque, vol. 222 (1994), pp. 259-283; (also preprint of Universite of Nice - Sophia Antipolis, PUMA TV0 309, Mai 1992). [Ko3] , On the Deligne-Simpson problem, C. R. Acad. Sci. Paris, t. 329, Serie I, p. 657-662, 1999. [Ko4] , On the Deligne-Simpson problem., Manuscript 47 p. Electronic preprint math.AG/0011013. [Ko5] , On some aspects of the Deligrie-Simpson problem, Manuscript 48 p. Electronic preprint math.AG/0005016. To appear in Trudy Seminara Arnol'da. [P] J. Plemelj, Problems in the sense of Riemann and Klein, Inter. Publ. New York-Sydney, 1964. [Si] C.T. Simpson, Products of matrices. Department of Mathematics, Princeton University, New Jersey 08544, published in "Differential Geometry, Global Analysis and Topology", Canadian Math. Soc. Conference Proceedings 12, AMS, Providence (1992), p. 157-185. Proceedings of the Halifax Symposium (Proceedings of the Canadian Mathematical Society Conferences), June 1990, AMS Publishers.

36

THEOREMS FOR EXTENSION ON MANIFOLDS W I T H ALMOST COMPLEX STRUCTURES

L.N. APOSTOLOVA

E-mail:

Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bontchev Street, Bl.8, 1113 Sofia, Bulgaria LtliaNAQm.ath.bas.bg, [email protected], [email protected] M.S. M A R I N O V Institute of Applied Mathematics and Informatics, Technical University, P.O.Box 384, 1000 Sofia, Bulgaria E-mail: [email protected] K.P. P E T R O V Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bontchev Street, Bl.8, 1113 Sofia, Bulgaria

The Hahn-Banach theorem for continuous extention of linear functional on subspace of linear normed space to whole space with preserving of the norm is extended for linear functionals depending smoothly on the point of a real manifold (i.e. when the functionals act on the fibers of a subbundle of vector bundle over a manifold and depend smoothly on the point of manifolds). This usually called Hahn-Banach theorem with parameter. Then such theorems are proved for vector bundle with one, two or tree anticommuting antiinvolutive automorphisms and invariant linear functional with parameter on an invariant subbundle with respect the authomorphism(s). The obtained results are Hahn-Banach type theorems for extention of functional on almost complex, almost quaternion or almost octonion manifolds when for fibre bundle is considered the bundle of tangent vectors invariant with respect to the given almost complex structure(s) on manifold. Corollary of the theorems for extention of invariant differential one-forms is given.

1

Introduction

The well known Hahn-Banach theorem for continuous extension of continuous linear functionals states that any continuous linear functional on a linear AMS 1995 Classification: Primary 53C15; Secondary 32D15, 46A22. Keywords: Hahn-Banach Theorem. Fibre Bundles. Partially supported by the contract MM-525/95 with the National Fund for Scientific Researches to the Ministry of Education and Sciences of Bulgaria.

37

subspace of a normed linear space has an extension to a continuous linear functional on the whole space with the same norm. It was first published by Hans Hahn [H] and Stefan Banach [B]. But yet in 1912 E. Helly [HI] proved the theorem for the moments for sequences in the space C\a, b] of continuous functions on a closed segment in the real line. In the more general context with a normed linear space E instead of C[a, b] and an arbitrary set of indices / instead the set of positive integers, this theorem is the Hamburger problem, which is equivalent to one of the versions of the Hahn-Banach theorem (cf. [Bs]). An extention of the theorem for linear normed spaces over the field of complex numbers was published by H.F. Bohnenblust and A. Sobczyk [BS] in 1938, where an example of a nonextendable complex linear functional on a real linear subspace of a complex linear normed space is given. The same year G.A. Suhomlinov [S] published a proof of this theorem for the complex and quaternion linear normed spaces. The case when the scalars are the octonions is considered in the paper of J.I. Lewis [L2] in 1988. A recent survey with wide circle of papers concerning Hahn-Banach theorem for extension is given by G. Buskes [Bs]. Here is proposed an extention of the Hahn-Banach theorem for linear continuous functionals "with parameter" and a theorem for extention of linear functional, invariant with respect to some antiinvolutive automorphisms on invariant subbundles. More precisely the cases of a paracompact manifold M and a bundle E on M with some antiinvolutive anticommuting automorphisms Ji, J2 etc. on E are considered. Very natural extention of the theorem claim that every continuous linear functional Fx : Ex —> R defined on a linear subspace Ex of the fiber Ex of the bundle E depending smoothly on the point x in M admits a continuous linear functional Fx : Ex —» R defined on the whole bundle E depending smoothly on the point x in M with the same norm (||F|| = \\F\\) and coinciding with the given functional F on E. This is a Hahn-Banach theorem "depending on parameter". In the case when on the bundle E has one or more antiinvolutive anticommuting automorphisms and the functional F is invariant with respect to them the extention of the functional F would be choosen to be invariant with respect to these automorphisms on the whole bundle E. The last case would be interesting for some goals of Clifford analysis on manifolds. The proofs of these two aspects of extention of the Hahn-Banach theorem use a decomposition of the vector bundle with metric as a direct sum of a given subbundle and its orthogonal subbundle in the case when the base manifold is a paracompact one. Application for extention of holornorphic one-forms from submanifolds of (almost) complex manifolds is a direct consequence of

38

the so obtained theorem if the bundle E is choosen to be the holomorphic tangent bundle of the (almost) complex manifold or the intersection of the holomorphic bundles with respect to the given antiinvolutive anticommuting automorphisms of the tangent bundle in the case of almost quaternion or almost octonion manifolds. Let us recall that a mapping || . || : Ex —> [0, +00) is said to be a seminorm if the following is fulfilled: a) the triangle inequality is true i.e. ||/ + g\\ < ||/|| + |j
\/f

eExy\eR.

In the case when the condition Jj/|| = 0 is valid if and only if / = 0 the seminorm is said to be a norm on the linear space Ex. , lljrll \F(f)\ \F(f)\ T , Ihe number ||.f || = supy e E = supj6B where B is the unit ball in E is said to be a norm of the linear functional F on Ex. Let us write some well known conditions for a linear real valued functional F on a linear seminormed space Ex over reals to be a continuous one. The following conditions are equivalent: • The linear functional F is continuous in the origin. • The linear functional F is uniformly continuous. • The linear functional F is continuous. • There exist number A>0

such that \F(f)\ < A\\f\\ for V/ from Ex.

• The linear functional F is bounded over the unit ball. • The set {/ e Ex : F(f) = 0} is closed. 2

Hahn-Banach type theorems

We will assume the manifold M to be a paracompact one. In the rest we consider only finite-dimensional manifolds and bundles. Theorem 1 (Hahn-Banach theorem with parameter). Let E be a vector bundle on a paracompact manifold M and E be a vector subbundle of E. Let F : E —• M x R be a morphisrn. of vector bundles. Then there exists a bundle morphisrn F : E -^ M x R such, that Fx(f) = Fx(f)

VfeEx.

Moreover, if a metric, depending smoothly on the point x <E M is fixed on Ex, then the extention morphisrn F would be choosen in such a way that the

39 linear Junctionals Fx on the fiber Ex oj the bundle E, depending smoothly on the point x e M to have the same norm as the given linear Junctionals Fx : Ex —> R for each point x G M: \\Fx\\ = \\FX\\. Here the norm on the fibers Ex is induced by the metric on

Ex.

A subbundle E of a vector bundle E with an antiinvolutive automorphism J (i.e. J : E —> E with J2 = —/) is called an invariant subbundle oj E with respect the. automorphism J, if Je G E for Ve € E. A bundle morphism F on a vector bundle E equipped with antiinvolutive automorphisms Ji,

J2,...,Jfc

2

to the bundle M x R equipped with imaginary units e\, e2, • • •, ek will be called an invariant morphism with respect the couples (Js,es) s = 1 , . . . , k on E if the following property holds: F(Jse) = esF(e), Ve s E and s = 1, 2, Now let us see how to construct hermitian metric on a given vector bundle E equipped with antiinvolutive automorphisms J i , J 2 , . . . , Jfe and a riemannian metric g. It is enough to put the hermitian metric with respect to J s , s = 1, 2, . . . , k to be the following h(X, Y) := ^{g(X, + g{JiJ2X, +

Y) + gMX, JXJ2Y)

JXY) + ... + g(JkX,

+ g{ J x J3X, Jx.hY)

JkY)

+ •••+ g{JxJkX,

(*) ,hJkY)

...+g(J1J2...JkX,J1J2...JkY)}.

We would like to verify t h a t h(JsX, JSY) = h(X,Y) for s = 1, 2, . . . , k. But in the right hand side of the equality (*) has two types of summands: 2k~l of t h e m have not t h e structure Js in t h e arguments of t h e riemannian structure g and the remaining 2 /> ' _1 have it. If we put in the sum (*) the vector fields JSX and JSY instead of X and Y respectively, we observe t h a t by the anticommutativity of the automorphisms J i , J2, . . . , Js we would arrange these automorphisms in the s u m m a n d s in increasing way of the indices without changing them. Then every s u m m a n d of the first kind give a s u m m a n d of the second kind. Also by the antiinvolutivity of the automorphisms every s u m m a n d of the second kind give a s u m m a n d of the first kind in the original sum (*). So the obtained sum is equal to

40

the original one in (*) and we prove that the obtained metric h(X, Y) is an hermitian one with respect to all automorphisms J i , J2, . . . , Jfc. For definition and some properties of riemannian and hermitian metrics on vector bundles would be seen Chapter 3 in [Hs]. Theorem 2. Let E be a vector bundle on a paracompact manifold M equipped with an antiinvolutive automorphism J. Let E be an invariant subbundle of E with respect to the automorphism J and F : E —> M x R 2 be an invariant morphism with respect to (J,i), i being the imaginary unit of the complex numbers and R" considered as a complex plane with standard complex structure. Then there exists a bundle morphism F : E —• M x R 2 which is invariant with respect to (J, i) with the property Fx(f)

= Fx(f)

VfeEx.

Moreover, if an hermitian metric g on the vector bundle E with respect to the automorphism J, depending smoothly on the point x e M is fixed, then the extention Fx would be choosen in such a way that \\F*\\ = \\FX\\, where the norm on the fibers is induced by the given metric. Theorem 3. Let E be a vector bundle on a paracom,pact manifold M equipped with antiinvolutive anticommuting automorphisms Jx, J-2Let E be an invariant subbundle of E with respect to the automorphisms Ji and J2 and F : E —> M x R 4 be an invariant morphism with respect to (Ji,i), (J2, j), i and j being imaginary units of the quaternions and R4 considered as a corps of quaternions with the standard quaternion structure. Then there exists a bundle morphism, F : E —> M x R 4 which is invariant with respect to ( i i , i ) and (J2,j) with the property Fx(f) = Fx(f)

VfeEx.

Moreover, if an hermitian metric g with respect to J\ and J2, depending smoothly on the point x £ M is fixed on the vector bundle E, then the extention Fx would be choosen in such a way that

\\FX\\ = II^H, where the norm on the fibers is induced by the given metric. Theorem 4. Let E be a vector bundle on a paracompact manifold M equipped with antiinvolutive anticommuting automorphisms J 1 ; J2, Js-

41

Let E be an invariant subbundle. of E with respect to the automorphisms Ji, J-2 and J3 and F : E —> M x R 8 be an invariant morphism with respect to (Ji,i), (J2,j), (J-3,k), i, j and k being imaginary units of the octonions and R8 considered as the algebra of octonions with standard octonion units. Then there exists a bundle morphism F : E —> M x R 8 which is invariant with respect to (Ji,i), (J2,j) and {J-i,k) with the property Fx(f) = Fx(f)

VfeEx.

Moreover, if an hermitian metric g with respect to J\, J2 and J3, depending smoothly on the point x 6 M is fixed on the vector bundle E, then the extention Fx would be choosen in such a way that \\FX\\ = \\FX\\, where the norm on the fibers is induced by the given metric. 3 3.1

Proofs Sketch of the proof of theorem 1

According the considerations in Chapter 3 in [Hs], for vector bundle with paracompact base would be constructed a riemannian metric g. For the vector subbundle E let us construct its orthogonal complementary H in E. Then it will be fulfilled E © x H = E. Then we define the extension Ffj of F on H to be equal to zero, and define linear functional FE on E (B1- H extending Fg and F^ by linearity. So we obtain a desired extension FE of the linear functional F to E ©-1 H = E. Let us remark that the idea for this natural proof was suggested by Daniel Lehmann in private communication for what he has our cordial thanks. 3.2

Sketch of the proof of theorem 2

Let now g be an hermitian metric with respect to J. We have E = E &1 H and g(Je, Jh) = g{e, h). Therefore J : E -> E, J : H -> H. Then F(Je + Jh) = iF{e) = iF(e + h), i.e. F(Je) = iF{e). Also \F(e + h)\ = \F(e)\ < \\F\\\\e\\ then \\F\\ < \\F\\ and ||F(e>)|| - \\F(e)\\ < \\F\\\\e\\ then ||F|| < ||F|| i.e. \\F\\ = \\F\\. 3.3

Sketch of the proofs of theorems 3 and 4-

Let us consider an hermitian metric g with respect to almost complex structures J\ and J2 or J i , J2 and J3 respectively.

42

The idea of the proofs is the same as above, because of E = E © x H, then J i , J2 : E —> E and J\, J2 • H —> H or ,7i, J 2 , J3 : E —> E and JUJ2,J3:H-+H. This follows from the property g(J\e, Jih) = g(e, h), g(J2e, Jih) = fif(e, /i) of the hermitian metrics with respect to the almost complex structures J i , J-2 or from the property g(Jie,J^h) = g(e,h), g(J2<3, J2I1) = g(e,h), g{ J^e, J3/1) = g(e,h), of the hermitian metrics with respect to the almost complex structures J\, J2, J3, respectively for the theorems 3 and 4. Other proof of the Theorems 2, 3 and 4 would be given using an idea in [BS], [S] and [L] for the case of availability of some antiinvolutive anticommuting automorphisms. Actually, the imaginary part(s) of the functionals F : E —> R2 (or F : E —» R4 or F : E —> R8 respectively) is (are) uniciuely determined by its real part. So, if we obtain an extention of the real part of F, this extension determines the corresponding extention(s) of the imaginary part(s), so would be obtained a desired extension of the given functional. 4

Extension of invariant differential forms

First let us give some definitions. An automorphism J of the real tangent bundle TM of a given smooth real manifold M is called an almost complex structure on M, if J 2 = —I, where I is the identity authomorphism. If on the manifold M is provided an almost complex structure, it is called an almost complex m.anifold. If an almost complex manifold M is finite dimensional, then this dimension is an even number. An almost complex manifold (M; J ) is called an almost quaternion manifold if it is equipped with a second almost complex structure K which anticommutes with the structure J, i.e. JK = —KJ. Then if the dimension of the manifold M is finite, it is equal to An, where n is a natural number. An almost quaternion manifold will be denoted by (M; J, K). An almost quaternion manifold (M; J, K) is called an almost octonion manifold when it is equipped with a third almost complex structure L, which anticommutes with the given structures J and K, i.e. JL = —LJ and KL = -LK. It will be denoted by (M;J,K,L). When the manifold M is finite dimensional, its dimension is of the kind 811, where n is a natural number. A complex-valued one-form w on an almost complex manifold (A/; J ) will be called a holomorphic one-form (or an invariant one-form with respect to the almost complex structure J) on M if the following property holds: w(Je) = iw(e) for Ve € TM, where i is the imaginary unit in C and TM is the real tangent bundle of the manifold M.

43

A quaternion valued one-form w defined on an almost quaternion manifold (M; J, K) is called a quaternion one-form (or an invariant one-form according the almost quaternion structure J, K) on M if the following properties hold: w(Je) = iw(e) and w(Ke) = jw(e) for Ve £ TM, where i, j are the standard quaternion units in the corps of quaternions Q and TM is the real tangent bundle of M. A quaterion one-form w on an almost quaternion manifold (A/; J, K) gives holomorphic one-forms on the almost complex manifolds (M, J ) and (M, K) which are determined on the manifold M when an almost quaternion structure (J, K) is given on M. These are u>i +i u>2, 103 + i W4 and Wi — i 1U3, u>2 — iwi respectively, where w = w\ + i w^ + j 'W3 + k W4 and u>,;, i = 1, 2, 3, 4 are real-valued one-forms. An octonion valued one-form w defined on an almost octonion manifold (M; J, K, L) is called an octonion one-form (or an invariant one-form according the almost octonion structure J, K, L) on M if the following properties hold: w(Je) = iw(e), w(Ke) = j w(e) and w(Le) = kw(e) for Ve € TM, where i, j , k are the standard octonion units in the algebra of octonions O and TM is the real tangent bundle of M. An octonion one-form w on an almost octonion manifold (M; J, K, L) gives 12 different holomorphic one-form on the almost complex manifolds (M, J ) , (M,K) and (M,L) which are determined on the manifold M when an almost octonion structure (J, K, L) is given on M. Theorems 2, 3 and 4 have as consequence Theorem 5 for extension of invariant differential forms with respect to the given almost complex structure^) from submanifold N of the manifold M to some neighborhood U of N in M. Theorem 5. Let the paracompact manifold M be an alm.ost complex, an almost quaternion or an alm.ost octonion manifold. Let N be a subm.anifold of M with tangent bundle invariant with respect to the. available almost complex structure(s) on M and w : N -> T}N°>1 (or N -> T}N°>1 n T^N0'1 or N -> T j N 0 ' 1 DT^N0'1 DTIN0'1, respectively) be an one-form which is a holomorphic, a quaternion or an octonion one respectively. Then there exists a holomorphic, quaternion or octonian one-form w on a neighborhood U of N w : U -> TjC/ 0 ' 1 (or U -> T*UQ<1 n T^U0'1 or U -> T*jU°<1 n l^U0'1 n TIU0'1, respectively) which is an extention of the given one-form., i.e. W\M = w. If an hermitian metric is fixed on the manifold, then the one-form w would be choosen to satisfy the property

I K 11 = \\wx ||

44

where the norm in the fiber of the tangent bundle is induced by the metric. In the end we would like to pose the following Problem. To pose and to prove the Hahn-Banach type theorem for manifold which is a base for a fibre bundle whith fibers admitting the structure of Clifford algebra (c.f. [GS]). References [B] S. Banach, Theorie des Operations Lineaires, Warsaw, 1932. [BS] H. Bohnenblust and A. Sobczyk, Extensions of functionals on complex linear spaces, Bull. Amer. Math. Soc. 44(1938), 91-93. [Bs] G. Buskes, The Hahn-Banah Theorem surveyed, Dissertationes Mathematical CCCXXVII, Warsaw, 1993. [GS] K. Giirlebeck, W. Sprossig, Quaternionic and Clifford Calculus for Physicisrs and Engeneers, John Wiley & Sons, 1997. [H] H. Hahn, Uber lineare Gleichiingen in linearen Raumen, Journ. fur reine und angev. Math. 157(1927), 214-229. [HI] E. Helly, Uber lineare Functionaloperationen, Sitzungsber. Akad. Wiss. Wien 121(1912), 265-297. [Hs] D. Husemoller, Fibre bundles, McGraw-Hill Book Company, New York, St. Louis, San Francisco, Toronto, London, Sydney, 1966. [L] J. Lewis, Extension of functionals on octonionic linear spaces, Acta Math. Hung. 52(1988), 249-253. [M] F. Murray, Linear transformations in Lp, p > 1, Trans. Amer. Math. Soc. 39(1936), 84-. [S] G. Suhomlinov, On extension of linear functionals in complex and quaternionic space, (Russian), Mat. Sbornik (Russian), 3(45)(1938), 353-357.

45

T H E T H E O R E M O N ANALYTIC R E P R E S E N T A T I O N ON H Y P E R S U R F A C E W I T H SINGULARITIES A.M. KYTMANOV* Krasnoyarsk State University, Russia. S.G. MYSLIVETS* Krasnoyarsk State University,

Russia

We consider a theorem on analytic representation of integrable CR functions on hypersnrfaces with singular points. Moreover, the behavior of representing analytic functions near singular points is investigated. We are aimed at explaining the new effect caused by the presence of a singularity rather than at treating the problem in full generality.

We begin by recalling a well-known theorem on analytic representation of CR. functions on smooth hypersurface [1, 2]. Let D be a domain in C n , n > 1, whose Dolbeault cohoinology with coefficients in the sheaf of germs of holomorphic functions vanishes at step 1, i.e., 11}(D, O) = 0. Such in the case, in particular, if D is a domain of holomorphy in Cn. Suppose that S is a smooth (of class C1) closed orientable hypersurface in D, dividing D into two open sets D+ and D~. As known, there exists a real-valued function p € C1 (D) such that S = { z e D : p{z) = 0 } and dp ^ 0 on S. We set D± = { z e D : ±p{z) > 0 } and orient S as the boundary of D~. Thus, D~ U S is oriented manifold with boundary. As usual, a function / 6 £} (S) is said to be a CR function on S if it satisfies / fdv = o Js for all differential forms V of bidegree (n,n — 2) with coefficients of class C°°(D) and a compact support in D.

* SUPPORTED BY RFFI. GRANT 99-01-00790 •SUPPORTED BY RFFI, GRANT 00-15-96140

46

T h e o r e m 1 ( A n d r e o t t i , Hill, C h i r k a ) . For any CR function f G Cl (S), there are functions h^ holomorphic in D±, respectively, such, that f = h+-h-

on

S.

(1)

±

(We write / i * e 0(D ).) More precisely, the equality (1) is interpreted, as follows: 1) if S e Ck+l. k e Z + , and f G C^(S), 0 < A < 1, then /i± £ U D±) and (1) is fulfilled at each point of S; 2) if S e C1 and f e £ f ( S ) , p>l, then for each point z° e S there is a neighborhood U such that

C^(S

lim

/

\(li+(C +

eu(O)-h-{C-eV(C)))-f(O\Pda=0,

where da is the Lebesgue measure vector to S at a point £ e S.

on S, and p(Q the unit outward

normal

It is worth pointing out (cf. [7, Ch. 2]) t h a t the boundary behavior of It*1 near hypersurface S coincides with t h a t of the Boclmer-Martinelli integral M(z) of / ,

zeD±,

-M(z) = JJ(OU((,z), where

and dC = dCi A . . . A d<„,

d([j} =d(1A...A

dCj-i A dCj + i A . . . A dCn •

T h e theorem on analytic representation is of crucial importance in the theory of CR functions (see, for example, [5, 10]). If the hypersurface S bears singularities, we may define t h e tangential Cauchy-Riemann conditions only at smooth points of S. Theorem 1 is no longer true for such hypersurfaces even in t h e case of point singularities. Let D be the unit bidisk in C 2 , D={z S=

= {z = (z1,z-2)eD:

(z1,z2)<=C2:\zl\
\Z2\}.

47

The origin 0(0,0) is a singular point of S. Indeed, S = {z G D ; p(z) = 0 } where p(z

l\Zi

-

Z2Z2,

and dp(z) vanishes at the only point z = 0 on S. Obviously, this is a conical point. Consider the open sets D^ = {z e D : ±p(z) > 0 } and the holomorphic function ^ 2

away from the planes Zj = 0, for j = 1, 2. The restriction of / is a smooth CR function on S \ {0}. Furthermore, we have / e Cl(S). To prove this, write zi = .X'i + i(/i,

22 = x2 + iy-i

x'i = rcos
x'2 = ?'cos
and parametrise S by

where 0 < r ^ 1 and 0 ^ <^i, ip2 ^ 27r. Then the Gramian has the form G =

/ 2 0 0' Or 2 0 \ 0 0 r2>

whence
=s/2 f dr I Jo Jo

dkpx I Jo

d^2 =

V^TT)2

is finite, as desired. Suppose that / meets the conclusion of Theorem 1, i.e., / = h+ — h~ on 6" \ {0}, where ^ £ 0(D±) are continuous up to S \ {0}. By the Cauchy theorem in one dimension it follows that

L=1 /2 /l± ^ A(te2 = 0 321 = 1/2

48 while L\

1/9

J\zi\ = l/2 M = l/2

Z\Z2

dzi A dz2 = {2ni)2

is different from zero. T h e contradiction shows t h a t / can not be represented as the difference of holomorphic functions on D^. T h e main obstruction to such a representation lies in the fact t h a t t h e cohomology H^(D \ {0}, O) is non-trivial. Yet another interpretation of this example consists in t h e following. T h e point 0 is not removable for t h e integrable CR functions on S, i.e., for a CR. function / on S \ {0}, the inclusion / 6 CX{S) does not guarantee in general t h a t / is a CR function on all of S. In t h e case of smooth hypersurfaces S, removable singularities of integrable CR functions are studied in [4, 8, 6, 10]. As a rule, isolated points are removable. T h e above example shows t h a t if z° ia a singular point of S then it can be unremovable for integrable CR functions. T h e purpose of this paper is to bring together two areas in which the problem of analytic representation can be studied. T h e first of the two is complex analysis with its explicit integral formulas which enable one to treat also problems of piecewise smooth "real" geometry. T h e important point to note here is t h e n a t u r e of singularities which are purely "real", namely conical points, power-like cusps, etc. T h e second area is the analysis of pseudodifferential operators on manifolds with singular points, cf. [9]. It introduces rather specific tools of real analysis in t h e complex problem, such as special weighted Sobolev spaces, asymptotic, "regularizations" of operators near singularities, etc. Using this approach we describe those locally integrable functions / on a hypersurface with singular points, which are still representable in t h e form (1). Moreover, we specify the asymptotic behavior of h±(z) close to every singular point. Consider a hypersurface S in D with one singular point z°, i.e., S \ {z0} is smooth. Assume moreover t h a t there exist a neighborhood U of 2 ° , a neighborhood V of 0 e C " and a diffeomorphism h : U —> V with the property that h{S nU) = {zeV:

where ip € C ^ e 0 ] satisfies ip(0) = 0 and ip(x„) > 0 if xn > 0, and tj = x3 +iyj,

j = 1,...,»;

x' = ( x i , . . . , x „ _ i ) .

49 Denote F = h{SnU)

and

re = {z e r: o^xn ^e}, s£ = h-\r£)} for 0 < e ^ £°. If / € ^ 1 1 oc (5) then the function

Sf(e) = —l—

vo\(dSe)

f

JdSc

\f\dwe

is, by Fubini theorem, finite for almost all 0 < e ^ e°, where du>£ is the Lebesgue measure on dSe, and vol(dSE) is the area of dS£. Theorem 2. Assume that f G £ | (S) is a CR function on S \ {z0}, satisfying Sf(e) — o(l/(p'2n~2(e)) as e —> 0. Then the Theorem 1 holds for f, more precisely, we have f = h+ — h~ on S\ {z0}, where h^ e C ( D ± ) and the boundary behavior of h^ near S \ {z0} is actually the same as that in Theorem 1. The important point to note here is the condition 5/(e) = o(l/if2n~2(e)) as e —> 0, on / . The "worse" the singularity of S at z°, the faster ip(e) tends to 0 as e —> 0, and so the wider the class of functions / meeting the additional condition. This is a general observation concerning the analysis on manifolds with cusps, cf. [9]. To prove Theorem 2 we may assume without loss of generality that D = U and h is the identity diffeomorphism. Thus, we have z° = 0, S = T and S£ = F£, for 0 < e ^ e°. In this case dS£ is a (2;i — 2)-dimensional sphere in R 2 "- 1 , namely 0S£ = { (x',y)

€ M 2 "" 1 : | 3 :'| 2 + \y\2 =
Consider the current /[S] 0 ' 1 in D \ {0}. Since / is a CR function on S \ {0}, this current /[S*]0-1 is 5-closed, i.e., d(f[S]0A) = 0 in D \ {0}. We next show that under condition of Theorem 2 / [ 5 ] 0 ' 1 extends to a enclosed current on all of D. Lemma 1. onS\{0}.If

Let D C RN, N > 2, and let f be a locally integrable function

/ Sf(e)
50

Lemma 2. Suppose f is a, locally integrahle CR function on S\ {0}. Then, for almost all e £ (0,s°), we have

fdv = S\SC

fv JdS.„

•whenever v is a differential form of bidegree (n, n — 2) with smooth coefficients and compact support in D. Having disposed of these preliminary steps, we can return to the proof of Theorem 2. By Lemma 2, <9(/[5]°- 1 ),i;>= lim / fdv = - lim / fv = 0 £ ^°+ Js\ss c->o+ JdSi for all (n, n — 2)-forms v with smooth coefficients and compact support in D. Indeed, /

fv^cf

\f\dLj£ =

JdSE

ca2n-d
JdSe

where c is a positive constant independent of e, and a-in-\ the area of the unit sphere in R 2 "" 1 . We have used the fact that vol(dS£)=a2n„1( 0, the desired equality follows. Thus, the current f[S]0A is 9-closed in D. The end of the proof is the same as in Theorem 6.1 in [7]. The rest part of the paper is devoted to the study of boundary behavior of the Bochner-Martinelli integral (and the functions h±) near singular point. Consider the defining function p(z) of the hypersurface S close to the singular point O ( 0 , . . . , 0) p(z) =
f(0jr%;.

^D\S,

where m > 0. In the sequel we restrict our attention to z <E D+.

51

T h e o r e m 3. Suppose that f € ^ ( S ^ ) anc/ ^/(e) = 0(l/ipN(e)) /or some A' $J 2n — 2. i ; If2n-m ^ iV, tfieri. P ( )= 0

"' ^

as

(MN))^-^)

|2H

'

as e -> 0+,

°-

£,) If In - m - 1 < N < In - m, then |jog^(|z|)|

p

M = ° IMI~D) T^wT^k+i)

3) If N < 2n-m-

<" N-o-

1, then Pm(z) = 0(l)

as

|z|->0.

Consider the Bochner-Martinelli integral Af (z) of the function / . Since \M(z)\ ^ c\P2n-i(z)\, with c a constant independent of z, Theorem 3 for m = 2?i — 1 implies the following statements. Corollary 1. Under the assumptions of Theorem 3, the following estimates hold for ze D\S: l)Ifl^ N, then 1 ,,.„,mJV )

M{z)=0[ 2) IfO^N

as

\z\-^0.

<1, then M{Z)

3j IfN<0,

Mm

= 0

\M\Z\))N)

' '

then M(z) = O(l)

as

|z| -» 0.

Corollary 2. Lef / 6 £ j (.9) 6e « CR function on S \ {z0} 67(e) = 0{l/ 0, /or some TV < 2?i - 2. Then, 1) For 1 ^ N, we have h±(z) = Ol

*

M\z-A))N

2) For 0 ^ AT < 1, we have .±,^-^|logy(l-s-* ft±

W = °

0

l)l

T ^ T o ^ F

Mk-z°l))"

**

satisfying

3) For N < 0, we have h±{z) = O(l)

as

z -, z°.

References 1. A. Andreotti and C D . Hill, E.E. Levi convexity and Hans Lewy problem. Part 1: Reduction to vanishing theorems, Ann. Scuola Norm. Super. Pisa 26(1972), no. 3, 325-363. 2. E.M. Chirka, Analytic representation of CR. functions, Mat. Sb. 98 (1975), no. 4, 591-623; English transl. in Math. USSR Sb. 27(1975). 3. R. Harvey and H.B. Lawson, On boundaries of complex analytic varieties. I, Ann. Math. 102(1975), 223-290. 4. B. Joricke, Removable singularities of CR functions, Arkiv Math. 26 (1988), no. 1, 117-142. 5. G.M. Henkin, The method of integral representation in complex analysis, Several Complex Variables I, Encyclopedia of Mathematical Sciences, Vol. 7, Springer Verlag, 1990, 19-116. 6. A.M. Kytnianov and C. Rea, Elimination of Cl singularities of Holder peak sets for CR functions, Ann. Scuola Norm. Super. Pisa 22(1995), no. 2, 211-226. 7. A.M. Kytnianov, The Bochner-Martinelli integral and its application, Novosibirsk, Nauka, 1992; English transl., Birkhauser Verlag, Basel, Boston, Berlin, 1995. 8. , On the removable of singularities of integrable CR functions, Mat. Sb. 136(1988), no. 2, 178 486; English transl. in Math. USSR Sb. 64(1989), no. 1, 177-185. 9. V.S. Rabinovich, B.-W. Schulze, and N.N. Tarkhanov, A calculus of boundary value problems in domains with non-Lipschitz singular points, Preprint 9, Univ. Potsdam, 1997, 1-57. 10. E.L. Stout, Removable, singularities for the boundary values of h.olomorphic functions, Proc. of the Mittag-Leffler Institute, 1987-1988, Princeton Univ. Press, 1993, 600-629.

53

P S E U D O G R O U P S T R U C T U R E S ON S P E N C E R M A N I F O L D S STANCHO DIMIEV Bulgarian

Academy of Sciences, Institute of Mathematics & Informatics, Acad. G. Bonchev Str., Bl. 8, 1113 Sofia, Bulgaria E-mail: [email protected]

This note is a continuation of the paper [2] (see references). We describe some natural pseudogroup structures on almost complex manifolds of type m. A kind of coherency is discussed for the sheaf of almost holomorphic functions.

1

Recall of definitions

Let (M, J ) be an almost complex manifold equipped with a smooth coordinates (x J ), j = 1, 2, . . . , 2n. In [1] complex self-conjugate coordinates (z J , 2-*), j = 1, 2, . . . , n, are used, where z^ = x2i~x + ix2^ and z-7 = . T 2 J _ 1 — ix2K A function / : U —> C, where U is an open subset of M, is called almost holomorphic if J*df = idf. Here by J* the action of J on differential forms of M is denoted, i.e. J*(u>)X := u(X), where w is a differential form on M, and X is a vector field on M. Let m be an integer, m < n. We say that a local Spencer coordinate system is defined on (M, J ) , iff (1) there exist an open subset U of M and in different functionally independent almost holomorphic functions fj : U —> C, j = 1, . . . , m, such that (2) the sequence / i , . . . , / m is a maximal sequence of functionally independent on [/ almost-holomorphic functions (3) the sequence (UJ1,...Jm,zm+\...,znJ1,...Jm, zm+\...,zn) determines a local self-conjugate coordinate system on (M, J). The condition (3) can be reformulate in terms of smooth coordinates as follows: (3')The sequence ([/, Re / , , Im / , , . . . , Re / m , Im fm, xm+1,ym+1,..., x2n,y2n) is a local coordinate system in smooth coordinates on (M, J ) . Setting Uj = Refj,Vj = Im fj, we have dvj = J*duj, j = 1, . . . , m. These are Cauchy-Riemann equations for fj. Clearly, each Spencer coordinate system on M defines a diffeomorphism of U in C™ x C , or equivalently in R 2 n . Complexifying the first 2m coordinate we can take C m x R 2 n - 2 m , i.e. R 2 n s C m x R2™"2™. We shall consider the smooth submersion R2™ —* Cm defined as a composition of the above mentioned diffeomorphism of U on R 2 n and the projection

54

of R 2 " on C m ( / l ) • • • ; Jrrn

z

i • • •iz

i 111 • • • i f mi

z

i---iz

)

* ( / l i • • • i Jm ) •

This composition is denoted by fu [more precisely by ( / i , . . . , fm)u] and the image of U by fu is denoted by Ucm. As it was proved in [2], each almost-holomorphic function h : U —> C is a superposition of a holomorphic function i / : Uc m —> C and the almostholomorphic functions / , , j = 1, . . . , m, i.e. /i = H(f\,..., f m ) , where H — H{wu...,wm)eO{Ucm). It was proved also that for each two systems of almost-holomorphic functions ( / i , . . . , fm) and (<7i,...,g m ) defined on U n V, there exists a bijective holomorphic transition mapping between the corresponding Spencer coordinate systems. As a consequence one obtain a theorem formulated by Spencer [1], namely Theorem. On each paracompact almost-complex manifold (M, J) of constant Spencer type m there exists a locally finite covering {Ua} constituted by self-conjugate Spencer's coordinate systems (Ua, wia,...), j = 1, .. ., m, such that on every intersection UanUp the Spencer coordinates (w^a) change biholomorphically in the other Spencer's coordinates (vJ@). 2

Pseudogroup structures

By X a topological space is denoted, respectively differentiable manifold (X = M), or vector space (X = R n , X = Cn). We recall the notion of pseudogroup r of local homeomorphisms of X, respectively — local diffeomorphisms of M, or local biholomorphisms of C m . Let U be an open subset of X, respectively of M, or of C m , and
X be a local mapping (homeomorphism, diffeomorphism, biholomorphism). The open set U is called the source of ip, and the image of ) e r and p'1) e I\ (3) If (U,ip) e T, and V C U, then (V,
r),

55

(4) If (U,(f) G F and every point of U admits a neighborhood on which the restriction of

r. Let (X,Tx) be a topological space X equipped with a pseudogroup of local homeomorphisms F\, and (Y, Ty) is another topological space equipped with a pseudogroup of local homeomorphisms Ty We say that (X,Tx) is defined over (Y,Ty) if for every source U of an element of T\, (U, such that the following diagram should be commutative

V

> ip{V) 1p

where V = fu(U) is the source of (V,tp)
i.e.

By T,i(M) the transitive pseudogroup of all local diffeomorphisms of the differentiable manifold M is denoted. In the case M is an almost-complex manifold, (M, J ) , we shall consider local almost-holomorphic diffeomorphisms of M, i.e. the diffeomorphisms / : [ / — > M, (/ being an open subset of M, and / satisfying the condition f * ° J = J ° f*, where /* is the tangent mapping (the differential) of the mapping / . The minimal transitive pseudogroup of local almost-holomorphic mappings will be denoted ^ahd (M). This pseudogroup is a subpseudogroup of Td(M). It is to recall here that the composition of two almost-holomorphic mappings is also an almost-holomorphic mapping where it is defined. The same is true for the inverse of an almost-holomorphic mapping. Accordingly to the theorem formulated in previous paragraph, we shall consider all local almost-holomorphic diffeomorphisms $ tt/ g : Un —> Up, and corresponding biholomorphisms ipnp : (Ua)cm —> (Ufj)cm in C m , where {Ua} are the sources of local Spencer coordinates systems. The family {$«/3} generates a subpseudogroup of r a /,d(M) which will be denoted by Tspci(M). Denoting by I \ ( C m ) the pseudogroup of all local biholomorphisms in C m , we consider the family {faf}} and the generated subpseudogroup of r / , ( C m ) which will be denoted by r s p /j(C m ). The pseudogroup Tsp(i(M) is over the

56

pseudogroup rsph(Cm) according to the above introduced definition. The pseudogroups Tspd(M) and r s p ^ ( C m ) are called Spencer pseudogroup of the almost complex manifold (M, J) of type in. This means that for every $ : U - V, (U,V) e rspd(M), we have fv o $ = ip o fr, where ( £ / c m , <^) G r s p /i(C m ) and also $ , o J = J o $». The following diagram is commutative.

[fv

'"I v

3

T-manifolds, integrability of (M, J )

Let r be a pseudogroup of local diffeomorphisms of one differentiable manifold M. It is said that a T-structure on the manifold M is defined if an atlas of local coordinates systems is introduced in such a way that all transition transformations between them to belong to the pseudogroup T. Let (M, J ) be an almost-complex manifold of type m. We shall consider the corresponding Spencer pseudogroups Tsp(i(M) and r s p / l (C T O ). Let us suppose that TV is an orientable differentiable (2m)-manifold equipped with a r s p /j(C m )-structure. We remark that the problem of the existence of restructures on a given manifold, especially r s p h(C m )-structures, is a difficult problem. Denoting / = {fu} a s the family of all local m-projections fu'-U—> C m defined by almost-holomorphic coordinates / i , /2, . . . , / m , we obtain the following diagram M =

M

1'

'{ Cm

<

atlas

A^

6

where F is defined locally as follows: F(U) = 0-1o

f(U),

U being an open subset of M. In the case m = n [M is a (2n)-manifold] the r s p / l (C")-structure on N is a structure of a complex manifold, and F is a diffeomorphism. So the almost complex manifold (M, J) is diffeomorphic to a complex manifold N. Taking

57

N = C n as differentiable manifold, we obtain that Tsph(Cn) defines an atlas of biholomorphisms on M. This implies that (M, J) is an integrable manifold. In the case of 4-dimensional almost-complex manifold M of type m = 1 there are Spencer pseudogroup structures r s p ( /(A/) and Tsph{C) The corresponding 2-dimensional manifold N, equipped with a r s p / l (C) must be a Riemann surface. 4

T-coherency

The notion of coherent sheaf is local. Having a pseudogroup F on a manifold M we can assign to each source U of F the set T(U) of different F-objects (functions, vector fields) defined on U. The mapping U —» r(U) defines a sheaf. Our purpose is to discuss a kind of coherency on almost complex manifolds using the introduced Spencer pseudogroup structures of type m. It is not difficult to see that almost-holomorphic functions on (M, J) define a sheaf. We denote this sheaf by Oah(M) and, respectively, by 0/j(C m ) — the sheaf of all holomorphic functions on C m . According to the famous Oka theorem Oh{Cm) is a coherent sheaf, the same is true for the sheaf Oit(G), where G is an open connected subset of C m . Proposition. For every source U e r s p c j(M), the sheaf Oah{U) is a coherent sheaf of almost holomorphic functions. Proof. It is enough to remark that Oa/,(U) is inverse image of O/^U^). 5

•

Perspectives

It seems that one can develope a deformation theory for almost complex manifolds of type m following some ideas of Donald Spencer (see [3]). References 1. D.C. Spencer, Potential theory and almost complex structures, Kaplan (ed.). Lectures on Functions of complex Variables, Univ. of Michigan Press, 1955. 2. S. Dimiev, R. Lazov, N. Milev, Spencer Manifolds, Meeting on quaternionic structures in mathematics and physics, Rome, 1999 (to appear). 3. D.C. Spencer, Deformations of structures on manifolds by transitive continuous pseudogroups, Ann. of Math. 76(1962), 306-445.

58

T Y P E - C H A N G I N G T R A N S F O R M A T I O N S OF HURWITZ PAIRS, Q U A S I R E G U L A R F U N C T I O N S , A N D H Y P E R K A H L E R I A N HOLOMORPHIC C H A I N S I JULIAN LAWRYNOWICZ Institute of Mathematics, Polish Academy of Sciences. Lodz Branch, Banacha 22, PL-90-238 Lodz, Poland Chair of Solid State Physics, University of Lodz, Pomorska 149/153, PL-90-236 Lodz, Poland E-mail: jlawryno krysia.uni.lodz.pl LUIS M A N U E L T O V A R Departamento U.P.A.L.M.,

de Matemdticas, E.S.F.M., Instituto Politecnico Edifico No. 9, Zacatenco, 07000 Mexico 14, D.F., E-mail: tovar esfm.ipn.mx

Nacional Mexico

J. Lawrynowicz (J.L.), R.M. Porter, E. Ramirez de Arellano, and J. Rembieliriski (1994) have proved that the correspondences between indices of pseudo-euclidean Hurwitz pairs in type-changing transformations related to their duality are represented by values of explicitly given quasiregular and quasimeromorphic functions. As observed by J.L. (1998) in 1996 these dualities appear to be complementary to the dualities of the related Hermitian Hurwitz pairs invented by S. Kanemaki and O. Suzuki (O.S.) (1989). The present authors study this extension in detail giving the complete list of the dualities concerned (part I of the paper, Theorem 3). Part II will include the related duality of holomorphic mappings, constructed via complex structures on isometric embeddings for some of the Hermitian Hurwitz pairs concerned so that the embeddings are real parts of those holomorphic mappings (J.L. and O.S., 2001) as well as the related duality of the almost hyperbolic quaternionictype pseudodistances connected with hyperkahlerian holomorphic chains (P. Dolbeault, J. Kalina, and J.L., 1999).

1

Introduction

We are dealing with pseudoeuclidean [12-15] and Hermitian [4, 5] Hurwitz pairs as well as holomorphic chains [1, 4] and almost hyperbolic quaternionictype pseudodistances [2]. Our study aims at establishing a counterpart of results of [8]. Research of the first author partially supported by the State Committee for Scientific Research (KBN) grant PB 2 P03A 010 17, and partially by the University of Lodz research grant no. 505/699. Research of the second author partially suportet by a grant of C.O.F.A.A.-I.P.N. The results of the paper have been included in the lecture by J.L. during the V International Symposium on Complex Structures and Vector Fields at St. Constantine near Varna on September 6, 2000.

59

Consider two finite-dimensional real vector spaces S and V, equipped with scalar products ( , )s ( , )v- We suppose that (b,a)s = (a,b)s, {jo,,b)s = j(a, 6)s,and (a, b + c)s = (a,b)s + (a,f')s whenever a, b, c G S and 7 G M. For f,geV, we suppose that (g,f)v = 5(f,9)v, 5 = 1 or - 1 ; the remaining postulates for ( , )y are the same as for ( , ),?. Hence, if (f„), a = 1, . . . , dim S = p, and (e^), j = 1, . . . , dim V = n, denote the bases of S and V, respectively, then for V = [Vaff] •= [(fa,e^)s], ]

a/3

and

K = [«jfc] := [(ej,ek)v\,

(1)

T

we get r/~ = [r/ ], r? = r/, detr] / 0, and an analogous relation for n. Without any loss of generality we can choose the basis (en) so that , . 5a = l for a = l , . . . , r + 1, ,0^ 7? = diag(dQ), r -, r , o 11 , (2) v , v ' i5„ = - l for a = r + 2 , . . , , r + l + s = p, ' and hence ??_1 = r\. It appears [9] that p < n + 2. We define a multiplication of elements of S by elements of V to mean a mapping 5 x V —> V such that, for / , g € V and a, 6 G S, we have (a + b)f = af + bf, a(f + g) = af + ag, (a,a)s(f,g)v = (af,ag)v, and for which there exists a unit element «o in S with respect to the multiplication: eof = /• Multiplication by a G S is an M-linear operation on V and scalar multiplication by 7 G M can be identified with the multiplication by 7^0- If the multiplication S x V —> V does not leave invariant any proper (nontrivial) subspace of V, the pair (V, S) is said to be irreducible, and is called a pseudo-euclidean Hurwitz pair [12] in contrast to (Euclidean) Hurwitz pairs. Now, let C " ( K ) be an n-dimensional Hermitian vector space with the metric of signature (r' + l,s'), r' + 1 + s' = n (/* denoting the hermitian conjugation of / ) : ((f,9))« : = / * « » ,

/,jeCn,

K = /,< + i,s< := T

and Rp(r^) — p-dimensional real vector space with the symmetric metric of signature (r + 1, s), r + l + s = p (J m denoting the ra-dimensional unit matrix): (a, 6),, := aT7yb,

a,b e W,

r/ = / r + l v

Ir+l 0

0

We consider a pair H = (C"(K),E p (r/)) with s ^ 0. For s = 0 the situation is different and rather easier. If there exists a bilinear mapping R p (r/) x Cra(«;) —> Cra(ft) called multiplication of elements W(T)) by elements of C"(K) such that,

60

for f,g e Cn(n) and a, b e Rp(/?), we have {a,a)„((f,f))K := {{af,af))K and, moreover, H is irreducible, i.e. there exists no subspace V of C™, V 7^ {0}, C n , such that f\W(n) x V —> I/, then W is called an Hermitian Hurwitz pair. 2

Dualities

Mathematically, the duality referring to the metric K of V, given in (1), is motivated and determined by the following result [8, 9] proved in 1989: Theorem 1. (I) Let (V, S) be a pseudo-euclidean Hurwitz pair with the scalar product in. S given by the matrix rj of equation (2). Choose t e { 1 , . . . ,p}. Then the equations Ca = iiaCt, 7a = -la, {la,lp}

CtCt = Vttln, re 7 a = 0,

t fixed, i e { l , . . . , p } ; a / t ;

where la := «7 Q K~1 , a = 0,...,p;

:=7a,7/3 +lg, Ja = 2fja>/lIn,

fja,0 = Va.p/'ht,

Vtt = 1 or - 1,

a, ft = 1,... ,p;

a/(;

a^t;

a, ft = 1 , . . . ,p; a ^ t

define a real Clifford algebra of order (r, s) chosen in the imaginary Maiorana representation. (1.1) By a further change of basis these generators can be chosen antisymmetric or symmetric according to f)Qfv = 1 or — 1. (1.2) The integers r satisfy (2) if rjtt = 1 and '

o~a = 1 for a = l , . . . , s , 5a = - 1 /or a = s + l , . . . , s + r + l = p .

XTiws two Clifford algebras are obtained unless r\ is positive definite, in which case only one Clifford algebra is obtained. (1.3) Further, if T)tt = 1, then via a change of the basis one may assume Ct —In- IfVtt = — 1, then via a change of basis one may assume Cp — In for an arbitrary p^t so that. Ct = —ilP(II) Consider a real Clifford algebra of order (r, s) generated by the matrices 7 Q in the imaginary Maiorana representation. Then there are exactly two pseudoeuclidean Hurwitz pairs (Vi, Si) and (V2, £2) which give rise to that Clifford algebra via the process described above. A Clifford algebra of order (r,s), r + 1 + s = p, which admits a representation (li,- • -,lP) with the condition 7Q = 7 a , a ^ t, t fixed, where 7 Q : = K 7 * K _ 1 , is called Hurwitz algebra and denoted by Hr,s whenever it is central. It defines in W(K) differential operators of the Dirac-Fueter type in

61

the following manner: by the use of (Sa) we define Dl

• = y ^ ( — i j a ^ — ) + InT.—TT

^r,s

:=

(^le generalized Fueter operator), (*'je generalized Dirac operator).

/J(~*7ap—) ,. oza

By [4, 16, 19] there exist canonical isomorphisms between: (A) Hr,s and W s _i i r + i whenever r + 1 and ,s, being positive, are not both odd, (B) Hri,s and 7i s ,r+i whenever r + 1 and s, are nonnegative and r + s is odd. Precisely, as far as the solutions of equations Z5FS*=0

resp.

£>°, = 0

(4)

are concerned, we have [4, 16, 19] the following result (1994) motivating and determining the duality referring to the differential operators (3): Theorem 2. Let S^-+ 'S(DD), E = F, resp. D, denote the linear space of equations (4) in an open set U. Then (k) S[js(D?) ~ S„~1,r+1(DF) (AB) S^;S(DD) ~ Sy~1,r+1(Dd) +1,S

D

r+1

forr + 1 and s positive, being not both odd; for r + 1 and s positive, r + l+s

being odd;

D

(B) Sjj (D ) ~ S*J (D ) forr + l and s nonnegative, r + l + s being even; in that case: ^•'(D0) ~ S&~1,r+1(£>F) © S* _ 1 ' r + 1 (£) F ) forr + l even and s^O, ^ + M ( D D ) ~ S[} +1,S - 1 (D F ) © S £ + M - 1 ( I > F ) /or s odd. Dualities referring to (mathttA) will thereafter be denoted by F G ( r + 1 , s), whereas those referring to (B) — by D G(r + 1, s); DFG(r + 1 , s) will denote the dualities referring to (AB), i.e. in the case where DFG(r + 1, s) = FG(r + 1, s); in that case we always have another duality D G(r + l,s) ^ DFG(r + l,s). Here G stands for the generation of the Hurwitz pair; in general it is not uniquely determined by (r, s). Following [2] we shall distinguish the following generations G of Hurwitz pairs: K— Kaluza-Klein, ^ T-, ., t, — Exciton,

P — Penrose, ,r ,, ,, . N—Neveu-Schwarz,

„ _ ,& , , ' , (J — Extended Octonion, . ,, „, TT H — other [16].

A mathematician may replace this by G\, G^, etc.

62 3

Complementarity

We are going to extend Theorem 3 in [6]. In order to compare the dualities referring to the differential operators (3) with those referring to the metric K of V, taking into account a possible change of symmetry of the inner product, we have to consider the generations G on two-sheeted Riemann surfaces [8], with the following distinction: Yh etc. — hyperbolic case, 1st sheet;

Ys etc. — symplectic case, 1st sheet;

Yh etc. — hyperbolic case, 2st sheet;

Ys etc. — symplectic case, 2st sheet.

Correspondingly, in the case of dualities referring to the metric K of V it is worth-while to distinguish between a type-reversing duality G = R, say, if the symmetry of the inner product is changed: symmetric — i > antisymmetric or antisymmetric >—» symmetric, and a type-preserving duality G = P, say, if the symmetry of that product is unchanged: symmetric ^—> symmetric or antisymmetric \-> antisymmetric. Here hyperbolicity and symplecticity always refers to the metric K of V, so it corresponds to the symmetry or antisymmetry, respectively. A direct calculation leads to Theorem 3. Within pseudo-riemannian Hurwitz pairs (V, S) all the dualities referring to the metric K of V are listed in Tables IR I H R - Within Hermitian Hurwitz pairs (C"(«),R p (f/)) all the dualities referring to the differential operators (3), except perhaps for the multiplicity of nonequivalent dualities of the same type F, DF or D between Hurwitz pairs of the fixed signatures: (r + l,s) — i > (r' + l,s') are listed in Tables IH-III//. Tables LR and LH, L — I, II, III, are complementary in the sense that Tables LRH formed by putting formally LH on RH (i.e., composing formally LR and LH) have no blank (empty) row in no column.

Table In'- Dualities referring to the metric K of V — • s (odd)

I r+1 RiV,(2,l)=iV(3,4) P7V,(2,1) = N.(l,2) Ry,(3,l)=y^(0,4) Ry,(3,1) = Yh(0,4) p y s ( 3 . i ) = y s (3,i) py,(3,i)=y5(3,i)

RA'"(4,1) = A,(l,4) RA-,(4,1)=A-"(1,4) PKh{4A) =-Kh{lA) PA'S(4,1)=A:S(1,4)

R£"(6,l)=£,(l,6) R£,(6,l) = Eh(l,6) PEh(6A) = ES{1,6) PEs(6A)=Eh(l,6) RO"(7.1) =o,(2,6) ROh(7,l) = Os(2,6)

Rys(l,3)=y/l(4,0) Ry,(i,3) = y"(4,o) pys(i,3) = y,(i,3) py,(i,3) = y s ( i , 3 )

RP,(2,3)=P"(l,0) PP,(2,3) = PS(3,2)

RM"(4,3) = A/s(1.2) PMh(4,3) = Mh{3,4) RO"(5,3) = Os(5,7) ROh(5,3) =Os(5,7) POfc(5,3) = Ofc(3,5) PO/l(5,3)=Ofc(3,5)

R£"(6,3) =£,(3,6) R£,(6,3)+£'l(3,6) P£h(6,3) = £''(3,6) P£,(6,3) = £,(3,6)

R RO P PO

RAT"(2,5) = As(5,2) RA',(2,5) = A''(5,2) PA"'l(2,5) = Ah(5,2) PA", (2,5)= AT, (5,2) RO"(3,5) =0,(7,5) ROh(3,5)=Os(7,5) POh(3,5) = Oh(5,3) POh(3;5) =Ofc(5,3)

PA

RP^(4,5) = P,(7,6) PP'l(4,5) =P'l(5.4)

R£ R£

RK RK PK

PE

RiV,(6,5) = JV"(7,8) P,V,(6,5)=iVs(5,6) RO*(7,5)= 0^(4,6) RO,(7,5) = Oh(4,6) POs(7.5) = Os(5,7) PO,(7,5)=0,(5,7)

P£ RO RO PO PO

Table 11^: h

PO (7A)=Oh(7,l) POh(7A) = Oh{7A)

RP"(8,1)=P S (3,10) Pph(8,1) = P f e (l,8) —> s (even) RA/ a (l,2) = iV(4,3) PiV s (l,2) == W5(2,1) s

RO (2,2) == Oh(4,4) RO s (2,2) == 0' l (4,4) PO s (2,2) == O s (2,2) s PO s (2.2) == O (2,2)

Dualities referring to the metric K of V,

I RA'ft(8,3) = ATS(3,8) RA' s (8,3)=A'' l (3,2) PA'' l (8,3) = A'' l (3,8) pA' 3 (8,3)=A' s (3,8)

R/3(8,5)=£s(5,8) R£ s (8,5) =Eh(5,8) h PE (8,5) = Eh(5,8) P£ 5 (8,5) = E s (5,8)

RAT"(1,4)

RKs{lA)=K (lA)

PA' fc (l,4) = AT/!(4,1) PA- S (1,4)=A' S (4,1)

R£"(l,6)=£s(6,l) R£ s (l, 6) = 7^(6,1) P£' l (l,6) = £''(6,1) P£s(l,6)=£,(6,l)

RA""(2,4) = A' s (4,2) RA"S(2,4) = 7:3(4,2) PA'^2,4) = 7 3 ( 4 , 2 ) PA- s (2,4)=A-,(4.2)

RO s (2,6) RO s (2,6) P0 S (2.6) PO s (2,6)

= 0 f c (7,l) = 0''(7,l) =Os(2,6) = <3S(2,6)

RJV*(3,4) = JVS(2.1) PNh{3A) =Nh(4,3)

R£"(3,6) RP S (3,6) P£ h (3,6) P£ s (3,6)

= £ s (3,6) =£''(3,6) = £''(6,3) = £ s (6,3)

=KS{1A) h

RP,(3,2) == P"(0,1) PP S (3,2) == P,(2,3) RA"(4,2) = RA'S(4,2) == PA''1 (4, 2) = PA-S(4,2) ==

A',(2,4) Kh{2A) Kh{2,4) KS(2A)

RO"(4,4) = O g ( 2 , 3 ) RO /l (4,4) = O s ( 2 , 2 ) PO f c (4,4)=Oh(4,4)

R£"(4,6) = £ s (6,4)

Table III/j:

Dualities referring to the metric K of V, cont

PO/l(4,4)=0''(4,4)

R£,(4,6)=£,(6,4) P£ h (4,6) = £ ' ' ( 6 , 4 ) P£,(4,6)=£,(6,4)

RA/"(5,2)=A-,(2,5) RAT,(5, 2) = Kh(2,5) PKh{5,2) = Kh(2,5) PA-,(5,2)=A",(2,5)

RP"(5,4) = £,(5,7) PP ft (5,4) = P h (4,5)

W(6,2)=Oh(l,7) RO,(6,2) =Oh(l,7) s PO (6,2) = 0 , ( 6 , 2 ) PO,(6,2) = O s ( 6 , 2 )

R£"(6,4)=£,(4,6) R£,(6,4) = £ / l (4,6) PE fc (6,4) = £ ^ ( 4 , 6 ) P£,(6,4) = £,(4,6)

R£"(7,2) = A',(2,7) RA',(7,2) = Kh(2,7) PKh(7,2) = Kh(2,7) P£,(7,2)=A/,(2,7)

R£"(7,4) = £ , ( 7 . 4 ) R£,(7,4) = £ ' ' ( 7 , 4 ) PEh(7,A) = £ h ( 4 , 7 ) P£,(7,4)=£s(4,7)

RP,(7,6) = £ " ( 4 , 5 ) PPS(7,6)=P,(6,7)

KKh(8,2) RAT,(8.2) PKh{8,2) PA-,(8,2)

Ry"(8,4) = y , ( 3 , 9 ) Ry/l(8,4)=ys(3,9) PYh(8A)=Yh(8A) py^(8,4) =Yh{8,-i)

R£"(8,6) = £ , ( 6 , 8 ) R£,98,6) = £ / l (6.8) PEh{8,6) = £ ' ' ( 6 , 8 ) P£,(8,6)=£,(6,8)

= Ka(2,8) = A'ft(2,8) = Kh{2,8) = A-,(2,8)

RiV,(5,6)=iV(8,7) PN,(5,Q)=NS(6,5) RO s (6,6)=O f t (6,6) ROs(6,6) =O f t (6,6) PO*(6,6) = 0,(6,6) PO,(6,6) = O s ( 6 , 6 )

Table IH'-

Dualities referring to the differential operato

F#(1,1)=A'(0,2)

F/V(1,5) = A'(4,2) DF7V(1,5) = K{4:2) DAT(1,5) = #(5,1)

DH(1,1)=A-(0,2)

DH(1,1) = H(1,1)

—> s (odd) ir + 1

FF(3,l)+^(0,4) 0FF(3,1)=^(0,4) Dr(3,i) = y ( i , 3 )

F#(5.1) = £(0,6) D F ff(5,l) DH(5A) = H(1,5)

FY(1,3) = 0(2,2) D F y(l,3) = 0(2,2) DF(1,3) = Y(3,1)

FH(3,3) = K(2,4) D F #(3.3) = A(2,4) Dff(3,3) = # ( l , l )

FO(5,3) = 0 ( 2 , 6 ) D F O ( 5 , 3 ) = 0(2,6) DO(5,3) = 0 ( 3 , 5 )

FO(3,5) = 0(4,4) D F O ( 3 , 5 ) = 0(4,4) D0(3,5) = 0(5,3)

F#(5,5) = £(4,6) D F #(5,5) = £(4,6) D#(5, 5) = # ( 5 , 5)

Table HH : Dualities referring to the differential operators (3), co

FO(7,1) = 0(0,8)

F#(7,3) = # ( 2 , 8 ) F D #(7,3) = # ( 2 , 8 ) D#(7,3) = # ( 3 , 7 )

FO(7,5) = y(4,8)

F # ( l , 4 ) = P(3,2)

F#(l,6) = #(5.2)

F#(2,4) = # ( 3 , 3 ) D F #(2.4) = # ( 3 , 3 ) D#(2,4) = # ( 4 , 2 )

D F O ( 2 , 6 ) = 0(5,3)

D F O ( 7 , 5 ) = y(4,8)

DO(7,5) = 0 ( 5 , 7 )

D F O ( 7 , 1) = 0 ( 0 , 8 )

DO(7,1) = 0(1,7)

s (even)

FiV(l,2) = 7V(1,2)

FO(2,2) = y ( l , 3 ) DFO(2.2) = y ( L 3 )

DO(2,2) + 0(2,2)

FO(2,6) = 0(5,3) DO(2,6) = 0(6,2)

FP(3,2) = # ( 1 , 4 ) FJV(3.4) = A/(3,4) F#(3.6) = P ( 5 , 4 ) | F#(4,2) = # ( 1 , 5 )

Table III//: D F tf(4,2) = # ( l , 5 ) DA'(4, 2) = AT(2, 4)

FAT(5,2) = £(1,6)

Dualities referring to the differential operators (3 FO(4,4) = 0(3,5) D F 0(4,4) = O(3,5) DO(4,4) = 0(4,4)

F£(4,6) = # ( 5 , 5 ) D F £(4,6) = # ( 5 , 5 ) D£(4,6) = £ ( 6 , 4 )

EP(5,4) = £(3,6) FiV(5,6) = JV(5,6)

FO(6,2) = 0 ( 1 , 7 ) D F 0(6,2) = 0 ( 1 , 7 ) DO(6,2) = 0(2,6)

F£(6,4) = # ( 3 , 7 ) D F £(6,4) = #(3,7) D£(6,4) = £ ( 4 , 6 )

FA"(7,2) = P(1,8)

F£(7,4) = A'(3,6)

FP(7,6) = £(5,8)

FA'(8,2) = P"(1,9) DFK(8,2) = # ( 1 , 9 ) DA'(8,2) = A'(2,8)

FY(8,4) = F(3,9) D F r(8,4) = y(3,9) Dy(8,4) = F(4,8)

F£(8,6) = £(5.9) D F £(8,6) = £(5,9) D£(8,6) = £(6,8)

FO(6,6) = 0 ( 5 , 7) D F O ( 6 , 6 ) = 0(5,7) D0(6,6) = 0 ( 6 , 6 )

T< FH(1,1) = K{0,2) 0 F #(1,1) = AT (0,2) Dff(l,l) = ff(l,l) RiVs(2,l) = W ft (3,4) PJVa(2,l) = JV,(l,2) Rys(3,l)=y/l(0,4) Ry s (3,i) = y fe (o,4) pys(3,i) = ys(3,i) py s (3,i) = y s ( 3 , i ) Fy(3, i) = y(o.4) D F y(3,i) = y(o,4) py(3,o) = y ( i , 3 ) RA*(4.1)+A' S (1,4) RKs(A,l) = Kh(l1A) PKh(AA) =Kh{lA) PA S (4.1) = A S (1,4) Ftf(5,l) = £(0,6) P F #(5,1) = £(0,6) Pff(5,l) = ff(1.5) R£ ft (6,l) = £ s ( l , 6 ) RJ5 s (6,l)=.E f c (l,6) P£>>(6,1)=£ 5 (1,6) P£,,(6,l) = £^(1,6) ROh(7,l)=Os(2,6) ROh(7,l)=Os(2,6)

!##: s

The formal co:

Ry (i,3) = y l (4,o) Rr s (i,3) = y' i (4,o) pys(i,3) = y ( i , 3 ) py s (i,3) = y s ( i , 3 ) Fy(l,3) = 0(2,2) D F y(l,3) = 0(2,2) py(i,3) = y(3,i) RP,(2,3) = P h ( l , 0 ) PP 6 (2,3) = P S (3.2) Fff(3,3) = A-(2,4) P F #(3,3) = A(2,4) PiJ(3,3) = g ( l , l ) RJVft(4,3) = Ns(l,2) PNh(A,3) = Nh{3,A) RO"(5,3) = O s (5,7) RO h (5,3)=0*(5,7) P0 h (5,3) = O h (3,5) PO h (5,3) = O h (3,5) FO(5,3) = 0(2,6) P F 0(5,3) = 0(2,6) PO(5,3) = 0(3,5) R£"(6,3)=£s(3,6) R£ s (6,3) =Eh(3,6) PEh(6,3) =Eh(3,6) P£ s (6,3) = £ s (3,6)

FiV(l,5)=A-(4,2) P F iV(l,5) = AT(4,2) DJV(1,5) = H(5,1) RATft(2,5) = A^(5,2) RATS(2,5) = ^ ( 5 , 2 ) PA' l (2,5) = Kh(5,2) PA S (2,5)=A%(5,2) ROft(3,5) = O s ( 7 , 5 ) RO h (3,5) = O s (7,5) PO h (3,5) = O ft (5,3) PO f c (3,5)=O h (5,3) FO(3,5) = 0 ( 4 , 4 ) P F 0(3,5) = 0 ( 4 , 4 ) PO(3,5) = Q ( 5 , 3 ) RP h (4,5) = P S (7,6) PP^(4,5) = Ph{5,4) F#(5,5)=£(4,6) P F #(5,5) = £ ( 4 , 6 ) Dff(5,5) = # ( 5 , 5 ) RjVs(6,5) = Nn(7,8) P7Vo(6,5)=7V s (5,6) RO s (7.5) = O h ( 4 , 6 ) RO.s(7,5) = 0 ^ ( 4 , 6 ) PO s (7,5) = 0 5 ( 5 , 7 ) POs(7,5)=Os(5,7)

Table llRH: POh(7A) = Oh{7A) PO fc (7,l) = 0 " ( 7 , l ) FO(7,1) = 0 ( 0 , 8 ) D F O ( 7 . 1) = 0 ( 0 , 8 )

DO(7,1) = 0 ( 1 , 7 ) RP ft (8,1) = P,(3,10) PPh(8,1) = Ph(l,8) —> s (even) RiV s (l,2) = iV"(4,3) PAT. (1,2) FAT(1,2) = N(l,2) s

RO (2,2) = RO,(2,2) = PO s (2,2) = PO,(2,2) = FO(2.2) =

Oft (4, 4) O fe (4.4) 0,(2,2) O s (2,2) Y(1,3)

D F O ( 2 , 2 ) = F(1,3)

DO(2,2) RP,(3,2) PP,(3,2) FP(3,2)

=0(2,2) = P"(0,1) = P,(2,3) = A"(l,4)

RA"(4,2) = A / l (2,4) RA,(4,2) = Kh(2A) PA' 1 (4, 2) = A"'1 (2, 4) PA,(4, 2) = A , (2, 4) FA(4,2) = # ( 1 , 5 )

The formal composition of Tables UR and I

Fff(7,3) == A(2,8) F D i/(7,3) = A(2,8) Dtf(7,3) == H(3,7) RA"(8,3) = A , (3, 8) RA,(8,3) = Kh(3,2) ?Kh(8,3) = A''(3, 8) PA, (8, 3) = A,(3,8) RA"(1,4) = A,(4,1) RA,(1,4) = A'"(l,4) PKh{lA) = A''(4,1) PA", (1,4) = A,(4,1) FA"(1,4) = £(3,2) RA'"(2.4) = A", (4, 2) RA,(2,4) = A' 1 (4, 2) PA"''(2, 4) = Kh(4,2) PA, (2, 4) = A , (4, 2) FA(2,4) == H(3,3) D F A(2.4) = H(3,3) DA"(2, 4) == A(4, 2) h

RN (3A)

= JV,(2,1) PAT (3,4) = iV h (4,3) FiV(3,4) == 7V(3.4) ROft(4,4) = 0,(2,3) RO, t (4,4) = O s (2,2) PO''(4,4) = Oh(4,4) h

FO(7,5) = ^ ( 4 , 8 ) D F O ( 7 , 5 ) = F(4,8)

DO(7, 5) = 0 ( 5 , 7) R£"(8,5) = £,(5,8) R£,(8,5) = Eh(5,8) PEh(8,5) = £''(5,8) P£,(8,5) = £ s (5,8) R £ " ( l , 6 ) = £,(6,1) R £ , ( l , 6 ) = Ek(6,1) P£ f e (l,6) = £''(6,1) P£,(l,6) =£,(6,1) F£(1.6) = A(5.2) RO s (2,6) = Oh(7,l) RO,(2.6) = 0' l (7,1) PO-'(2.6) = 0 , ( 2 . 6 ) PO,(2,6) = 0*(2,6) FO(2,6) = 0(5,3) D F O ( 2 , 6 ) = 0(5,3)

DO(2,6) = 0(6,2) R£ h (3,6) = £,(3,6) R£,(3,6) = £''(3,6) P£ h (3,6) + £' l (6,3) P£,(3,6) + £,(6,3) F£(3.6) = P(5,4) R£"(4,6) = £,(6,4

Table I I I H H : D F A(4,2) = # ( 1 , 5 ) DA(4,2) = A(2,4) RA"(5,2) = A,(2, 5) RA,(5,2) = Kh(2,5) ?Kh(5,2) = A h (2,5) PAT,(5,2) = A,(2,5) FA(5,2) = £(1,6) ROs(6,2)=Ofc(l,7) RO,(6,2) = 0 ' ' ( l , 7 ) PO s (6,2) = 0 , ( 6 , 2 ) PO,(6.2) = O s (6,2) FO(6,2) = 0 ( 1 . 7 ) D F O ( 6 , 2 ) = 0(1,7)

DO(6,2) = Q(2.6) RA" l (7,2) = A s (2.7) RA,(7,2) = A h (2.7) PA h (7,2) = Kh(2J) PA,(7, 2) = A , (2, 7) FA'(7,2) = P(1,8) RA""(8,2) = A,(2, 8) RA,(8,2) = Kh(2.8) PKh(8,2) = Kh{2,8) PA,(8, 2) = A , (2, 8) FA(8,2) = # ( 1 . 9 ) DFA"(8,2) = # ( 1 . 9 ) DK(8,2) = A(2,8)

The formal ,l

_

PO/,(4,4) = 0 (4.4) FO(4,4) = 0(3,5) D F O ( 4 , 4 ) = 0(3,5)

DO(4,4) = Q(4,4) RPft = P,(5,7) p p h ( 5 , 4 ) = P h (4,5) |FP(5,4) = £(3,6) R£"(6,4) = £,(4,6) R£,(6,4) = £^(4,6) P£'"(6,4) = £''(4,6) P£,(6,4) = £,(4,6) F£(6,4) = # ( 3 , 7 ) D F £(6,4) = # ( 3 , 7 ) D£(6,4) = £(4,6) R£"(7,4) = £,(7,4) R£,(7,4) = £ fe (7,4) P£ h (7,4) = £''(4,7) P£ S (7,4) = E,(4,7) F£(7,4) = A(3,6) RYh(8;4) = F,(3,9) RYh(8,4) = Y s (3,9) PYh(8,4) = Yh(8,4) PV h (8,4) = Yh(8,4) FY(8,4) = F(3,9) D F F(8,4) = K(3,9) DY(8,4) = Y(4,8)

Dsition of Tables IIIR R£,(4,6) = £,(6,4) PEh(4,6) = £ h ( 6 , 4 ) P£,(4,6) = £,(6,4) F£(4,6) = # ( 5 , 5 ) D F £(4,6) = # ( 5 , 5 ) D£(4,6) = £(6,4) RJVS(5,6) = Nh(8,7) PiV,(5,6) = iV,(6,5) FiV(5,6) = iV(5,6) RO s (6,6) = Oh{6,6) RO,(6,6) = 0 ^ ( 6 , 6) PO s (6,6) = 0 , ( 6 , 6 ) PO,(6,6) = 0-"(6,6) FO(6.6) = 0(5.7) DFO(6,6) =0(5,7)

DO(6.6) = Q(6,6) RP,(7,6) = P"(4,5) PP,(7,6) = P,(6,7) FP(7,6) = £ ( 5 , 8 ) R£ h (8,6) = £,(6,8) R£,(8,6) = £ h ( 6 , 8 ) P£ h (8,6) = £''(6,8) P£,(8.6) = £,(6,8) F£(8,6) = £(5.9) D F £(8,6) = £(5,9) D£(8,6) = £(6,8)

T h e dualities encircled in Tables 11;/, IIIH and. consequently, also in Tables URH, IIIRH give rise to four kinds of new mathematical objects [17, 18, 20-22]: F P ( 3 , 2) = A'(l, 4), F A'(l, 4) = P ( 3 , 2) = >

Hurwitz twistors (?• + 1 + .s = 5).

F P ( l , 8) = K{7,2),

pseudotwistors (r + l+s

F A'(7,2) = P ( l , 8 ) = >

= 9).

F P ( 7 , 6 ) = £ ( 5 , 8), F £ ( 5 , 8) = E{7, 6) ==» bitwistors (r + 1 + s = 13), F P ( 5 , 4) = E(3, 6), F £ ( 3 . 6) = P ( 5 , 4) = >

pseudobitwistors (r + 1 + s = 9).

A study of these objects will lead in P a r t II to the related dualities of holomorphic mappings, constructed via complex structures on isometric embeddings for some of the Herniitian Hurwitz pairs concerned so t h a t the embeddings are real parts of those holomorphic mappings [22]. It will also lead to the related duality of the almost hyperbolic quaternionic-type pseudodistances connected with hyperkahlerian holomorphic chains [1-3].

References 1. G. Boryczkaand L.M. Tovar, Hyperbolic-like manifolds, geometrical-properties and holomorphic mappings. Generalizations of Complex Analysis and their Applications in Physics. Ed. by .1. Lawrynowicz (Banach Center Publications 37), Inst, of Math., Pol. Acad, of ScL, Warszawa 1996, pp. 53-06. 2. P. Dolbeault, J. Kalina, and J. Lawrynowicz, Quaternionic-type structures and almost hyperbolic pseudodistances, Aspects of Complex Analysis, Differential Geometry, Mathematical Physics and Applications. Ed. by S. Dimiev and K. Sekigawa, World Scientific, Singapore 1999, pp. 183 192. 3. and J. Lawrynowicz, Holomorphic chains and extendability of holomorphic mappings. Deformations of Mathematical Structures. Complex Analysis with Physical Applications. Ed. by J. Lawrynowicz, Kluwer Academic Publ. Dordrecht 1989, pp. 191 204. 4. I. Furuoya, S. Kanemaki, J. Lawrynowicz and O. Suzuki, Hermitian Hurwitz pairs. Deformations of Mathematical Structures II. Hurwitz-Type Structures and Applications to Surface Physics, Kluwer Academic Publ. Dordrecht 1989, pp. 135-144. 5. S. Kanemaki and O. Suzuki, Hermitian pre-Hurwitz pairs wad the Minkowski space, Deformations of Mathematical Structures. Complex Analysis with Physical Applications. Ed. by J. Lawrynowicz, Kluwer Academic Publishers, Dordrecht-Boston-London 1989, pp. 225 232.

73

6. J. Lawrynowicz, Type-changing transformations of pseudo-euclidean Hurwitz pairs, Clifford analysis and particle lifetimes, Clifford Algebras and Their Applications in Mathematical Physics. Ed. by V. Dietrich, K. Habetha and G. Jank, Kluwer Academic Publishers, Dordrecht-BostonLondon 1998, pp. 217-226. 7. J. Avendano Lopez, F.L. Castillo Alvarado, R.A. Barrio and Anna Urbaniak-Kucharczyk, Clifford analysis, Riemannian geometry and the electromagnetic field, Essays of the Formal Aspects of Electromagnetic Theory. Ed. by A. Lakhtakia, World Scientific Publ. Co., Singapore 1993, pp. 533-558. 8. , R.M. Porter, E. Ramirez de Arellano and J. Rembelinski, On dualities generated by the generalized Hurwitz problem, Deformations of Mathematical Structures II. Hurwitz-Type Structures and Applications to Surface Physics. Ed. by J. Lawrynowicz, Kluwer Academic Publishers, Dordrecht-Boston-London 1994, pp. 189-208. 9. , E. Ramirez de Arellano and J. Rembelinski, The correspondence between type-reversing transformations of pseudo-euclidean Hurwitz pairs and Clifford algebras, Bull. Soc. Sci. Lettres Lodz 40 Ser. Rech. Deform. vol. 8 (1990), 61-97 and 99-129. 10. and J. Rembieliriski, Hurwitz pairs equipped with complex structures, Seminar on Deformations, Proceedings, Lodz-Warsaw 1982/84. Ed. by J. Lawrynowicz (Lecture Notes in Math. 1165), Springer-Verlag, Berlin 1985, pp. 184-195. 11. , Supercomplex vector spaces and spontaneous symmetry breaking, Seminari di Geometria 1984. Ed. by S. Coen, Universita di Bologna, Bologna 1985, pp. 131-154. 12. , Pseudo-euclidean Hurwitz pairs and generalized Fueter equations, Clifford Algebras and Their Applications in Mathematical Physics. Ed. by J.S.R. Chisholm and A.K. Common (NATO-ASI Series C: Mathematical and Physical Sciences 183), D. Reidel Publ. Co., DordrechtBoston-Lancaster-Tokyo 1986, pp. 39-48. 13. , Complete classification for pseudoeuclidean Hurwitz pairs including the symmetry operations, Bull. Soc. Sci. Lettres Lodz 36, no. 29 Ser. Rech. Deform. 4, no. 39 (1986), 15 pp. 14. J. Lawrynowicz and J. Rembieliriski, Pseudo-euclidean Hurwitz pairs and the Kahiza-Klein theories, J. Phys. A: Math. Gen. 20 (1987), 5831-5848. 15. , On the composition of nondegenerate quadratic forms with an arbitrary index, Ann. Fac. Sci.Toulouse Math. (5) 10 (1989), 141-168 [due to a printing error in vol. 10 the whole article was reprinted in vol. 11 (1990), no. 1, of the same journal, pp. 141-168]

74

16. 17. 18.

19.

20. 21.

22.

and O. Suzuki, Hurwitz duality theorems for Fueter and Dirac equations, Advances Appl. Clifford Algebras 7, no. 2 (1997), 113-132. , The tuiistor theory of the Hermitian Hurwitz pair (C 4 (/ 2 , 2 ),R 5 (/ 2 ,3)), ibid. 8, no. 1 (1998), 147-179. , An approach to the 5-, 9- and 13-dimensional complex dynamics I. Dynamical aspects, II Twistor aspects, Bull. Soc. Sci. Lettres Lodz 48 Ser. Rech. Deform. 25 (1998), 7-39 and 26 (1998), 23-48. , Hurwitz-type and space-time-type duality theorems for Hermitian Hurwitz pairs, Complex Analysis and Related Topics. Ed. by E. Ramirez de Arellano, M. Shapiro, L.M. Tovar and N. Vasilevski (OperatorTheory. Advances and Applications 114), Birkhauser Verlag, Basel 2000, pp. 155169. , An introduction to pseudotwistors — Spinor equations, Rev. Roumaine Math. Pures Appl., 12 pp., forthcoming. , An introduction to pseudotwistors — Spinor solutions vs. harmonic forms and cohomology groups, Clifford Algebras and Their Applications in Mathematical Physics I. Algebra and Physics. Ed. by R. Ablamowicz and B. Fauser (Progress in Physics 18), Birkhauser Verlag, Basel 2000, pp. 393-423. , Pseudotwistors, Internat. J. Theoret. Phys. 39 (2001), 9 pp., forthcoming.

75

E M B E D D I N G OF T H E MODULI SPACE OF R I E M A N N SURFACES W I T H IGETA S T R U C T U R E S INTO T H E SATO G R A S S M A N N MANIFOLD YOSHITAKE HASHIMOTO Department of Mathematics and Physics, Graduate School of Science, Osaka City University, Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan E-mail: [email protected] KIYOSHI OHBA Department of Mathematics, Faculty of Science, Oc.hanomizu University, Otsuka 2-1-1, Bunkyo-ku, Tokyo 112-8610, Japan E-mail: ohba&math. ocha.ac.jp Dedicated

to Professor

Takuo Fukuda on his sixtieth

birthday

We introduce the moduli space of closed Riemann surfaces with "Igeta structures", which is embedded in the Sato Grassmann manifold through the Krichever map. An Igeta structure on a closed Riemann surface R means the triple of a point p on R, a 2-jet at p, and a Lagrangian lattice, a certain subgroup of the first homology of R. The group IGLi(C) = Cxi C x acts on the moduli space and also on the Sato Grassmann manifold, and the embedding is equivariant.

1

Introduction

In our previous paper [1] we introduced a method of constructing closed Riemann surfaces with one marked point, and named the method the Igeta construction. (An Igeta means a pair of disjoint parallel line segments with the same length in the Gauss plane. Given g disjoint Igeta, we cut the plane along the segments and paste each side of one segment and the opposite side of the other segment by a parallel translation for each Igeta. We thus obtain a Riemann surface of genus g with one marked point at infinity.) When we construct a closed Riemann surface R with one marked point p by using the Igeta construction, we naturally obtain a local coordinate function £ around p and a Lagrangian lattice A of R. (A Lagrangian lattice of R means a subgroup A of Hi(R; Z) satisfying A = A x , where A 1 denotes the orthogonal complement with respect to the intersection form on H[(R;Z).) The local nierornorphic This research was partially supported by Grant-in-Aid for Scientific Research (No. 12740036), Ministry of Education, Science and Culture, Japan.

76

1-form dC x around p extends to a global abelian differential u> of the second kind, which has a pole of order 2 at p and has no other poles. The differential u> is a generator of the kernel, which is always one-dimensional, of the homomorphism H°(R;ttl(2p))

—>Hom(A,C)

given by abelian integrals. Then ( is reconstructed from the integral of a generator of the kernel of the homomorphism above and the 2-jet data of (. We therefore call the triple (p, A, j) of a point p on a closed Riemann surface R, a Lagrangian lattice A of R, and a 2-jet j represented by a local coordinate function around p an Igeta structure of R. Let MXg be the moduli space of closed Riemann surfaces of genus g with Igeta structures. It is easy to see that MXg is a complex V-manifold of dimension 3g. M. Sato and Y. Sato [6] introduced the infinite-dimensional Grassmann manifold (the universal Grassmann manifold or the Sato Grassmann manifold) to solve the KP-hierarchy. Let IGLi(C) = C x C x be the group of affine automorphisms of the affine line over C We naturally define actions of IGLi(C) on the moduli space M1g and on the Sato Grassmann manifold, and show that the moduli space M.Tg is embedded in the Sato Grassmann manifold SGM through the Krichever map equivariantly with respect to these actions. The authors wishes to express their gratitude to Professor Akio Hattori and Professor Yukio Matsumoto for continuous encouragement and to Professor Kenji Fukaya for useful comments. They are also grateful to Professor Koei Sekigawa and Professor Hideya Hashimoto for having given the second author an opportunity to give a talk at the workshop held at St. Constantine in Bulgaria in 2000. 2

Preliminaries

In this section we recall the definition of the Sato Grassmann manifold and the Krichever map. Let V 0 , V^°\ and V be complex vector spaces such as

n>0

n<0 0

(O

V := V ®.V \ where en (n € Z) are formal symbols. Then the Sato Grassmann manifold is defined as follows (see [6], [7]):

77

Definition ([6], [7]). The Sato Grassmann manifold of charge d, SGM^d\ is the set of linear subspaces U of V such that the natural projection au from U to V/V (0) is a Fredholm operator of index d. The Krichever map gives an embedding of the moduli space A^oo of triples (C,p, £) in the Sato Grassmann manifold SGM^~9\ where C is a closed Riemann surface of genus g, p a point of C. and £ a local coordinate function around p with £(p) = 0. The Krichever map is given as follows: Let / be a meromorphic function on C that is holomorphic on C — {p}. Then / is described around p with the local coordinate function £ as a Laurent series oo

/(o= E a^n («» eC ) n=~N

for some integer N, and determines a point J2^L~ N anen of the space V. Note that V is isomorphic to the vector space C((£)) of the formal Laurent series of one variable £ with complex coefficients:

c((0)<-^v £> n r~E a " e ")We then obtain an embedding fa :

limH0(C,O(Np))^V.

The image of £ turns out to be a linear subspace of V with the natural projection to V/V^0' of index —g, i.e., to be a point of SGM^~g\ In this way we obtain the Krichever embedding (see [3], [5]): K : MOO ^ SGM{~9\

3

{C,p,i) ^

^(\\mH0{C,O{Np))).

Embedding and IGLi(C)-actions

In this section we first define natural actions of the group IGLi(C) = C x C x on the moduli space MXg of Riemann surfaces with Igeta structures and on the Sato Grassmann manifold. (The product on C x C x is defined as (a, \)(b, n) := (a + Xb, A/i).) We then show the following result: Theorem. The moduli space MIg is embedded in the Sato Grassmann manifold SGM^~9^ through the Krichever map, and the embedding is equivariant with respect to the IGLi(C)-actions.

78

Let (R, (p,A, [£])) be a closed Riemann surface with an Igeta structure, where £ is a local coordinate function around p with ((p) = 0. For (a, A) e IGLi(C) = C x C x we set (a, A) • (R, (p, A, [C])) := (R, (p, A, [(AC"1 + a)- 1 ])). It is easy to check that (A( _ 1 + a ) - 1 = ((X + a £ ) _ 1 is a local coordinate function around p with (AC^1 +a)~x(p) = 0, and that the 2-jet [(AC-1 +a)~1} is well-defined for [(}. We thereby obtain a free action of IGLi(C) on MIg. We next define an action of IGLi(C) on the Sato Grassmann manifold SGM{d) by identifying V with the space C((£)) of the formal Laurent series of one variable £. For E™=-NanC € C ( ( 0 ) and (a, A) e IGLi(C) we set OO

(a,A)- ]T anC~ n=-N

OO

E

MAr'+fl)""

n=-N oo

n=-yv The right-hand-side of the equation above turns out to be a formal Laurent series of £. We then obtain a linear action of IGLi(C) on V fixing V' 0 ', and hence an action on SGM^ for each d. We now show that the moduli space MX(J is embedded into SGM1-"^ equivariantly. Let (p,A,j) be an Igeta structure on a closed Riemann surface R. The Lagrangian lattice A and the 2-jet j uniquely determines a local coordinate function £ around p as follows (see also [1]): Consider the homomorphism H°(R;n1(2p))

—>Hom(A,C)

given by abelian integrals. Then the kernel Z of the homomorphism above is one-dimensional by the Riemann-Roch formula and the bilinear relations of Riemann. If u; is a generator of Z, the integral of u induces a local coordinate function £ around p with £(p) = 0 such that ri£~! extends to u). The function £ is determined up to the affine automorphisms

for (p,A). The IGLi(C)-action on the set of 2-jets at p is equivalent to the one on the set of 2-jets at oo of the complex projective line Pi (C) = C U {oo}. It is easy to check that this action is principal (i.e., transitive and free). We thereby obtain the unique local coordinate function £ around p for the Igeta structure (p,A,j).

79 Considering the local coordinate function £ associated to a given Igeta structure (p, A,j) on a closed Riemann surface R, we obtain an embedding of MXg into SGAf(~9^ through the Krichever map: i: Mlg

--> Moo ^

SGM{-g).

The embedding is equivariant with respect to the actions of IGLi(C) because (A£ _1 + a ) - 1 = £(A + a £ ) _ 1 is the coordinate function around p associated to the Igeta structure (p, A, (a, A) • j) for any (a, A) € IGLi(C). Remark 1. The result above implies that the drawing of each set of disjoint Igeta on the Gauss plane determines a point of the Sato Grassmann manifold. Remark 2. When we consider abelian differentials ur of the second kind having a unique pole at the marked point p of order 2 instead of Lagrangian lattices, a similar result holds: the moduli space of Riemann surfaces with (p,<x>,j)-structures is embedded in SGM^~9' equivariantly with respect to the actions of IGLi(C). In this case the dimension of the moduli space is 4g. References 1. Y. Hashimoto and K. Ohba, Gutting and pasting of Riemann surfaces with abelian differentials, I, Int. J. Math. 10(1999) 587-617. 2. , On the anti-parallel Igeta construction of Riemann surfaces, Aspects of Complex Analysis, Differential Geometry, Mathematical Physics and Applications, World Scientific, (1999) 60-76. 3. N. Kawamoto, Y. Namikawa, A. Tsuchiya and Y. Yamada, Geometric realization of conformed field theory, Comni. Math. Phys. 116(1988) 247308. 4. I.M. Krichever, Methods of algebraic geometry in the theory of non-linear equations, Russian Math. Surveys 32:6(1977) 185-213. 5. J.M. Muiioz Porras and F.J. Plaza Martin, Equations of the moduli of pointed curves in the infinite Grassmannian, J. Differential Geom. 51(1999) 431-469. 6. M. Sato and Y. Sato, Soliton equations as dynamical systems on infinitedimensional Grassmann manifold, Nonlinear partial differential equations in applied science (Tokyo, 1982), North-Holland Math. Stud., 81(1983) 259-271. 7. G. Segal and G. Wilson, Loop groups and equations of KdV type, Inst. Hautes Etudes Sci. Publ. Math. 61(1985) 5-65.

80

ON T H E Q U O T I E N T SPACES OF S 2 x S2 U N D E R T H E N A T U R A L ACTION OF S U B G R O U P S OF D 4 KAZUNORI KIKUCHI Department of Mathematics, Osaka University, Toyonaka, Osaka 560-0043, Japan E-mail: [email protected] Dedicated

to the memory

of Professor

Katsuo

Kawakubo

There are 10 quotient spaces of S2 x S2 under the natural action of subgroups of the 4th dihedral group DA. 5 of the 10 quotient spaces are known to be diffeomorphic to familiar 4-mauifolds. We prove that 4 of the remaining 5 quotient spaces are diffeomorphic to CP 2 , RP 2 x S 2 , S2 xKP 2 , and the Grassmannian manifold G'2(R4) of all unoriented 2-planes through the origin in R4. We also pose problems on the last quotient space and on G*2(R4).

1

Introduction

The 4th dihedral group D\ naturally acts on S2 x S2. In fact, the map interchanging the two coordinates and the antipodal maps on eigher factor generate a group isomorphic to D4. There are exactly 10 subgroups of D4. Thus there are 10 corresponding quotient spaces of S2 x S2 under the natural action of subgroups of D,\. Massey [1] used a diagram of 5 quotient spaces (and 5 quotient maps) among the 10 quotient spaces above, to prove that the quotient space of CP 2 under complex conjugation is diffeomorphic to S4. In the course of his proof, he showed that the 5 quotient spaces in his diagram are S2 x S2, CP 2 , RP 2 x RP 2 , RP 4 , and S4. The purpose of this talk is to study the remaining 5 quotient spaces of S2 x S2, find familiar manifolds diffeomorphic to the quotient spaces, and pose problems on the quotient spaces. To achieve our purpose, we use the complete diagram of all the 10 quotient spaces and all the 15 quotient maps. Our main result is that 4 of the remaining 5 quotient spaces are diffeomorphic to CP 2 , RP 2 x S2, S2 x RP 2 , and the Grassmannian manifold G 2 (R 4 ) of all unoriented 2-planes through the origin in M4. We have not succeeded in finding a familiar manifold diffeomorphic to the last quotient space. This talk goes as follows. In §2 we give notation. In §3 we give a brief survey to Massey's diagram and results. In §4 we study the remaining quotient spaces of S2 x 6'2. In §5 we conclude this talk by posing two problems.

81

2

Notation

We identify the 4th dihedral group D\ with the set of all orthogonal matrices in 0(2) which fix the set of the 4 points (±}) in R 2 : precisely, D4 =

{ ( o i ) ' ( i o ) ' ( - i o ) ' ( o i)'

(o-i)'(o -v'v °M- i( vr DA naturally acts on S2 x S2 = { (x,y) £ I 3 x I 3 | ||x|| = \\y\\ = 1 } as matrix multiplication: e.g.,

Each subgroup of D4 acts on S2 x S2 in the same way. There are exactly 10 subgroups of D4: one of order 1, five of order 2, three of order 4, and one of order 8. For the subgroups of D4 of order 2 or 4, i.e., for those neigher the trivial one nor D\ itself, we use the following notation: G2

'i:=\(io))'

G2

-2:=\(o 1))'

G2 3:=

< \{o

-1))'

G 2

'+

G2 2

'<

<e.-°o>-

G4J

where a pair of ( , ) means that the subgroup is generated by the elements in ( , ). Note that the first subscript of each subgroup above coincides with the order of the subgroup. There are 10 corresponding quotient spaces of S2 x S 2 under the natural action above of subgroups of D\. For the quotient spaces except S2 x S2 itself, we use the following notation: XiJ:=S2xS2/Gi,j,

XlJ:=S2xS2/G'r:j,

X8 := S2 x

S2/D4.

Note that the first subscript of each quotient space above coincides with the degree of the quotient map from S2 x S2 onto the quotient space.

82

3

The quotient spaces in Massey's diagram

In order to prove that the quotient space of CP 2 under complex conjugation is diffeomorpliic to S'4, Massey [1] used the following diagram of inclusion •relations between subgroups of D4 <-*2,l

G44

D4

{1} G4.J 4,2

and the following diagram of corresponding quotient maps between quotient spaces of S2 x S2 X'2,\

•*- Xj.l

S2 x S2

X*

X4:2

In the course of his proof, he showed the following: • X'2,1, the 2-fold symmetric product of S2, is diffeomorpliic to CP 2 . • X-2,\ —> X4.1 is equivalent to the quotient map under complex cojligation. • X4,2 is diffeomorphic to MP2 x RP 2 . • Xs, the 2-fold symmetric product of RP 2 , is diffeomorphic to RP 4 . • X.\,i —* ^ 8 is a 2-sheeted unbranched covering map. • X4J is diffeomorphic to S4. Note that S2 x S 2 -> X2.i ^ CP 2 , X2,i * CP 2 -> X4A =* S4, and X4.2 ~ MP2 x RP 2 —> Xg = RP 4 are double branched coverings whose branched loci are connected closed surfaces (S2 or RP 2 ): see also [2, §6].

83 4

O t h e r q u o t i e n t s p a c e s in t h e c o m p l e t e d i a g r a m

Massey's diagrams are incomplete in t h a t they include only 5 subgroups or quotient spaces out of t h e 10 ones. T h e complete diagram of inclusion relations between all t h e subgroups of D4 is

and the complete diagram of corresponding quotient maps between all t h e quotient spaces of S2 x S2 is

We study the other quotient spaces of S'2 x S2 not in Massey's diagram but in the complete diagram above. P r o p o s i t i o n 4 . 1 . X2 1 is diffeornorphic with the opposite orientation.

to the complex projective plane C P 2

Proof. Both T (? o ) P r e s e r v e the orientation of S2 x S2, and they are conjugate to each other since ( _°x ~J ) = ( ° - 1 ) _ 1 ( « ' ) (» - 1 ) , while ( ° " ' ) reverses t h e orientation of S2 x S2. According t o Massey [1], X2,i = S2 x S2/{( ° J ) ) is diffeornorphic t o C P 2 . Therefore, X!2l = S2 x S2/(( ^~01)) is diffeornorphic to C P 2 . ' D Proposition 4.2. respectively.

X2.2 and X'22 are diffeornorphic

to MP 2 x S2 and S2 x MP 2

Proof. T h e generator ( "Q1 ° ) of G2t2 acts antipodally on t h e first factor and trivially on t h e second factor of S2 x S2. Hence, X2,2 is diffeornorphic t o

84

RP 2 x S2. The proof for X^2 goes the same way. Proposition 4.3. X2.3 is diffeomorphic to the Grassmannian G2(R 4 ) of all unoriented 2-planes through the origin in R4.

D manifold

Proof. S2 x S2 is diffeomorphic to the Grassmannian manifold G2(R 4 ) of all oriented 2-planes through the origin in R 4 , since G 2 (R 4 ) is identified with the set of all 2-vectors ui G [\ R 4 such that ui = £ A •// for some 1vectors £, 77 and ||u;|| = 1, which turns out to be equal to the set of all pairs (ui+, uj-) £ / \ + R 4 ® A - ^ 4 °f self-dual and anti-self-dual 2-vectors such that || w + || = ||a;_|| = l/>/2: for details, see [3]. The action of (~0l \ ) on S 2 x S2 is equivalent to that of —1 on G2(R 4 ), which is nothing but reversing the orientation of each 2-plane in G2(R 4 ). Therefore, ^2,3 is diffeomorphic to G 2 (M 4 ). ' • We should remark here that Massey [1] stated without proof that R P 2 x S2 and G2(R 4 ) occur among the quotient spaces of S2 x S2 under the natural action of subgroups of D4. It is not difficult to verify that X4.3 is also a smooth 4-manifold. We have not yet succeeded, however, in finding a familiar 4-manifold diffeomorphic to -^4.3-

5

Problems

We conclude this talk by giving two problems on X ^ and X2,3The first problem is on the last quotient space remaining unfamiliar. Problem 5.1.

Find a familiar 4-manifold diffeomorphic to X4 3 = S2 x

^/((f-o1))To state the second problem, we take a smooth embedding of S2 x S2 in CP : namely, the smooth quadric surface Q = { Yl'j=o zj = ^ ) m ^ P 3 - The smooth map 3

J : CP —> CP ,

J[zo, z\_, Z2, 23] = [—z\, zo, — 23, Z2],

is an orientation-preserving fixed-point free involution on C P 3 which restricts to the quadric surface Q. Let Y be the quotient space Q/(J). Note that Y and ^2,3 have the same fundamental group Z 2 and the same homology groups, and that they are possibly homeomorphic. That is why we pose the following:

85

Problem 5.2. Is Y diffeomorphic to X2,3 = S2 x S2/{( V _0X)} = G 2 (K 4 )? Or are they another pair of not diffeomorphic but homeomorphic 4-manifolds? Acknowledgments The author would like to thank the organizers for having invited him to give a talk at the workshop. He is also grateful to Professor Stancho Dimiev and Professor Kouei Sekigawa for their hospitality, and to Professor Hideya Hashimoto and Professor Kiyoshi Ohba for helpful discussions. References 1. W.S. Massey, The quotient space of the complex projective plane under conjugation is a A-sphere, Geom. Dedicata 2(1973), 371-374. 2. S. Nagami, Existence of spin structures on double branched covering spaces over four-manifolds, Osaka J. Math. 37(2000), 425-440. 3. I.M. Singer and J.A. Thorpe, The curvature of ^.-dimensional Einstein spaces, in "Global Analysis, Papers in honor of K. Kodaira", ed. D. Spencer and S. Iyanaga, Princeton University Press, Princeton, 1969, 355-365.

86 EXISTENCE OF SPIN STRUCTURES ON CYCLIC B R A N C H E D COVERING SPACES OVER FOUR-MANIFOLDS

SEIJI N A G A MI Department of Mathematics, Osaka University. Toyonaka, Osaka 560-0043. Japan E-mail: [email protected]. osaka-u.ac.jp We study whether an m-fold branched covering space X over a given oriented connected closed smooth 4-manifold X with Hi(X;Z2) = 0 branched along an oriented connected closed surface F smoothly embedded in X is spin or not. We prove that when m is odd, X is spin if and only if X is spin, and that when m is even, X is spin if and only if the modulo 2 reduction of the homology class [F]/m coincides with the second Stiefel-Whitney class w-2(X) of X.

1

Introduction

Our aim of this paper is to study whether a branched covering space over a given oriented connected closed smooth 4-manifold X with H\(X;7,2) = 0 branched along a connected closed surface F smoothly embedded in X is spin or not. As for this problem, we have obtained the following theorem in [3]. Theorem ([3]). Let X he the double branched covering space over an oriented connected closed smooth 4-manifold X with Hi(X;2i2) = 0 branched along a connected closed surface F smoothly embedded in X. Then X is spin if and only if F is orientable and the modulo 2 reduction of [F]/2 S H2(X; Z) coincides with the second Stiefel-Whitney class 'U^(X) £ H2{X\ Z2) of X for a fixed orientation of F. Here, [F] £ H2(X;Z) denotes the homology class that is represented by F with coefficients in Z, and [F]/2 £ H2{X; Z) the homology class that satisfies 2([F]/2) = [F] £ H2(X; Z). We wish to generalize the theorem above to the case when the covering degree is greater than two. First, note the following. Suppose that F is non-orientable. Then the order of the homology class represented by a meridian to F is either 1 or 2. Hence 'm-fold branched covering spaces over X branched along F with in > 3 do not exist. Therefore we shall consider the case when the branched locus is orientable. Our result is the following.

87

Theorem 1.1. Let X be an m-fold branched covering space over an oriented connected closed smooth A-manifold X with H\(X; Z2) = 0 branched along an oriented connected closed surface F smoothly embedded in X. When rn is odd, X is spin if and only if X is spin. When in is even, X is spin if and only if the modulo 2 reduction of [F]/m £ H2(X\ Z) coincides with the second Stiefel-Whitney class w2(X) £ H2(X;Z2) of X, where [F]/m £ H2(X;Z) is a homology class that satisfies m([F]/m) = [F] £ H2(X\Z). Remark 1.1. [F] is divisible by rn (§2 Propositon 2.1) and \F\/m is not uniquely determined in general. If m is a power of 2, then [F]/m is uniquely determined, since we have assumed that Hi(X; Z2) = 0. Example 1.1. Let C be a non-singular algebraic curve of degree run on the complex projective plane C P 2 . It is known that the m-fold branched covering space X over C P 2 branched along C is spin if and only if 771 is even and n is odd [2, p. 22]. Note that C P 2 is not spin and that the modulo 2 reduction of [C\/m = n[CP'] £ # 2 ( C P 2 ; Z) coincides with u; 2 (CP 2 ) if and only if n is odd. In §2 we give notation and prove a proposition, and in §3 we prove Theorem 1.1.

2

Preliminaries

We first give some notation. N a t a t i o n 2 . 1 . (I) For a Z-module G with no 2-torsion and an element g £ G such that g is divisible by 2* with i £ N, we denote by g/2l the element that satisfies 2l(g/2l) = g. Note that g/2l is uniquely determined, since G has no 2-torsion. (II) Let Y be an oriented closed smooth 4-manifold and in a natural number. (i) For elements xm,ym £ H2(Y; Z m ) (resp. x,y £ H2{Y; Z)), we denote by xm • ym £ Z m (resp. x-y £ Z) the intersection number of xm and ym (resp. x and y). (ii) Let H be an oriented closed surface smoothly embedded in Y. We denote by [H]m £ H2{Y; Zm) (resp. [H] £ H2{Y\ Z)) the homology class represented by H with coefficients in Z m (resp. Z). (iii) Let L be a closed tubular neighborhood of H in Y. Then dL has an .S^-bundle structure over H. We call an embedding of a 1-sphere onto a fiber of dL a meridian (to H).

88

The following proposition is fundamental in the theory of branched covering spaces. Proposition 2.1. Let Y be an oriented connected closed smooth 4-manifold and H an oriented connected closed surface smoothly embedded in Y. Then an m-fold cyclic branched covering space Y over Y branched along H exists if and only if the homology class [H] £ H2(Y; Z) represented by H is divisible by m. Proof. We have only to show that Y exists if and only if there exists a surjection p : H\(Y — H;Z) —> Zm that sends the homology class represented by an oriented meridian to if to 1 £ Z m . Consider the following commutative diagram, where the vertical maps denote the Lefschetz duality isomorphisms, the upper row the connecting homomorphism of the cohomogy exact sequence for the pair (Y, Y — H) with coefficients in Z m , and the lower row a homology exact sequence for the pair (Y, H) with coefficients in Z m .

Hl{Y-H-Zm)

• H2(Y,Y - H;Zm)

•h H3(Y,H;Zm)

I" -^->

H2(H;Zm)

- ^ H

H2(Y; Z m )

Here, dm denotes the connecting homomorphism and (i/f*) m the homomorphism induced by the inclusion in : H -^ Y. The diagram above shows that such a homomorphism p e H1(Y - H; Z m ) = Hom(H\ (Y - H; Z), Z m ) exists if and only if there exists an element p £ Hs(Y, H;Zm) such that dm(p) = 1 e Z m = H2{H;Zm). By the exactness of the lower row, such a p exists if and only if (iH*)m([H]m) = 0 <E H2{Y;Zm). Clearly, (iH*)m{[H]m) = 0 € H2(Y;Zm) if and only if [H] is divisible by m. This completes the proof. D R e m a r k 2.1. In the situation of Proposition 2.1, suppose that H\{Y; Z 2 ) = 0. Then, by using the homology exact sequence for the pair (Y, Y — H) with coefficients in Z, we can easily see that [H] is divisible by 2 if and only if HX{Y - H; Z 2 ) = Z 2 (see [3^ Proposition 2.8]). Hence we observe that a double branchd covering space Y exists if and only if Hi (Y — H; Z 2 ) = Hl (Y — H; Z 2 ) = Z 2 . Therefore, a double branched covering map over Y branched along H is uniquely determined up to isomorphism. Let / : Y ~» Y be a double branched covering map branched along H. Set H = f~[(H). If [H] £ H2(Y:Z) is divisible by 2l+\ then [H] e H2(Y;Z) is divisible by I1. This fact can be shown as follows. Let Y2i+i -* Y be a 2 ;+1 -fold branched covering

89 m a p branched along H. Let a transformation r : Y2i+\ —> Y2i+i generate the covering transformation group. Let Y2I+-I/(T2) denote the qutient manifold of y 2 '+i by the cyclic group ( r 2 ) generated by r 2 . Define p : Y2I+I/{T2} —> Y by p([a:]) = /(x') for x G Y2i.+i, where [:r] G F 2 ; + i / ( r 2 ) denotes the equivalence class of x. Then p is a double branched covering m a p branched along H and hence isomorphic to / : Y —^ Y by t h e argument above. Since the quotient m a p Y2i+i —> Y2I+I/(T2) = Y is a 2 ; -fold branched covering m a p branched along H, [H] is divisible by 2l by Poposition 2.1. 3

P r o o f of T h e o r e m 1.1

We divide the proof into two cases: the case when in is odd and the case when m is even. Let T : X —> X be a generator of the covering transformation group. Let 7r : X —> X denote the covering m a p . Case 1: in is odd. First suppose t h a t X is spin. Recall t h a t (n*)2 : H2(X; Z 2 ) -> -H"2(X; Z 2 ) is an isomorphism onto the subgroup of H2(X;Z2) t h a t consists of (T*) 2 2 invariant elements { i 2 <E i f ( X ; Z 2 ) ; (T*) 2 (5 2 ) = z 2 } (see [1, p.121]). Then by the fact above, for any z2 G H2(X;Z2), there exists an element y2 G i J 2 ( X ; Z 2 ) such t h a t (7T*)2(y2) = z2 + (T*)2(z2) + ••• + ( T * ) ™ " ^ ^ ) . Note t h a t we have (z2 + {T*)2(z2) + ••• + (T*)2n-\z2)) • (z2 + (T*)2(z2) + ••• + (T*y2n-\z2)) = z2 • (z2 + (T*)2(~z2) + ••• + (T*)™-\z2)) = ~z2 • z2 G Z 2 . On the other hand, we have t h a t (ir*)2(y2) • (7r*) 2 (y 2 ) = Vi • V'l since the covering degree is odd. Hence we have z2 • z2 = y2 • y2 = 0 for all z2 G H2(X; Z 2 ) since X is spin. Therefore X is spin. Next suppose t h a t X is not spin. Then there exists an element y2 G H2(X; Z 2 ) such t h a t y2 • Wi = 1. Since (7r*) 2 (y 2 ) • (7r*) 2 (j/ 2 ) = V2 • Vi = 1, X is not spin. This completes the proof in the case when in is odd. Note t h a t Case 1 holds without the assumption t h a t H\(X\ Z 2 ) = 0. Case 2: m is even. Let p and q denote the integers such t h a t 2pq = m with q odd. Let l X/(T ) denote the quotient manifold of X by the cyclic group (T ? ) generated l by T . T h e n we have t h e following sequence of quotient maps, which are cyclic branched covering maps with a connected branched locus: X=X/{Tm)

^

••• - ^ ^

^

X/{T2") 2

X/(T ")=X

^ U

X/{T2"~l)

^

^

90 Here, 717 _: X/(Tl) - • X/(T^>) denote the map defined by 7r/([x|t) = [x]j for .7; G X, where [x]». G X/(T') denotes the equivalence class of a; by the quotient map X —> X/(T%). Then the degree of the branched covering map •Kq : X —> X/(T2 ) is odd. Hence X is spin if and only if X/(T'2''} is spin by Case 1. The map 7r2.+ i : X/(T'r+i) - • X/(T'r) is the double branched 2 covering map for i = 1, ..., p. Let F 2 . C X /(T ) denote the branched locus. By Proposition 2.1, [F] e # 2 ( X ; Z) is divisible by m = 2pg and hence by 2P. Therefore [F2,] G H2(X/(T2'); Z) is divisible by 2 ^ by Remark 2.1. To complete the proof in Case 2, we need the following key lemma. L e m m a 3 . 1 . Let Y be the double branched covering space, over an oriented connected closed smooth 4-manifold Y with H\{Y\ Z 2 ) = 0 branched along an oriented connected closed surface H smoothly embedded in Y with covering projection f : Y —* Y. Suppose that the homology class [H] e H2{Y\ Z) represented by H with coefficients in Z is divisible by 2*+1 with I > 1. Then we have that Hl(Y;Z2) = 0 and that [H]/2l £ H2{Y;Z) is characteristic if 1+1 and only if [H}/2 G H2{Y; Z) is characteristic, where H = f~l{H). Remark 3.1.

Hx(Y; Z2) = 0 holds even if / = 0. See [3, Lemma 3.8].

Proof of Lemma 3.1. Let L be an closed tubular neighborhood of H in Y. Let W denote the closure of the complement of L in Y and L (resp. W) the inverse image of L (resp. W) under / . We claim that H\(W\Z2) = Z2 and the generator of Hi(W;Z2) is the homology class represented by a meridian. To seejdiis^consider the following Gysin exact seqence for the double cover / ' = f\W : W —> W: 0

• H°(W;Z2)

-^-^

H°(W;Z2)

z2 l

> H (W;Z2)

z2 ^ ^

l

H {W-Z2)

z2 > H2{W;Z2) First note that H°(W;Z2)

•

H°(W;Z2)

z2 X

> H {W-Z2)

z2 ^—^ = Hl{W\Z2)

H2(W;Z2)^ Z 2 by the connectedness of W

91 and Remark 2.1. Therefore, by the diagram above, H1(W;Z2) is the trivial group or isomorphic to Z2. Hence we have only to show t h a t the homology class represented by a meridian to H in W with coefficients in Z2 does not vanish. Since [H] is divisible by 4, [H] is divisible by 2 by Remark 2.1. Hence a double branched covering space branched along H exists by Proposition 2.1. Therefore, the homology class represented by a meridian in W with coefficients in Z2 does not vanish. To see t h a t Hi(Y;Z2) = 0, consider the following Mayer-Vietoris exact sequence, where (ej.*)2 and (e-i*)2 denote the homomorphisms induced by the inclusions e.y : 8L —> W and c2 '• dL —> L respectively. Hi(dL;Z2)

lei hm,>2 h

'

':

H1{W;Z2)iBH1(L;Z2) • Hi(Y;Z2)

- 0.

Since H\(W\ Z2) is generated by the homology class represented by a meridian, cu (& c-2* is a surjection. Hence we have Hi(Y\ Z2) = 0 by the exactness of the sequence above. Consider the homology exact sequence for the pair (Y, Z) with coefficients in Z 2 . H2(Y;Z2)

> H2(Y,W;Z2)

> H1(W;Z2)

> Hl(Y;Z2)

2

= 0.

2

We have H2{Y,W;Z2) ^ H2{L,dL;Z2) ^ H (L;Z2) ^ H {H;Z2) ^ Z2 by excision and Poincare-Lefschetz duality. Combining this fact with Hl(W;Z2) = Z2, we obtain t h e following commutative diagram, where t h e lower row is exact and (iw*)2 is the homomorphism induced by the inclusion iw : W - • Y: H2(W;Z2)

•

(f.h

(f,h H2(W;Z2)

H2(Y;Z2)

-^^>

H2{Y;Z2)

• 0.

Note t h a t ( / ' * ) 2 : H2(W; Z2) -> H'2(W; Zj) is an injection by the Gysin exact sequence above. Therefore, ( / ^ 2 : H2(W;Z2) —• H2(W;Z2) is a surjection. Moreover, (iw*)2 is a surjection by the exactness of the lower row of the diagram above. Using these facts, we see t h a t (f*)2 is a surjection by the commutativity of the diagram above. Let a homology class x £ H2(Y;Z) and a torsion element ( e H2{Y\Z) (resp. a homology class x 6 H2{Y;Z) and a torsion element C, G H2(Y;Z)) be such t h a t [H] = 2l+1x + ( (resp. [H] = 2lx + C). Then we have 2[H] • f] =

92

[H] • f*(i}) for all f] G H2(Y; Z), and hence we have x • f] = x • f*(fj). Since H\(Y;Z2) = 0, every 772 G H2{Y\ Z 2 ) has an integral lift . Hence we have x2 • 772 = x2 • (f*)2(fj2) for all 7/2 G ^2(5^; Z2), where x 2 and x 2 denote the modulo 2 reductions of x and x respectively. Suppose t h a t [H}/21+1 is characteristic, i.e., x is characteristic. Then we have^x-2 • 7>2 = x2 • (f*)2(V2) = {f*h{m) • (f*h(V2) = m • m for all 7>2 G H2{Y;Z2). Hence x is characteristic. (The equality {f*)?^) • (/*)2(%) = ca 772 • V2 n be shown as follows. Since [H] is divisible by 4, [H] is divisible by 2 by Remark 2.1. Hence we have [if]2 = 0. Therefore, we have ( / . ^ ( ^ h ) • (f*h(m) = m • m + [H}2 • m = m • m- For a proof of the equality (/*) 2 ('?2) • (f*h(m) = m • m + [H}2 • fj2, see [3, Sublmma 3.5].) Coversely, suppose t h a t [if]/2' is characteristic, i.e., x is characteristic. For any element 772 G H2{Y; Z2), there exists an element 772 such t h a t (f*h(V2) = V2, since (/«) 2 : if2(-X"; Z 2 ) —> H2(X: Z 2 ) is a sujection. T h e n we have .T2 - 772 = £'2 • ?72 = V2 • f]2 = V2 • m for all 772 G H2(Y] Z2). Hence x is characteristic. This completes t h e proof of Lemma 3.1. Since Hi(X/{T2')\ Z2) = 0 for i < p — 1 by Lemma 3.1, we have t h a t X/{T "} is spin if and only if [ F 2 , - i ] / 2 G H2(X/(T2"~1);Z) is characteristic by Theorem [3]. By virtue of Lemma 3.1, [F2;—1]/2 is characteristic if and only if [JF 2 I--2]/2 2 G H2(X/{T'2'' ");Z) is characteristic. Iterating t h e process above, we have t h a t X is spin if and only if [F]/2P G i f 2 ( X / ( T 2 " ) ; Z) is characteristic. Since m = 2pq with q odd, [F]/2P is characteristic if and only if [F]/m is characteristic. This completes the proof in the case when m is even. 2

We have completed the proof of Theorem 1.1. References 1. G.E. Bredon, Introduction to compact transformation press, New York and London, 1972. 2. N. Hitchin, Harmonic spinors, Adv. Math. 14(1974), 3. S. Nagami, Existence of spin structures on double spaces over four-manifolds, Osaka J. Math. 37(2000),

groups, Academic 1-55. branched covering 425-440.

93 L E N G T H S P E C T R U M OF GEODESIC S P H E R E S IN R A N K ONE S Y M M E T R I C SPACES TOSHIAKI ADACHI Department of Mathematics, Nagoya Institute of Technology Nagoya 466-8555, Japan E-mail: [email protected] We study closed geodesies on geodesic spheres and tubes around hyperplane in rank one symmetric spaces. We give a condition on the radius that all the length spectrums are simple. Also in view of classical number theory, we study the asymptotic behaviors of multiplicity and the number of length spectrums.

1

Introduction

The aim of this paper is to study the length spectrum of geodesic spheres and tubes around hyperplanes in rank one symmetric spaces. It is well-known that geodesic spheres in complex and quaternionic projective spaces and in a Cayley plane are counter-examples to the odd-dimensional extension of Klingenberg's lemma, which assures that on a compact simply connected evendimensional Riemannian manifold whose sectional curvatures lie in the interval (0, K] the length of each closed geodesic is not less than 2n/y/~K. In this paper we throw a new light on these classical objects and study how all length of closed geodesies are distributed on the real line. For the cases of complex and quaternion we already investigated in the preceding papers [AMY] and [AM], so what we have to make mention here is the case of Cayley algebra. But for the sake of readers' convenience we here summarize all the cases. Throughout this paper, we denote by K either the field of complex numbers C, the field of quaternion numbers H or the Cayley algebra O. We should note that IK does not indicate the field of real numbers R. So by KPn we denote a K-projective space which is either a complex projective space C P n , a quaternionic projective space HP™ or a Cayley projective plane O P 2 , and by KHn we denote a K-hyperbolic space which is either a complex hyperbolic space CHn, a quaternionic hyperbolic space MHn or a Cayley hyperbolic plane OH2. 2

Geodesic spheres and tubes around hyperplanes

We shall start by briefly recalling geodesic spheres and tubes around hyperplanes in K-projective and hyperbolic spaces from the viewpoint of submani-

94 fold theory. Let w : s « « n + i ) - i ( i ) _> K P n ( 4 ) be a Hopf fibration of a standard sphere of radius 1 onto a K-projective space of maximum sectional curvature 4, where d = dim(K), which is 2, 4 or 8 according to K = C, M or O. We consider a real hypersurface

Mg(r)

(w0,...,wn)eKn+l

w =

K-i|2 +

|tuo| = cos r, \wn\2 = sin 2 ?-

in S d ( " + 1 ) - 1 for 0 < r < TT/2. Then Ms(r; 4) = w(Mg(r)) of radius r in K P r a ( 4 ) . We also denote by w : nf71*^1 space Tjd(n+1)-1

ti]

r

i

\ _

= \w = (w0,...,wn)

e

-» KHn(-4)

rn+1

is a geodesic sphere

a fibration of a de Sitter

|'ty 0 | 2 + h i | 2 + -

-1}

onto a K-hyperbolic space of minimum sectional curvature —4. If we consider the following real hypersurfaces in H1 for 0 < r < oo, Ms{r)

= { w G K n + 1 | |to 0 | = coshr,

Mt(r) = Mh =

liveKn+1 w e

rn+l

\Wl\2

+ |tu n | 2 = s i n h 2 r },

u' 0 | 2 + |«;i| 2 + --- + | w n _ j | 2 sinh r wn \Wo -

W]

• cosh ?',

|= 1

then the real hypersurfaces Ma(r;-4)=w(Ma(r)),

Mt(r; - 4 ) = w(Mt(r)),

Mh(-4)

=

w{Mh)

in KHn(—4) are a geodesic sphere of radius r, a t u b e of radius r around hyperplane KHn~1, and a horosphere respectively. Let M denote one of these real hypersurfaces in KPn or KHn. We have a natural orthogonal decomposition V®VL of the tangent bundle TM of M: T h e subbundle V is the maximal subbundle of TM which is invariant under the action of K. If we denote by MM the unit outer normal of M in I P " or in KHn, then t h e subspace P ^ at each point x e M is generated by this normal; V^ = (A/A/(X) • K) n T^A/. We denote by AM the shape operator of M with respect t o NM- It satisfies t h e following table for each u e V and

95 7] e

vL.

M s (r;4)

Mh(-4)

A/,(r; - 4 )

Mt(r; - 4 )

AMU

(cot r)u

u

(coth r)u

(tanhr)w

A Mn

(2cot2r)?7

27,

(2 coth 2r)r,

(2coth2r)r,

T h e y are the only examples of real hypersurfaces in IK-projective and hyperbolic spaces which have only two principal curvatures. 3

L e n g t h s p e c t r u m from qualitative viewpoint

A smooth curve a : R —> M on a complete Riemannian manifold M which is parametrized by its arclength is called closed if there exists a non-zero constant T with a{t) = a{t + T) for all t. T h e minimum positive T with this property is called length of a and is denoted by length(cr). For an open curve a, a curve which is not closed, we p u t length(cr) = oo. It might be more usual to use the terminology "length spectrum" for the meaning of the set of length of all closed geodesies. Here consider a standard sphere of radius 1. There are infinitely many closed geodesies of length 2ir and there are no other geodesies. Of course, this shows t h a t a standard sphere is highly symmetric, but still lose some information. In order t o get rid of the influence of the group of isometries, we consider congruency classes of geodesies. Two smooth curve <7i,<72 on M parametrized by their arclength are said to be congruent if there exist an isometry ip of M and a constant to satisfying either (0, oo] defined by CM{[I}) = length(7), where [7] denotes the congruency class containing a geodesic 7. We also call its image Lspec(Af) = £A/(Geod(A/)) n R on the real line length spectrum of M. For a compact symmetric space of rank one, all geodesies are closed and congruent each other. Therefore length spectrum of a standard sphere of radius 1, of a real projective space R P r a ( l ) of constant sectional curvature 1 and of a IK-projective space K P " ( 4 ) are as follows: L s p e c ( 5 n ( l ) ) = {2TT},

L s p e c ( R P " ( l ) ) = Lspec(KP™(4)) = {TT}.

We are hence interested in length spectrum of non-symmetric homogeneous Riemannian manifolds. In this paper, we restrict ourselves to geodesic spheres

96

and tubes around hyperplanes which are nice classical objects as submanifolds in rank one symmetric spaces. In the first place, we study set theoretical properties of the length spectrum of our homogeneous manifolds. Proposition 1. (1) The length spectrum of a geodesic sphere in a Kprojective space K P n is a discrete unbounded subset of R. (2) The length spectrum of a geodesic sphere and a tube around Khyperplane in a K-hyperbolic space KHn are also discrete unbounded subsets of R. For readers' good guide, we here give the length spectrum of a flat torus T2 = [ 0 , l ] x [ 0 , A ] / ~ ( A > 1 ) : 2 Lspec(T ) = (I VP2 + Q2A2 v

P

Q&

^ positive Z ^ integers f ^ ]j IWJ{l, prime i ' A}. >

It is also a discrete unbounded subset of R. Our objects are quite similar to this set. 4

Length spectrum from quantitative viewpoint

Next we study quantitative properties of length spectrum of geodesic spheres and tubes around hyperplanes in rank one symmetric spaces. We are interested in the following problems: i) Whether we can determine a congruency class of geodesies only by their length spectrum or not. ii) How many congruency classes of geodesies are there? To attack these problems we introduce the notion of multiplicity. For nonnegative A we set TOM(A)

= (the cardinality of the set

£M(A)~

).

We call this function rtiM • [0, oo) —• N u {oo} multiplicity of length spectrum. When the multiplicity JUMW of A is 1, we call A simple. For example, the length spectrum 2-n of a standard sphere is simple. The condition that a length spectrum A is simple means that we can determine the congruency class with length A. So when all the length spectrums are simple, we can classify congruency classes only by their length spectrum. For multiplicity of length spectrum of our manifolds, we find the feature depends on the rationality of the value of a function of radius.

97 Theorem 1. The multiplicity of length spectrum of geodesic sphere M — Ms(2r/^/c;c) of radius IrJ^fc in a ]K-projective space KPn(c) of maximum sectional curvature c satisfies the following properties. (1) When coth r is irrational, all length spectrums are simple. (2) / / coth 2 r is rational, then i) 771M(A) < oo for each A, ii) it is not uniformly finite; limsup^^^?/^/!^) = oo, iii) its growth order is not so rapid; lim^oo A_*rriA/(A) = 0 for all 8 > 0. Theorem 2. Let M denote either a geodesic sphere Ms(2r/y/c; -c) of radius 2r/y/c or a tube Mt(2r/y/c; —c) of radius 2r/y / c around K-hyperplane in a ^-hyperbolic space WLHn(—c) of minimum sectional curvature —c. Its length spectrum satisfies the following properties. (1) 7/coth 2 r is rational, then i)TOM(A) < oo for each A, ii) it is not uniformly finite; l i m s u p ^ ^

TOM(A)

= oo,

(5

iii) its growth order is not so rapid; lirn^oo A~ TOM(A) — 0 for all S > 0. (2) When coth r is a irrational number of the form either ^{p2

+ q2 - 1 - v V ~(q-

1 ) W ~ (q + I) 2 } }

for some relatively prime positive p, q with even pq and p > q + 3 or J _ {p2 + q2 _ 4 _ y { p 2 _{q_

2)2}{p2

_

{q

+ 2j2j }

for some relatively prime positive p, q with oddpq andp > g + 4, only the multiplicity of (2ir/^/e) sinh 2r is two and other spectrums are simple. (3) When coth r is irrational and is not the case of (2), all length spectrums are simple. Next we study the asymptotic behavior of the number of length spectrum. We set for a nonnegative A ?ijv/(A) = (the cardinality of the set { [7] e Geod(M) |

£M([I])

< A }).

It is well-known that for a compact Riemannian manifold N of negative curvature riiv(A) ~ (ft;vA)_1exp(/i/vA), where hu denotes the topological entropy of the geodesic flow on the unit tangent bundle and for two functions / , g : M —> R we denote by / ~ g if they satisfy l i m ^ o o f/g = 1-

98 T h e o r e m 3 . (1) For a geodesic sphere M = Ms(2r/y/c;c) in a ^.-projective space K P " ( c ) we have nM(X)

~

of radius

2r/y/c

. A2. 7TJ sin 2r

(2) Le£ M denote either a geodesic sphere Ms{2r/y/c; —c) of radius 2r/<Jc or a tube Mt(2r/y/c; —c) of radius 2r/'sfc. in a K-hyperbolic space KHn(—c). We then have n

I\\

M(\)

3 c r

~

2

1 . , - A . 7r* sinhi?'

4

It might be a great help for readers to make mention of quantitative properties of length spectrum of a flat torus T2 = [0,1] x [0, A ] / ~ (A > 1). Applying our argument we can obtain t h e following. (1) When A 2 is irrational, all length spectrums are simple. (2) If A 2 is rational, then i) mq^(X)

< co for each A,

ii) limsup A _ > 0 0 m ^ A ) = oo, but l i r n \ _ 0 0 X~5mT2(X) (3) For t h e asymptotic behavior we have nT>(\)

~

= 0 for all S > 0. 2

j^X -

In t h e following sections we give precise representations of length of all closed geodesies on geodesic spheres and tubes around hyperplane in Kprojective and hyperbolic spaces. 5

Structure torsion of geodesies

Let M be either one of t h e following real hypersurfaces; a geodesic sphere in KPn(c), a geodesic sphere, a horosphere or a t u b e around KH71'1 in n KH (—c). In order to study geodesies on these hypersurfaces we introduce an invariant in this section. For a geodesic 7 on M we define its (fiber) structure torsion r 7 by r 7 = | | P r o j p i (7(1)) ||, where P r o j p i : TM —> V1 denotes t h e projection. One can easily see t h a t it is constant along 7 in t h e following way. In a complex projective space C P " ( c ) and in a complex hyperbolic space w tn CHn(—c), we have r 7 = \(i{t),0\ i t h e characteristic vector field £ = — JJ\FM on M, where J denotes t h e complex structure on CPn(c) or CHn(—c), We define t h e structure tensor
= noJ\rM with t h e canonical projection

99 ix : TCPn(c)\M —> TM or n : TCHn(-c)\A1 formula we have the following:

| ( 7 ( * ) , 0 = (i(t)^AMi{t))

—> TM. By using Weingarten

= -(
=

-{AAI^(t),i(t))-

Since the shape operator AM and the structure tensor
o AM, this computation shows that the structure torsion r 7 is constant along 7 in this case. In a quaternionic projective space i P " ( c ) and in a quaternionic hyperbolic space MHn(—c), we choose a local basis {J\, J2, J3} of the quaternionic structure on these spaces. Denoting by V the Riemannian connection of these spaces we have local 1-forms qi (i = 1, 2, 3) satisfying V Y Ji = q i + 2(^)./,+i - ^ + i(X).7 i + 2

(i mod 3)

for every local vector filed X and i = 1, 2, 3. We define local characteristic vector fields on M by 6 = — JjA/jw and local structure tensors on M by 4>i = % o Ji\TM- As we have

r7 = y/mUi)2

+ W)&)2

+ (7«,&> 2 .

we see l_d_

2

2dT 7

(7,93(7)6 - 9 2 ( 7 ) 6 + (7,9I(7)6

+0I-4A/7)(7.6)

-93(7)6

+ 2^M7)(7,6)

+ (7,92(7)6 ~ 9i(7)6 + U/7)(7,6) (7, ^ 1 ^ / 7 } (7,6) + ( 7 , ^ 2 ^ / 7 ) (7,6) + ( 7 , ^ 3 ^ / 7 } (7,6) and T 7 is constant because fa o AM = ^4j\/ ° ?: (i = 1, 2, 3). In order to show that the structure torsion is constant for the Cayley case, we reduce our case to the complex case. Let M be either a geodesic sphere in a Cayley projective plane OP 2 (c), a geodesic sphere, a horosphere or a tube around OH1 in a Cayley hyperbolic plane OH'2(—c). By the representations in section 2 and by Lemma 1 in the next section, for a unit tangent vector v 6 TM, we have a totally geodesic complex projective plane CP 2 (c) in OP2(c) or a totally geodesic complex hyperbolic plane CH2(—c) in OH2(—c) with the following properties:

100

i) v is tangent to this totally geodesic plane. ii) M n CP 2 (c) or M n CH2(—c) is a corresponding geodesic sphere, tube around hyperplane or horosphere in CP 2 (c) or
Congruency of geodesies

In this section we shall show that two geodesies on our manifold M are congruent if and only if they have the same structure torsion. First we show the "if" part. If two geodesies 71, 72 have the same structure torsion T, their initial vectors are of the form 7i(0) = Vl — T2 U{ + rr/i with unit tangent vectors «,; € T> and r\i G T>L (i = 1, 2). The following lemma guarantees the existence of an isometry Lp of M with dcp7l(o)(7i(0)) = 72(0). Thus 71 and 72 are congruent. Lemma 1. (1) Let M be a geodesic sphere in K P n . For any tangent vectors Ui +r]i S TX.M = VXi ®T>£. ( c TX:KPn) with \\ui\\ = ||r?,:|| = 1 at arbitrary points Xi (i = 1, 2) on M, we have an isometry (p ofKPn such that i)
= u2 and dipXl(r)i) = r)2,

iii) d
101 Here, for v = Yll=o aiei e ® ( a * e R ) w e s e t ^ = a ° e ° ~ S,:=i a J e « a n d Re(i/) = ao- We usually identify a real number ao with aoeo- At a point w G Mt{r),

t h e tangent space of Mt(r)

G {w} x O 3 | Re(-uJ 0 -u 0 + Wivi)

Tt„Mt(r) = { (w,v) Let Mw G (TwMt{r)) J\fM{w(w))

= R e ^ i ^ ) = 0}.

(~l W w denote the horizontal lift of t h e normal vector

at a point w(w), in TwHf3.

of TwMt(r)

is represented as

where (TwMt(r))

is the orthogonal complement

At a point w = (WQ, u / i , s i n h r ) we see t h a t Mw

=

(w, (iuo t a n h r , w\ t a n h r , c o s h r ) ) , hence we find t h e horizontal lifts of Z>TO(,„) and 2 ? i , > are of the following forms; G {w} x O 3 | -w0v0

T>w = { {(w,(v0,Vl,0))

+ w1vl

=0},

V^ = { (tu, (io 0 t a n h r, wi t a n h r, cosh r) • v) G {w} x O 3 | Re(i/) = 0 }. P u t 2 = ( c o s h r , 0 , s i n h r ) (G Mt(r)). ((if, (wo,ui,0)) G !?,„, we find t h e m a t r i x

For each unit tangent

vector

^ u ; o ( c o s h r ) _ 1 VQ 0 \ I U I ( c o s h r ) " 1 vi 0 G M a t ( 3 , 0 ) 0 0 1/ induces an isometry 0 of Hf3 i) ip(z) = to and tp(Mt(r))

such t h a t = Mi(r),

i i ) d ^ ( ( z , ( 0 , l , 0 ) ) ) = (u;,(^,,T;i ) 0)), iii) dipw>(Dw>) = V^{wl) and dipw>(Sfw, • v) = A ^ ( w / ) • v for every jy G O at an arbitrary point w' G

Mt(r).

Next for each /x = ]Cj=i ^ e « / I

e

® w-ith |/J| = 1, we consider a matrix

0 G Mat(24,

0

(b-i * b2 *

\

A

GO(7), \b7 *

^/

*/

where A is an orthogonal matrix and Is G G L ( 8 , K ) denotes the identity matrix. This induces an isometry ty of Hf3 such t h a t i) &{z) = z and !P(M t (r)) =

Mt{r),

ii) d!? z ((z, ( 0 , 1 , 0 ) ) ) = (z, ( 0 , 1 , 0 ) ) and d$z(Nz

• ei) = A4 • /x,

102 iii) G W V ( I V ) = T>j,(w>) and d\Pw>(Afw,) = Af,p(wl)

at an arbitrary point

w' G Mt(r). We therefore obtain desirable isometry (p of OH2.

O

We now show the "only if" part. Let i denote an isometric embedding of M into K P n ( 4 ) or KHn(~4). We study the shape of /. o 7 for each geodesic 7 on M and show t h a t it depends on the structure torsion T 7 of 7. As t is equivariant, t h a t is for each isometry if of M there exists an isometry (p of KPn or KHn with p i = 10
= Ki-iXi-i{t)

+KiXi+1(t),

X\ = &, X0 = Xm+i

= 0,

i =

l,...,rn,

with a field of orthonormal frame fields {Xi,..., Xm} and positive constants Kj. A curve is called circle if it is a helix of order 2 (i.e. of proper order 1 or 2). T h e constants K$ are called the geodesic: curvatures of a. By Lemma 1 we are enough to study the complex case. For a helix of proper order m satisfying (H) on a Kahler manifold with complex structure J, we define its complex torsions by r ^ = (Xj.,JXj), 1 < i < j < m. By use of GauB and Weingarten formulae and by the table of principal curvatures in section 2, we obtain the following result on the shape of 1 0 7 . L e m m a 2. (1) For a geodesic 7 on Ms(r;4), 0 < r < ir/2 in KPn(4), shape of t o 7 as a curve in K P " ( 4 ) is as follows. l a ) When n/4
the

ir/2 and r 7 = c o t r , the curve 1 0 7 is a geodesic.

l b ) When r ^ 7r/4 and T1 = 1, the curve I.OJ is a circle of geodesic curvature 2| c o t 2 r j . In the case K = C, its complex torsion is T\I = ± 1 . lc) When r 7 = 0, the curve 1 0 7 is circle of geodesic curvature c o t r . In the case ofK = C, its complex torsion is T12 = 0. Id) When 0 < r 7 < 1 and r 7 ^ cotr, the curve t o 7 is a helix of proper order 4 whose geodesic curvatures are described as tan7'|, KI = Il cc oo tt rr -—Tr1 tan7'|,

K2== TT-y\/l —, 7T2 t a n r , K2 1^^

In the case K = C, its complex torsions

K3 = c o t r .

are

T 1 2 = ± T 7 , TU = ± v / l - T^, T23 = ± A / 1 - T"2, T34 = ± T 7 , T 1 3 = T24 = 0.

103

(2) LetM denote either Ms(r; -4), Mh(-4) or M,.(r; - 4 ) inKHn(-4). For a geodesic 7 on M the shape 0 / 1 0 7 as a curve, in KHn(—4) is as follows. 2a) When T 7 = 1, the curve t o ^ is a circle whose geodesic curvature is 2 if M = Mh{-4) and is 2coth2r if M = Ms(r;~4) orMf(r;-4). In the case K = C, its complex torsion is T\% = ± 1 2b) When T 7 = 0, the curve 107 is a circle whose geodesic curvature is cothr if M - Ms(r; -4), is 1 if M = h(-4), and is t a n h r if M = Mt(r; - 4 ) . In the case K = C, its complex torsion is ryi = 0. 2c) When 0 < T 7 < 1, the. curve t o 7 is a helix of proper order 4 whose geodesic curvatures are as in the following table. Mh(-4) Kl

1+T72

«2

Tyy/1-^

^3

1

Mt(r; - 4 )

Ms(r;-4) coth r + T^ tanh r

tanh r + T% coth r

T 7 wl — r 2 t a n h r TjJl

— T'* cothr tanh r

cothr

In the case K = C, its complex torsions are 1~12 = - T 3 4 = ± T 7 ,

T14 = - T 2 3 = ±\Jl

~ T2,

T13 = T 2 4 = 0.

Here, the signature of complex torsions depend on the signatures of cot r — r 7 t a n r and (7, £) (see [AMY] for more detail). This lemma guarantees that two geodesies on our manifold are not congruent if they do not have the same structure torsion. It is known that on CPn and on CHn every helix with constant complex torsions is generated by some Killing vector field on these spaces(see [MO]). Therefore Lemma 2 also guarantees the following. Corollary. Every geodesic is a simple curve on a geodesic sphere in KP" and on a geodesic sphere, a tube, around KHn~l and horosphere in KHn. 7

Sketch of Proofs

In order to investigate length of closed geodesies on our manifold, we use the fiber structure of K-projective and hyperbolic spaces. By Lemma 1 we are enough to study the complex case. For a geodesic 7 on a geodesic sphere Ms(r;4) in CP n (4), we consider a horizontal lift 7 on a standard sphere 5 2 r l + 1 ( l ) of the curve 107 on CP"(4).

104

Regarding 7 as a curve in C n + 1 we find by Lemma 2(1) that when r 7 ^ cotr (71-/4 < r < 7r/2) it satisfies the following ordinary differential equation 7" ± 2 v / r T I cot 2r|-y' + 7 = 0, 4

7<

2

> + ((smr)" -r

2

2

+ r tan

T-y = 2

r)f'

+ r 2 ( ( c o s r ) - 2 - r 2 t a n 2 r ) 7 = 0,

1,

0 < r 7 < 1,

hence is of the form 'exp(±vc:lt|cot2r|) {Aexp(Vr^lt(sin2ry1)

x

+ B 7(«)

J 4exp(v

/

if

T7

= 1,

1

exp(-\^t{sin2r)~ )},

^TtT 7 tanr) + Bexp( — \/—Hr 7 tanr) tyj'(sinr)~2

+ Cexp [y^l

- r2)

+ £>exp ( - V ^ I ^ |sinr)

2

—r

if 0 < r 7 < 1, 2

•

with some non-zero vectors A, B, C, D s C n + 1 . Here, in the case r 7 — 1, the signature depends on the sigunature of complex torsions of t o 7. Since 1 0 7 is a simple curve, we can easily obtain the following. Proposition 2. Let 7 be. a geodesic on a geodesic sphere Af s (r;4) of radius r (0 < r < TT/2) in KPn{4). (1) W/ien r 7 = 1, it is closed with length 7rsin2?\ (2) When T 7 = 0, it is also closed and its length is 27rsinr. (3) When 0 < T 7 < 1, it is closed if and only if r

t

=

q ,

sin r \Jp2 tan 2 r + q2

with some relatively prime positive integers p and q with q < p t a n 2 r . In this case, its length is length (7)

27rvp 2 sin r + q2 cos2 r,

if pq is even,

TT\/p2 sin2 r + q2 cos2 r,

if pq is odd.

This proposition assures that the length spectrum of a geodesic sphere Af s (2r/Vc;c) of radius 2rjsfc (0 < r < TT/2) in KP n (c) is of the following

105

form; 2r

Lspec I M.

2TT

••'))-{

U{^ sinr }

sin 2r

I ) < 47r i / - (p2 sin 2 r + g 2 cos 2 r)

p and g are relatively prime 1 positive integers which satisfy > pq is even and q < p t a n 2 r J

M < 2TTJ-(P2

p and
sin

r + g2cos2r)

For example, the length spectrum of a geodesic sphere Ms(ir/4;4) TT/4 in KPn(4) is as follows:

of radius

Lspec (A/,, (J; 4^ = {TT, V/2TT, V ^ T T , V/TOTT, \ / 1 3 7 T , V ^ T T , 5 T T , \/26 7r, V ^ T T , \/34 7r, V ^ T T ,

VHn,

J50ir,

V537T, V ^ T T , \ / 6 1 TT, V ^ T T , V ^ T T , V ^ T T , \/82 7r, \/85 7r,

%/89TT, V ^ T T , V/IOITT, VTOGTT, V H W T T , vTlSlTr, \ / l 2 2

7r, S V ^ T T , V ^ O T T , . . . } .

Here TT and \ / 2 7r are the length of geodesies with structure torsion 1 and 0 respectively, \/657r is the first double spectrum which is the common length of geodesies of structure torsion 3 / \ / 6 5 and 7 / \ / 6 5 corresponding (p,q) = (11, 3) and (9,7), and JISETT, JT30TT are the second and the third double length spectrums corresponding (p,q) = (11,7), (13,1) and (p,q) — (7,4), (8,1). By the expression of t h e length spectrum we find there exists a natural mapping of the "union" of length spectrum of geodesic spheres of radius 2r and (n — 2 r ) / v / c 2r Lspec I Ms I —7= ; c

\{2~sin 2?' :

onto t h e length spectrum of a flat torus with A = c o t r : It is given by

{

x{(2ir/y/c) s i n r } - 1 ,

for length spectrum corresponding odd pq,

x{(47r/y / c) s i n r } - 1 ,

otherwise.

It is a 2-to-l correspondence except for (4-7r/ JC) sin r and (4ir/J~c) cos r. This is why the first length spectrum (2n/y/c) sin 2r has an interesting property.

106

Next we consider geodesies on a geodesic sphere in KHn. Let 7 be a geodesic on M s (r;—4) in C i / n ( - 4 ) . For the curve [ 0 7 on
if r 7 = 1, 1

+ £exp(-v H(sinh2r)- )}, 7«

^4exp(\/--TiT 7 tanhr) + Bexp(~ v / —Ttr 7 tanhr'

+ Cexp (V^ltJ(smhr)-'2

+ T2) 2

+ Dexp (-y/^ltJ(smhr)-'

if 0 <

T7

< 1,

+ T'}\ ,

with some non-zero vectors A, B, C, D e C r a + 1 . Here, in the case r 7 = 1, the signature depends on the sigunature of the complex torsion of 1. o 7. Thus we can easily obtain the following. Proposition 3. Letj be a geodesic on a geodesic sphere Ms(r; —4) of radius r ( 0 < r
0, ?i is also closed and its length is 27rsinh?\

(3) When 0 < r 7 < 1, it is closed if arid, only if

sinh r \/p2 tanh r — q2 with some relatively prime positive integers p and q with q < ptanh r. In this case, its length is

length(7)

27r\/p 2 sinh" ?'— q'2 cosh r,

if pq is even,

r\/p 2 sinh r — g 2 cosh r,

«/p<7 is odd.

This proposition assures that the length spectrum of a geodesic sphere Ms(2r/s/c; —c) of radius 2 r / v / c in K i l n ( —c) is of the following form;

107

Lspec

u

M,(—;-(

{^sk,l„jU{^»mh2-j

47TA / - ( p 2 sinh r — q2 cosh ?•)

U 2.

- ( p 2 sinh r — q2 cosh r ' c

p a n d q are relatively prime positive integers which satisfy pq is even a n d q < p t a n h r p and q are relatively prime "j positive integers which satisfy ^ 2 pg is odd and q < p t a n h r- I

For example, t h e length spectrum of a geodesic sphere M s ( l o g ( l + A/2); —4) of radius log(l + y/2) in KHn(-4) is as follows: Lspec(M,(log(l + v/2);-4)) = {2TT, V^TT, 2V2 n, A/23TT, A / 3 1 TT, A/47TT, 2 \ / l 4 7 r , 2V / T7TT, A/TITI-, A / 7 9 7T, A / 1 0 3 TT, \ A 1 9 T T , A/1277T, 2V / 34TT, A / 1 5 1 TT, 2 \ / 4 1 T T , \ / l 6 7 7 r , 2 \ / 4 6 7r, \/T9l7r, 2\/497r, \Z199TT, A / 2 2 3 TT, A/239TT, 2A/62TT, A / 2 6 3 TT, A / 2 7 1 TT, A / 2 8 7 TT, . . . } .

Here 2-zr a n d 2A/2-7T are t h e length of geodesies with structure torsion 0 and 1 respectively, A/1197T is t h e first double length spectrum which is t h e length of geodesies of structure torsion 5-^/2/119 and A / 2 / 1 1 9 corresponding iPil) = (13,5) and (11,1), a n d A / 2 8 7 TT is t h e second double length spectrum corresponding (p, q) = (17,1), (23,11). Finally we study geodesies on tubes around K-hyperplanes in KHn. We call a smooth curve a on ~KHn unbounded in both directions if b o t h c([0, oo)) and CT((—oo, 0]) are unbounded sets. Considering t h e ideal boundary 8KHn of ~KHn as a H a d a m a r d manifold, we can define its limit points cr(oo) = lim a(t), t—>oo

cr(-oo) =

lira a(t) e

dKHn

f—» — o o

at infinity if they exist. We shall call a smooth curve a on KHn horocyclic if the following conditions hold. i) It has single point at infinity; CT(OO) = cr(--oo). ii) If a geodesic p on KHn with p(oo) = cr(oo) crosses a , then they cross orthogonally at their crossing point in KHn. These conditions are equivalent t o the condition t h a t a is unbounded in both directions a n d lies on a horosphere.

108

Let 7 be a geodesic on Mt(r; —4) in CHn(—4). For the curve i o 7 on Ci/ (—4) we consider its horizontal lift 7 on H^n+1. Regarding this as a curve in C r a + 1 we find by Lemma 2(2) that n

Aexp(\/—ltT-y + 5exp(-

cothr) 1 tTj coth r)

+ Cexp (tJ(coshr)-2

if 0 < r 7 < 1/coshr,

- T'A

+ Dexp ( — tJ(coshr)~2

— rij

,4 + Bt + Cexp( v ^ f / s i n h r ) ,

m

if r 7 = 1/coshr,

Aexp(\/—I
if 1/coshr < r 7 < 1,

- (coshr)" 2 )

+ £>exp f — v / ^ i!Wr 2 - (coshr) exp(±v / z Ttcoth2r) x |^exp(v/=:lt(sinh2r)"1)

if T 7 = 1,

+ Bexp(-v/::lt(sinh2r)"1)|, with some non-zero vectors A, B, C, D G C™+1. following.

Therefore we find the

Proposition 4. Let 7 be a geodesic on a tube Mt(r; —4) of radius r (0 < r < 00) around a hyperplane KHn~x in KHn(—4). (1) When Tj — 1, it is closed with length 7rsinh2r. (2) When r 7 < 1/coshr, it is unbounded in both directions, and has two distinct limit points as a curve on KHn. (3) When r 7 = 1/coshr, it is horocyclic as a curve on K / / n . (4) When 1/coshr < T 7 < 1, it is bounded. Under this situation, it is closed if and only if 7"

V

=z

— . 2

cosh r\/p'

—

— q2 coth r

with some relatively prime positive integers p and q with q < ptanh r.

109 In this case, its length is length(7)

M

2ir\/p2 sinh 2 r — q2 cosh r,

if pq is even,

7rvp 2 sinh r — g2 cosh r,

if pq is odd.

We obtain by this proposition that the length spectrum of a tube Mt{2r/y/c;-c) of radius 2r/'-Jc around a hyperplane KHn~l in KHn(-c) is of the following form; Lspec ( Mt [ — ; —c

{^Sinh2r}

M < 4.n \ - (p2 sinh r — q2 cosh

p and q are relatively prime "j positive integers which satisfy > pq is even and q < ptanh r)

M < 2n \ - (p2 sinh2 r — q2 cosh2

p and q are relatively prime ^ positive integers which satisfy > . pq is odd and q < ptanh r)

In particular, we find that 2r Lspec ( Ms ( —=; — c ) ) = Lspec ( M t

^~C))U{^Sinhr}-

For example, the length spectrum of tubes Af t (^log(2 + \/3); — 4) and M t (log(v/2 + N/3); - 4 ) in KH" n (-4) are as follows: Lspec \Mt Q log(2 + \/3); - 4 J j = {V^TT, VTl 7T, \/23TT, V2Qir, V39n, V47TT, %/59TT, V66TT, VTITT, Vmir,

Vfin,

A/107TT, VTTTTT, V T ^ T T , \ / l 3 1 TT, \/138TT, \/T43TT, \/l46 7 r , . . . } ,

Lspec(Af t (log(V2 + v % - 4 ) =

{\^T:,2VEw,V2^7r,2V6Tr,y/^n,Vn^,V8fTT,y/95TT,2V297r, 2\/387r, \/l597r, \/167TT, \/T91 TT, 2\/50TT, 2\/53 7r, \/215 7r, \/2397r, \/2637T,2\/69TT, \/303 7r, \/3lTTT, \/335TT,2^86TT, \ / 3 5 9 T T , . . . } .

Here, in the former case, \/3 TC is the length of a geodesic of structure torsion 1 and A/143 IT is the first double length spectrum which is the common length of geodesies with structure torsion 19/\/429 andl7/\/429 corresponding (p, q) = (19,5) and (17,1). In the latter case, 2\/6TT is the length of a geodesic of

110

structure torsion 1, v95 7r is the first double length spectrum which is the common length of geodesies with structure torsion 7 -^2/285 and i 1 \/2/285 corresponding (p,q) = (7,1) and (11,7), and \/215 7r, \/335 7r are the second and the third double length spectrums corresponding (p,q) = (11,3), (17,11) and (p,q) = (13,1), (17,9). Proposition 1 is a direct consequence of these propositions. We can obtain our Theorems 1, 2 and 3 by these propositions and some consideration on integers (see [AMY] for more detail). We here give the first and the second length spectrums of geodesic spheres and tubes around hyperplanes. Proposition 5. (1) The first and the second length spectrum.s of Ms(2r/y/c;c) inKPn(c), Ms{2r/y/c;-c) and Mt(2r/s/c; -c) in K P n ( - c ) are simple. (2) For Ms(2r / y/c; c), the first length spectrum is (2it/y/c)s'm2r, which is the length of geodesies of structure torsion 1. The second length spectrum is

{

(4n/y/c) sin?',

if 0 < r < 7r/4,

2n/y/c,

ifn/4
< TT/2,

which is the length of geodesies with structure torsion 0 and cotr, respectively. (3) For Ms{2r Jsfc\ —c), the first length spectrum is (Ait/yfc) sinh. r, which. is the length of geodesies of null structure torsion. The second length spectrum is 2 7 i y ( 8 s i n h 2 r - l)/c, (2TT/VC)

sinh 2r,

if r > (1/2) log(2 + \/3), if r < (1/2) log(2 +

A/3)

,

which is the length of geodesies with structure torsion l / ( s i n h r \ / 9 tanh r — l) and 1, respectively. (4) For M t (2r/y / c; —c), the first length spectrum is 27r^/(8sinh 2 r - l)/c,

if r > (l/2)log(2 + s/3),

(2ir/y/c) sinh 2r,

if r < (1/2) log(2 + %/3),

which is the length of geodesies with structure torsion. 3/(coshr\/9 — coth r)

111

and 1, respectively. The second length spectrum, is ' 471-^(3 s i n h 2 r - l ) / c

if r > log(l + N/2),

< (27r/v^) sinh2r, 2itJ{Am(m

«/ (1/2) log(2 + \/3) < r < log(l + \/2),

+ 1) sinh 2 r - l}/c,

where rn = 2, 3, . . . .

z/ \/2»ii - 1 < cothr < y/2m + 1,

It is the length of geodesies of structure

2

2/(coshr\/4 - coth r ) , 1 and (2m + l)/(coshrJ(2m

2

torsion 2

+ l ) - coth r ) , re-

At the last stage we make mention of geodesies on a horosphere in a K-hyperbolic space. For a geodesic 7 on A//,.(—4) in CHn(—4), we find a horizontal lift 7 on JH"12n+1 of the curve 1. o 7 on C i f " ( - 4 ) satisfies j 7 ( 4 ) + 2 r 2 7 " + r47 = 0, \7"±2v

C:

/

T 7 - 7 = 0,

if 0 < r T < 1, i f r 7 = l,

by Lemma 2(2). Hence it is of the form 'A + Bt + Ct2, j(t) = I (A + Bt)exp(y^ltr7) >exp(±v

/r

Ti)(^ +

+ (C + Dt) exp(-x/^TtT 7 ), Bi

)>

with some non-zero vectors A, B, C, D e C" following.

ifr7=0, if 0 < r 7 < 1, ifr 7 = l,

+1

. Therefore we obtain the

Proposition 6. There are infinitely many congruency classes of geodesies on a horosphere A//l(—c) in KiJ ra (—c). Every geodesic on this horosphere is unbounded in both directions, hence is horocyclic as a curve on KH™. Acknowledgment This research is partially supported by Grant-in-Aid for Scientific Research(C) (No. 11640073), The Ministry of Education, Science, Sports and Culture. References [AMY] T. Adachi, S. Maeda and M. Yamagishi, Length spectrum of geodesic spheres in a non-flat complex space form, to appear in J. Math. Soc. Japan 54.

112

[AM] T. Adachi and S. Maeda, Curvature-adapted real hypersurfaces in quaternionic space forms, Kodai Math. J. 24(2001), 98-119. [K] W. Klingenberg, Contributions to Riemannian geometry in the large, Ann. of Math. 69(1961), 645-666. [MO] S. Maeda and Y. Ohnita, Helical geodesic immersions into complex space forms, Geom. Dedicata 30(1983), 93-114. [W] A. Weinstein, Distance spheres in complex projective spaces, Proc. A.M.S. 39(1973), 649-650.

113

G R A S S M A N N G E O M E T R Y OF 6-DIMENSIONAL S P H E R E , II HIDEYA HASHIMOTO Nippon Institute of Technology, Miyashiro, Saitama 345-850J, Japan E-mail: [email protected] KATSUYA MASHIMO Department of Mathematics, Tokyo University of Agriculture and Technology, Koganei, Tokyo 184-8588, Japan E-mail: [email protected]

Introduction Let (M, g) be a Riemaimian manifold. We denote by GP{TXM) the Grassmann manifold of all oriented p-dimensional linear subspaces of the tangent space TXM of M at x G M and by GP{TM) the Grassmann bundle UX€MGP{TXM). Let E be a subbundle of GP(TM). A p-dimensional submanifold iV of M is called a E-submanifold if TXN G E holds for any x G N. If a Lie group G acts transitively on M, the action is naturally extended to the action of G on GP{TM). It seems to be an interesiting problem to study E-submanifold for an orbit E c GP(TM). Let J be the standard almost complex structure of the 6-dimensional sphere S6 and (,} the standard Riemannian metric. It is well-known that the group of automorphisms of ( 5 6 , J, (,)) is isomorphic to the compact exceptional simple Lie group G
Preliminaries Cayley algebra

Let H be the skew field of all quaternions. The Cayley algebra £ over R is € = H © H with the following multiplication: (q,r) • (s, t) - (qs - tr,tq + rs),

q,r, s . f e H

114

where " " means the conjugation in H. We define a conjugation in £ by (q,r) = (q, —r),q,r € H, and an inner product (,) by {x,y} = (x-y + y -x)/2,

x,ye£.

We use the canonical orthonormal basis E0 = (1,0), E\ = (i,0), E2 = (j, 0), E3 = (A;,0), E4 = (0,1), E5 = (0,i), E6 = (0, j ) , £ 7 = (0,fc), which satsifies EXE2 = E3, E1E4=E5,

E^^Ee,

E2E5 = E7,

E2E4 = EQ, E3E4 = E7, E:iEct = E5. We put £0 = {'« G £ :

M+ M

= 0}.

We identify Co = JZi=] R-E't with the set of all 7-dimensional column vectors in a natural manner and consider G2 as a subgroup of SO(7). Lemma 1.1. For a pair of mutually orthoyonal unit vectors a\. 04 in £0 put as = ai • 04. Tafce a «m( vector a2, which is perpendicular to 04, «i anrf as. / / we put, a-i = a\ • a2, a^ = a2 • «4 and a 7 = a$ • a\ then the matrix g = (a-i, a2, a 3 , a 4 , a 5 , a 6 , a 7 ) G <SO(7) zs an element of G2. The unit sphere S6 C Co centered at the origin has an almost complex structure J defined by JP(X) =p-X,

pes6,

X e TPS6.

Let G%j (1 < i ^ j < 7) be a skew-symmetric transformation on £ defined by Gij(Ek) = SjkEi - StkEj. The Lie algebra Q2 of G2 is spanned by the following vectors in the Lie algebra so(7) of 50(7): ' aG2i + bG4s + c.G76, aG31 + bG4G + CG57, aGi2 + 6G47 + cG 6 s, < 0G51 + 6G73 + cG62, aGi4 + 6G72 + CG36, a G i 7 + 6G24 + cG 53 , k aGei

+ bG34 + cG25,

where a, b, c are real numbers with a + b + c = 0.

115

1.2

Structure equation

Define a frame (e,E,E) = ( £ , E i , E 2 , E 3 , E i , E 2 , E 3 ) of C & £o by e = E4 = (0,1) G H © H , = iN,

E2 = jN,

E 3 = -fcN,

E i = iN,

E2 = jN,

E 3 = -fcN,

EI

where N = (1 - J=\e)/2, N = (1 + y/^e)^ £ C £. A frame ( u , / , 7 ) of C (g> Co is said to be an admissible frame, if there exists an element g of G2 such that (u,f,f) = (e,E,E)g. We identify an element of Gi with an admissible frame by the injection t

: G 2 ^ G L ( 7 , C ) ; g ~ (e,E,E)g.

We denote by Af pX9 (C) the set of p x q complex matrices. Let [a] be the element given by

(

0 a3 -a2\ -a3 0 oi e M 3 x 3 (C) a2 - a i 0 /

for a = *(aj a 2 a 3 ) e M 3 x i ( C ) . Then we have [a]6+[6]a = 0, where a, 6 e Af 3 x i(C). Proposition 1.1 (cf. Bryant [1]). form of GL(7, C) is of the form

The pull-back $ of the Maurer-Cartan

o -4^1 fd yf=i le\ $ = -2^=T 6 K [0] , V 2^/^T6> [0] K / (

where K = ( « / ) (1 < i,j < 3) (resp. 6 = t(61,92,63)) is an su(S)-valued (resp. M3xi(C)-valued) left invariant 1-forms. The Maurer-Cartan equation d$ = - $ A $ reduces to d6 = - K A 0 + [0] A 8, dn= - K A K + 30 A * 0 - ('0 A 0) J 3 . The Maurer-Cartan form $ of G 2 C SO(7) is

116

where P is the transformation matrix (e,E,:E) = 2

(EuE2,---,E7)P.

J-holomorphic curves

Let f be an oriented 2-plane of TpS6. The Kahler angle K(£) of £ is cos^ 1 (J(ei),e 2 ) where ei, e 2 is a oriented orthonormal base of £. The orbit decomposition of the Grassmann bundle G2(TS6) is 1)O<0<XT,Q^ where we put

42) = {{;eG2(TS6)

: m)

= 9)

A Eg -submanifold is a surface of constant Kahler angle 9. On the existence of S^ -submanifold we refer to [10]. An isometric immersion x : M —> S6 of a Riemann surface M into SG is said to be a J-holomorphic curve if it satisfies

£) x x{z) = ^~lx* (i)'

u X*

where z is a local conformal coordinate of M. A J-holomorphic curve of S6 is a minimal surface in S6. We denote by a the second fundamental form of a J-holomorphic curve x : M2 —> S6. If m e M is not a geodesic point (i.e., c|m 7^ 0), CT(ei,ei) and cr(ei,e2) are mutually orthogonal and the length of them are equal to each other for any orthonormal frame e\, e 2 = J(ei) of TmM. We take an (local) orthonormal normal frame field along x as follows e3~a(ei,ei)/X,

e 4 =
e 5 = eie 3 ,

e 6 = J(e 5 )

where we put A = |a(ei,ei)|. By lemma 1.1, [x, ei,e 2 , e3,e4,e5, — ee] is an element of G 2 . The third form

= Y, ((VX
Z), et) ei, X, Y, Z e

TmM.

A J-holomorphic curve x : M —> S6 is said to be superminimal if the holomorphic 6-differential (
x a(dz,dz))

(dzf

is identically 0 on M. A J-holomorphic curve x is superminimal if and only if there exists a holomorphic horizontal lift E : M —> Q5 of x ([!}) with respect

117

to the 5Lr(3)-connection. We refer to [5], for the detail of local structures of J-holomorphic curves of S6. Define an admissible frame field (u, / , / ) by ' / 3 = (l/2)(ei

Uei),

h = (l/2)(e 3

lJe3),

./i=(l/2)(es

Ue5).

We shall give the explicit (local) solution of holomorphic horizontal curves of Q5 associated to J-holomorphic curves of S 6 . From this formula, we can calculate the 1st fundamental form and the Gauss curvature of superminimal J-holomorphic curves, explicitely. Theorem 2.1. Let E : U —> Q5 be the holomorphic horizontal curve with S(0) = Ei. Then S :£/-—> Q5 can be represented as follows, where U is a simply connected open set of C which contains the origin 0 e C. 3(C) = e ai(C) + E i • 1 + E 2 a 2 (C) + E 3 a 3 « ) + E i a 4 « ) + E 2 a 5 « ) + E 3 C where ai(C) = (v/^T/2) [c(/"(C) + /"(0)) - 2(/'(C) - /'(0))], a2(C) = (l/2)C 2 /"(0 " C(2/'(C) + /'(0)) + 3(/(C) - /(0)), a3(C)

= (l/2)/"(O(/'(0)-/'(C)) + /"(0) [(1/2)(C/"(C)) - /'(C) + /'(0) + (l/4)(/"(0)C)] + (3/4) /" (/"(*)) 2 ^, JO

«4(C) = /"(C)[-(3/2)/(C) + (1/2)C/'(C) + /'(0)C + (l/4)(/"(0)C2) + (3/2)/(0)]

+ /'(C) . (/'(C) (3/4) C

/"(0)C - 2/'(0)) + (3/2)/"(0)/(C)

l\r{z))'

dz

Jo

- (l/2)(/'(0)/"(0)C) " (3/2).f(0)/"(0) + (/'(0)) 2 , a5(C) = (l/2)(/"(C)-/"(0)), for an arbitary holomorphic function /(£) on U.

118

Remark 2.1.

We note that the quadric Q5 C P 6 ( C ) is defined as follows

Q5 = { [wo : W\ : w2 : u>3 : u>4 : w5 : w&] G P 6 ( C ) I (wo) 2 + W1W4 + W2W5 + W3WQ = 0 }

where [ ] denote the homogeneous coordinate system of P 6 ( C ) . Take two real numbers t (> 0) and 9. We apply theorem 2.1 to a function /(C) = (v/15i/36)e?6'C4. After a change of variable C, = \/6z, the corresponding holomorphic horizontal curve S t : P*(C) —» QB is as follows ( is/Wte^z3

\

1 \/l5iel0z4 Et(z) =

-^/6te™z5

(e,E,E)

te

V

eC.

2i0 - 6

V6z

J

The curve Et : P J ( C ) —> Q 5 yields a superminimal J-holomorphic curve by the formula x t (z)

(5 t (z),l^))'

The explicit expression of the curve xt(z) is /(1/2)(1 - 67; - 15tp2 + Ihtp4 + 6 i V - * V ) \ je-^N/lfW r 3 (1 + 3p + 3ty 2 + tp 3 - i V 6 r(l - 5fp2 - 5tp3 + t2p5) 1 ~t(z)>Z't(z

(e,E,E)

%y/\Mel9 z2(l +2pi0. VI Ot z\l -ie°"

2tp3 - tp4)

+3p + 3ty 2 + tp3)

3 5fy22 - 5tp -J- +t2p- 55) iy/6z{l - Up htp +

-i\/l5t c"'% 2 (l + 2p - 2£//3 - tp 4 )

y

119

where p — zz = \z\'2 and (Et(z),St(z)}

= (1/2) (1 + 6p + 15tp2 + 20tpi + 15ip4 + 6t2p5 + t2p6) > 1/2.

It is possible to calculate the Gauss curvature of this 1-parameter family of the immersions, we obtain the following Theorem 2.2. Let x4 : P X (C) —• S° be a 1-parameter family of superminimal J -holomorphic curves of a 6-dimensional sphere defined as above. Then xt is a deformation up to the action of G-2, for 0 < t < 1. 3

3-dimensional submanifolds

In this section, we study Grassmann geometry of 3-dimensional submanifolds of S6. 3.1

Decomposition of the Grassmann bundle

We denote by u)\, ..., io7 the orthonormal coframe dual to E], . . . , E7. The complex volume form to = (LO2 + V-IU3)

A {u>4 + V ^ I w s ) A (LO7 +

V^luJe)-

of the tangent space TE1S6 is extended to a G2-invariant 3-form on 5 6 , which we also denote by u>. For a complex number p (\p\ < 1), we put X
{t<=G3(TS6):u>(S)=p}.

From the following proposition, S p is an orbit of the action of G2 on G.$(TS6). Proposition 3.1. Let£, £' be elements of'G^(TS6). There exists an element g € Gi such that g(£) = £' if and only ifu>(£) = u>(£'). For a 3-dimensional subspace £ of TPS6 the condition J(£) _L £ is equivalent to |w(£)| = 1. From theorem 2.3 in [12], u(TpN) = ±1 (p G N) holds for a 3-dimensional totally real submanifolds, namely a totally real submanifold is nothing but a E^j-submanifold. Recently we have the following Theorem 3.1.

If a compact T,p

submanifold exist then p is a real number.

120 3.2

Tubes over J-holornorphic

curves.

In [3], Ejiri obtained a 3-dimensional totally real subrnanifold, which was the first example of 3-dimensional £ p -subrnanifold (p = ± 1 ) . Recently, Dillen and Vrancken [2] studied 3-dimensional totally real submanifolds which are obtained as a t u b e over a J-holomorphic curve. Ejiri's example is obtained also as a t u b e over a J-holomorphic curve S L —> S 6 . Here we study T,p subrnanifold which is obtained as a t u b e over a J-holomorphic curve. We denote by a the second fundamental form of a J-holomorphic curve ip : M2 —> S6. For any orthonormal frame e i , e2 = J(e\) of TmM, a(e.\, e i ) j m and cr(ei,e2)| TO are mutually orthogonal and the length of are equal to each other if the point m is not a geodesic point. We take an (local) orthonormal normal frame of ip as follows e-i = cr(ei,ei)/A,

e 4 = cr(ei,e 2 )/A,

e5 = e i e 3 ,

e6 = J ( e 5 )

where A = |
:

S6 ; (p,X)

~ (cos7)^(p) + ( s i n 7 ) ^ ( X ) .

UN2 —> S6 is defined in a similar fashion.

T h e o r e m 3 . 2 . Let ip : M2 —> S6 be a J-holomorphic curve without geodesic point. If a tube
2,j over ip of radius 7 is a T^p-subrnanifold, then one of the following holds (i) 7 = 7r/2 and p = 1, (ii) ip is a superminimal J-holomorphc curve and 9 sin" 7 — 8 4 — 3 sin 7 Especially, (^2,7 is a totally sin 2 7 = 1 or 4 / 9 .

real subrnanifold

(i.e.,

p = ± 1 ) if and only if

We put h% = (Ve.-ej, efc) (1 < i < 2, 3 < j , k< 6). T h e o r e m 3 . 3 . Let tp : M2 -^ S6 be a J-holomorphic curve without geodesic point. If a tube ip\n over ip of radius 7 is a E p -subrnanifold, then one of the following holds

121

(i)
SG, 0 < 7 < ir/2 and 6 cos 2 (7) - 8 sin 2 (7)

. , P = sin(7)

A/9COS 4 (7) — 3 cos 2 (7) sin 2 (7) + 4 sin 4 (7) (ii) 7 = 7r/2, /if3 = 0, h\j, /ii 4 are constants 3 and 2 \2

(hW

C'fl)

'•13

2

2

\ / ( ( M l ) + (^13) )((^ll) 2 + ( ^ 3 - l ) 2 ) (hi) 7 = TT/2, /if3 = 0, {h3n)2 = h5u(h5u - 1) and p = 0. Example 1 (Tubes over a J-holomorphic flat torus).

The subgroup

T = { exp(aG 2 3 + bG45 + cG7e) • a, b, c e R, a + b + c = 0 } is a maximal torus of G 2 . The mapping
S6 defined by ip : T - • 5 6 ; (a, 6, c) .-> exp(aG 23 + 6G 45 + cG76){Po) where we put po = (E2 + E4 + EQ)/-\/3,

is a J-holomorphic curve. We have

' J,4 —_ /,4 —_ 1,3 —_ "12 "21 "ll (,3 _ L 3 _ L 4 " 1 2 — "21 — " 2 2

"13

h 14

=

"14

=

"23

—

"23 *'

=

;,3 "22

-hj! = 0, ~"24

"13

—

=

0)

"24

• The tube in the direction of Res + Re6 is a totally real submanifold, • The tube in the direction of first normal space is a S_ 1 ,^-submanifold. Example 2 (Tubes over the Veronese surface). We denote by C/4 the subgroup of G2 corresponding to the Lie subalgebra generated by ( Xi = 4 G 3 2 + 2G 5 4+6G 7 6 ,

X2 = V6(G37 + G26 - 2G15) + VW(G42 - G35), X3 = \/6(G 6 3 + G 27 - 2G 4 i) + v/10(C?25 - G 34 ). The subgroup U4 is isomorphic to SO (3) and its action on R 7 is irreducible (cf. [12]). The orbit of the subgroup C/4 through E\ is the Veronese embedding 5 2 , 6 —> S 6 . The length of the second fundamental form is 5/3. Image of
122

Ms in [12]) of the subgroup £/4 through the point E2. The tube ipin with (COS7)2 = 5/8 is congruent to A/3 and the tube
Homogeneous

CR-submanifolds

From Mal'cev's classification [11], we have 4 isomorphism classes of subgroups of G 2 ; Ui:q- (x, y) = (x, yq), q E Sp(l), U2-q- (x, y) = (qxq, yq), q£ Sp(l), U3:q- (x, y) = (qxq, qyq), q G 5p(l), and U4. T h e o r e m 3.4 ([8]). (1) Let N be the orbit of U2 through the point p0 = ( l / 3 ) £ 2 + (2 v / 2/3)£ , 4 . Any ^-dimensional CR submanifold of SG, which is an orbit, of U2 in S6, is congruent to N under the action of G2 on S1'. (2) There does not exist any ^-dimensional CR submanifold of S6 which is an orbit of the subgroup U3. The subgroup U4 is isomorphic to 50(3). Since the tangent space of the orbit M through E7 is RE2 + REs + REQ , a neighbourhood of E7 in the great sphere S3 = { x\E\ + X4E4 + X5JE5 + x7E7 : x\, X4, x 5 , X7 6 R } n S 6 can be considered as a portion of the orbit space. Proposition 3.2 ([8]). / / the dimension of the. orbit N = U4 • p through a point p of the great sphere { x\Ei + X4E4 + x5E5 + x7E7 : xi, x 4 , ;r5, x 7 e R } n S6 is 3, then it is a CR-submanifold if and, only if f(x\,X4,x§,x7)

= 0 where

/(xi,X4,x 5 ,x 7 ) = -5.T4 - IOX4X5 - 5.T5 + 42x4X7 + 72x^X7 + 42x|xy - 9x^ - 24VT5x|x5X7 + 4

sVl5xlx7.

4-dimensional submanifolds

The orbit decomposition of the Grassmann bundle G2(TSh) is Uo<e<7r * Xfl where * is the Hodge star operator. On the existence of ^Eg-manifolds, we have the following

123

Theorem 4.1 (A. Gray [4]). morphic subm,anifold.

There does not exist any 4-dimensional holo-

On the other hand, there exists many 4-dimensional CR-submanifolds of S6. Proposition 4.1 ([9]). Let 7 : / —> S2 C ImH be a regular curve in the unit 2-sphere. The following immersion ij> : I x Sp(l) —> S6 is a 4-dimensional CR-submanifold of S6; 1>(t,q) =(a~/{t),bq), where a, b are positive real numbers satisfying a2 + b2 — 1. For an element (z,q) of U(l) x Sp(l), we have an automorphism i(z,q) of the Cayley algebra defined by (i(z,q))(r,s)

= {qrq,zsq),

r, s G H, r + f = 0.

(4.1)

We denote by L the image of the Lie group homomorphism i : U(l) x Sp(l) —> G-2- It is easily verified that on each orbit of the action of L on 5 6 , there exists a point of the form (ai, b + cj) with a > 0, b > 0, c > 0 and a2 + b2 + c2 — 1. Proposition 4.2 ([9]).

For any positive numbers a, b, c satisfying a2 + b2 +

2

c = 1, i/ie 07'6ii

{aqiq, z{b + cj)q),

z e C/(l), g £ 5p(l),

is a 4-dimensional CR-submanifold of S6. The problem whether there exists a *£ e -submamfold for 6 ^ n/2 is open. References 1. Bryant, R. L., Submanifolds and special structures on the octonians, J. Diff. Geometry 17(1982) 185-232. 2. Dillen, F. and Vrancken, L., Totally real submanifolds in S6(l) satisfying Chen's equality, Trans. Amer. Math. Soc. 348(1996), 1633-1646. 3. Ejiri, N., Equivariant minimal immersions of S2 into 5 2 m ( l ) , Trans. A. M. S. 297(1986), 105-124. 4. Gray, A., Almost complex submanifolds of Six sphere, Proc. A. M. S. 20(1969) 277-279. 5. Hashimoto, H., J-holomorphic curves in a 6-dimensional sphere, Tokyo J. Math. 23(2000), 137-159.

124

6. Hashimoto, H., Explicit representation of super-minimal J-holomorphic curves in a 6-dimensional sphere. Kodai Math.J. 20(1997) 241-251. 7. Koda, T. and Hashimoto, H., Grassmann geometry of 6-dimensional sphere, in "Topics in Complex Analysis, Differential Geometry and Mathematical Physics", eds. Dimiev S. and Sekigawa K., World Scientific Publ., Singapore, 1997, 136-142. 8. Hashimoto, H. and Mashimo, K., On some 3-dimensio7ial CR submanifolds in S6, Nagoya Math. J. 45(1999), 171-185. 9. Hashimoto, H., Mashimo, K. and Sekigawa, K., On 4-dimensional CRsubmanifolds of a 6-dimensional sphere, to appear. 10. Li, X., A classification theorem for complete minimal surfaces in S6 with constant Kahler angle, Arch. Math. 72(1999), 385-400. 11. Mal'cev, A.I., On semi-simple subgroups of Lie groups, A.M.S. Transl, Ser. 1 9(1950), 172-213. 12. Mashimo, K., Homogeneous totally real submanifolds of S6, Tsukuba J. Math. 9(1985), 185-202. 13. Sekigawa, K., Some CR-submanifolds in a 6-dimensioanl sphere, Tensor(N.S.) 6(1984), 13-20.

125 HYPERSURFACES IN EUCLIDEAN SPACE WHICH ARE O N E - P A R A M E T E R FAMILIES OF S P H E R E S G. GANCHEV Bulgarian Academy of Sciences, Institute of Mathematics and Informatics, Acad. G. Bonchev Str., bl. 8, 11 IS Sofia, Bulgaria E-mail: [email protected] V. MIHOVA University of Sofia, Faculty of Mathematics and Informatics, 5 James Bonder Str., 1164, Sofia, Bulgaria E-mail: [email protected] In the present paper we find the second fundamental tensor of a hypersurface in Euclidean space, which is a one-parameter family of spheres. We prove a characterization theorem for such hypersurfaces in terms of their second fundamental tensors. 1

Introduction

T h e envelope Mn of a one-parameter family of hyperspheres {iS"(s)}, s S J , defined in an interval J is said to be a canal hypersurface (cf Chen and Yano [1]). Any canal hypersurface Mn is a Riemannian manifold (Mn,g, £) endowed with a unit vector field £ (and with the one-form r\ corresponding to £). A canal hypersurface (Mn,g,£) with second fundamental tensor h is characterized by the equality [Ganchev and Mihova 2] h = ag + /??/ r], where a =£ 0 and j3 are functions on Mn. T h u s every canal hypersurface Mn can be characterized as a regular oneparameter family of spheres of codimension two: Mn = { 5 n _ 1 ( . s ) } , s G J, i.e. the tangent space TpSn~1(s) to every sphere Sn~1(s) at an arbitrary point n 1 p G S ~ (s) is an eigen space of the second fundamental tensor of Mn. Hence the Riemannian geometry of a hypersurface Mn = {Sn~1(s)}, s € J , which is a general one-parameter family of spheres of codimension two occurs to be more general t h a n the geometry of canal hypersurfaces. In this paper we study hypersurfaces Mn = {Sn~1(s)}, s G J , which are one-parameter families of spheres with respect t o their second fundamental tensors. We prove a characterization lemma for a sphere S1™-1 of codimension two. We show t h a t any hypersurface Mn under consideration is a Riemannian

126

manifold (Mn,g, W,£) endowed with an orthonormal frame field {W, £} (and with their corresponding one-forms to and ?;, respectively). The basic result is the following Theorem 1.1. Let (Mn,g,^) be a hypersurface. in En+1 endowed with a unit vector field £ (and with the one-form, r; corresponding to £). Then AP is locally a part of a non-regular one-parameter family of spheres (integral submanifolds of r\) iff there exists a unit vector field W, orthogonal to £, (and a one-form ui corresponding to W'), so that: i) The second fundamental tensor h of Mn satisfies the equality h = Kg + fi(u 03 i) + r\
Spheres of codimension two in Euclidean space

In this section we consider hyperspheres of codimension two in Euclidean space En+l. We denote the standard metric in En+1 by g and its Levi-Civita connection by V . For an arbitrary Riemannian manifold (M,g) we denote by TPM the tangent space to M at an arbitrary point p £ M and by XM the algebra of all C°° vector fields on M. Let Mn~l = 5 " " 1 be a sphere in a hyperplane En and I be the unit normal vector field to En. Denote by n the unit normal to Mn~l in En. The orthonormal frame field {l,n} at the points of S71^1 is said to be canonical. With respect to the canonical frame field {l,n} we have

V"j.n = — kx

for an arbitrary vector field x € XMn~x.

Here k — -, where r is the radius

71 1

of the sphere S " . Consider a.n arbitrary orthonormal frame {./V,£}, normal to Sn~l.

Such

127

normal frame and the canonical normal frame are related as follows N = cos (p I + sin if n,

(2)

£ = — sin f I + cos if 71, where f = Z(N,l). From (1) and (2) we find VXN = -k sin f x + df{x) £, V U = -kcos f x - df(x) N;

x

eXSn~l.

A sphere 5 n _ 1 of codimension two can be characterized as follows: Lemma 2.1. Let Mn~l be a surface in En+i Then Mn~l lies (locally) on a sphere iff

with normal frame field {N, £}.

VXN = -ax + a(x) ^ V;T£=

-bx-a(x)N,

'

for some functions a and b on Mn~l satisfying the condition a2 + b2 > 0 and a one-form a on Mn~l. Proof. I. Let M n _ 1 = Sn~l be a sphere with radius r. With respect to the canonical frame field {l,n} we have (1). If {N,£} is an arbitrary normal frame field, related to {l,n} by the equalities (2), we have (3). Putting in (3) a — k sin f, b = k cos f and a = df, we obtain (4). II. Let M n _ 1 be a surface in En+l, satisfying (4) and let us denote by R' the zero curvature tensor of V . Taking into account (4) we calculate R'(x,y)N and R'(x,y)£, for x, y 6 XMn~x. Thus we obtain the following integrability conditions for (4): {da(y) - ba(y)}x - {da(x) - ba{x)}y + da(x, y)£ = 0, {db(y) + aa(y)}x - {db(x) + aa{x)}y - da(x,y)N = 0. These equalities imply da = 0; da = 6(7, db = —aa.

(5) (6)

From (5) it follows that there exists locally a function f' such that o = df'. Substituting a = df1 into (6) we get da = bdf':

db = ~adf'.

(7)

128 From (7) we find d(a2 + b2) = 0. Hence a 2 + b2 = k2 = const > 0. Integrating (7), we obtain a = k sin((p' + a),

b = — k cos(tp' + a),

where a = const. Putting ip = ip' + a and I = cos ip N — sin ip£, n = sin

D

One-parameter families of spheres of codimension two in £ n + 1

Let g and V be the standard metric: and its Levi-Civita connection in the Euclidean space En+1. In this section we characterize hypersurfaces M " in En+1, which are oneparameter families of spheres of codimension two, in terms of their second fundamental tensors. The induced metric and its Levi-Civita connection on Mn are denoted by g and V, respectively. Definition. Let Mn be a hypersurface in En+L, which is a one-parameter family of surfaces {Qn"1(s)}, s G J, of codimension two. The family {Qn~1(s)}, s G J, is said to be regular if the tangent space TpQn~1(s) to every surface <3 n_1 (s) at an arbitrary point p £ Qn~1(s) is an invariant space of the second fundamental tensor of Mn. Let Mn = {5" _ 1 (.s)}, s 6 J, be a regular one-parameter family of spheres and £ be the unit vector field, orthogonal to the spheres Sn^1(s). If N is the unit normal to Mn, then from the definition it follows that V^TV = -KX,

x ± £.

(9)

The assumption K = 0 implies that 5" _1 (,s) are planes. Hence we can assume that K ^ O .

129

Let h(X, Y) = g{AX, Y), X, Y e XMn be the second fundamental tensor of Mn. If X S XMn, then the vector field x = X - r/(X)£ _L £. Substituting x into (9) we find AX = KX + ui]{X)i and h = Kg + is7]<8 T],

where K ^ 0 and (/ = /i(£,£) — K are functions on Mn. Applying Proposition 3 [Ganchev and Mihova 2] we obtain Proposition 3.1. A hypersurface Mn in En+[ is a regular one-parameter family of spheres iff Mn is a canal hypersurface. Further we shall characterize non-regular one-parameter families of spheres. Furst we obtain some integrability conditions for hypersurfaces with special second fundamental tensor. Let (Mn,g, W, £) be a hypersurface in En+l endowed with an orthonormal frame field {W, £} and let {LJ, TJ} be the one-forms corresponding to W and £, respectively. The distribution, which is orthogonal to W and £, is denoted by

A„. Consider a hypersurface (Mn,g,W,£)

with second fundamental tensor

h = Kg + /J,{LJ ® T) + T] <& U)} + L>1] ® T),

where K / 0, /J / 0 and v are functions on Mn. Replacing h from (10) into the Codazzi equation (Vxh)(Y,Z) = (Vy/i)(X,Z), X,Y,Z e

(10)

XMn

and substituting any of the vector variables X, Y and Z by io G Ao, W and £, we obtain the following Integrability conditions of first order 1) dn(xo) = 0; 2)VXaW=^K)-^K)X0

+

3) V*„£ = ^ - x

0

-

l M

j(x0)W-

M

4) g(VwW,x0)

=

d

-^l +

%{xo);

;

130

5) #(Vw'£,.-co) = -27(2:0); 6) g(VsW,x0)

= d (j)

(.T0) +

7
2 /

* 2 f 2 7(*o);

7(^0); - ^ ( « ) + 2^(/i.) + ^ ( / x ) - 2/iiy(K + ^) . I^T~2 -KM

^ 2/J£(K) + 2/iVy(/x) + VW(K + v)

9)
:

^

r

-

,

2

where 2 0 & Ao and 7(20) = 77 + i) to] + vi] r), where K / 0, /j / 0 and v are functions on AIn. ii) The distribution A of n is involutive. Hi) W(K) = div£. Proof. I. Let Mn = {S' n ~ 1 (s)}, s 6 J, be a hypersurface in En+1, which is a one-parameter family of spheres, and £ be the unit vector field, orthogonal to the spheres Sn'1(s). Then the spheres Sn~1(s) are integral submanifolds of the distribution A of £, which implies ii). If N is the unit normal to A/™, then the orthonormal frame {iV,£} is normal to every sphere Sn~1(s). From Lemma 2.1 we have VxN

= -KX - dtp(x)£,

x e A.

(11)

Since the one-parameter family of spheres is non-regular, then we can assume that d(f(x) ^ 0. We denote by w the unit one-form such that dip(x) = fi(j(x),

x e A,

fi = \\dip\\

and by W the unit vector field corresponding to the one-form u. Then from (11) we obtain: h{x, y) = Kg(x, y),

h(x, £) = /xw(.r),

x, y e A.

(12)

131

For arbitrary 1 , 7 6 XMn the vector fields x = X-i)(X)(, are in A. Substituting x, y in (12) we obtain

and y =

Y-r)(Y)£

h = Kg + /;.(w 0$ t] + V 05 u>) + vrj rj,

where v = /i(£, £) — K, which is i). Let { e i , . . . , e n - 2 } be an orthonormal basis for Ao at a point p G M". We have the formula n.-2

div^ = Y, ff(Vei£, ei) + g ( V ^ , W)-

(13)

?.=i

From the integrability condition 3) and the equality (13) it follows that d i v £ = ( n - 2 ) ^ ^ + <;(V M ^,W0.

(14)

Applying Lemma 2.1 we get g(V'Xt,Z,x0)=g(V'wZ,W),

x0eA0)

||.x- 0 ||=l.

(15)

Taking into account the integrability condition 3) and (15), we find g(VwS,W) = y ^ i .

(16)

Now from (14) and (16) we obtain W{K) = - A - d i v * , n- 1 which gives iii). II. Let ( M n , g , £) be a hypersurface in En+l satisfying the conditions i) — iii)- Then the integrability conditions l ) - 9 ) hold good. From the condition ii) we have dv{xo,yo) = 0,

dij(x0, W) = 0,

x 0 , y0 G A 0 .

The integrability conditions 3) and 5) imply that dr](x0, W) = 7(.x'o), XQ G A 0 . Hence 7(2:0) = 0, ;r0 G A 0 . Since the distribution A of r; is integrable, then Mn is a one-parameter family of submanifolds Mn = {Qn~1{s)}, s G J. If AT is the unit normal to Mn, then the orthonormal frame {N, £} is orthogonal to the submanifolds Qn^1(s). From the condition i), the equality (11) and the integrability

132

condition 3) we have V^JV = -KX0, x0 e A 0 ; V'WN = -KWH£; V ^ S = ——'-co, A*

x0 € A 0 ;

Vw£ = jdiv£ - (n - 2)^^\w

+ »N.

Taking into account the condition iii) we get A*

Now from Lema m2.1 it follows that Qn~1(s) is (lies on) a sphere S"~1(s). Hence Mn is locally a part of a one-parameter family of spheres {5™ _1 (s)}, s e J. D Up to now the spheres Sn^1(s) were considered as integral submanifolds of the distribution A of the one-form r/. We note that £ is not an eigen vector field of the second fundamental tensor A of Mn. Now we shall consider hypersurfaces Mn = {Sn~l(s)}, s G J, with respect to their diagonalized second fundamental tensor. In order to diagonalize the second fundamental tensor AX =

KX

+ H{LU(X)£ + v(X)W}

+ wi(X)£,

X e

XMn,

we have to find the eigen values of A on the distribution A = span {W, £}. If we denote these eigen values by K + k\, n + k2, then k\, k2 satisfy the equation k2 -vk-

ii2 = 0.

Let £i, £2 be the corresponding eigen vectors of A and r)\, r]2 be the one-forms corresponding to £1, £2, respectively. Then we have AX = KX + fcir,i W 6 + k2m(X)&, where fci + k2 = v, k\k2 = -p? < 0. Up to the choice ±W we can assume \i > 0. Denoting £1 = cos (3W + sin 0£, £2 = -sin(3W + cos/3£,,

X e

XMn,

133

from the system — cos (3 k + sin (3 \x = 0, cos (3 n + sin (3 {v — k) = 0 we obtain tan/3 = — = W - T - . M V «2 Using the first part of Theorem 3.2 we have Corollary 3.3. Let Mn = {Sn~l{s)}, s e J, be a hypersurface in E"-+1, which is a non-regular one-parameter family of spheres. Then we have h = Kg + kit]i r)i + k2r]2 <8> V2,

k\k2 < 0, (ki > 0).

If P = arctan \/ — j - , and W = cos/3£i — sin/3 £21 £ = sin/3^i +cos/3^2, ij £/ie distribution A of ^ is involutive; ii) W(K) =

div£. 71 — 1

Using the second part of Theorem 3.2 we obtain Corollary 3.4. tensor

Let Mn be a hypersurface in En+l

with second fundamental

h ~ ng + kiiji ® 771 + k2r\2 &> >]2, n ^ 0,

kik2 < 0.

Let ip = arctan \ — — , and V ^2 W = cos/3£i — sin/3^2, £ = sin/?£i + cos/3£2jj tte distribution /S. of £ is involutive;

.., ,.,

11) TW(K)A =

\J-k\k2 ..d i v £ ,

tften M n is locally a part of a non-regular one-parameter family of spheres.

134

References 1. B.-Y. Chen and K. Yano, Tohoku Math. J. 25(1973), 177. 2. G. Ganchev and V. Mihova, J. Reine Angew. Math. 522(2000), 119.

135 H Y P E R S U R F A C E S OF C O N U L L I T Y T W O IN E U C L I D E A N SPACE W H I C H A R E O N E - P A R A M E T E R SYSTEMS OF TORSES

G. G A N C H E V Bulgarian

Academy of Sciences, Institute of Mathematics and Acad. G. Bonchev Str. bl. 8, 1113, Sofia, Bulgaria E-mail: [email protected]

Informatics,

V. M I L O U S H E V A Institute

of Civil Engineering, Department of Mathematics, Suhodolska Str., 1373, Sofia, Bulgaria E-mail: [email protected]

32,

In the present paper we consider some classes of hypersurfaces of conullity two in Euclidean space. We prove a characterization theorem for the hypersurfaces of conullity two which are one-parameter systems of torses in terms of the second fundamental tensor of these hypersurfaces. We prove that every ruled hypersurfa.ee is a hypersurface of conullity two and find characterization conditions for ruled hypersurfaces in terms of their second fundamental tensor.

1

Preliminaries

It is well-known that a Riemannian manifold (M, g) with Riemannian curvature tensor R is said to be a semi-symmetric space if (SSS)

R(X,Y)-R

= 0.

In 1968 K. Nomizu [3] conjectured that in all dimensions > 3 every irreducible complete Riemannian semi-symmetric space is locally symmetric. In 1972 H. Takagi [7] constructed a complete irreducible hypersurface in E4 which satisfies the condition (SSS) but is not locally symmetric. In 1972 K. Sekigawa [4] proved that hi Em+1(m > 3) there exist complete irreducible hypersurfaces satisfying the condition (SSS) but are not locally symmetric. In 1982 Z. Szabo [5] gave a classification of Riemannian semi-symmetric spaces. According to his classification there are three types of classes: 1) "trivial" class, consisting of all locally symmetric Riemannian spaces and all 2-dimensional Riemannian spaces; 2) "exceptional" class of all elliptic, hyperbolic, Euclidean and Kaehlerian cones;

136

3) "typical" class of all Riemannian manifolds foliated by Euclidean leaves of codimension two. Complete semi-symmetric hypersurfaces in En+1 were classified by Szabo [6]. Riemannian manifolds of the "typical" class were studied under the name Riemannian manifolds of conullity two by Boeckx et al. [1] with respect to their metrics. In this paper we study some classes of hypersurfaces of conullity two with respect to their second fundamental tensor. The basic tools in our investigations are the integrability conditions for a hypersurface of conullity two, which give an approach to further classifications, geometric descriptions and constructions of these hypersurfaces. We prove a characterization theorem for hypersurfaces of conullity two in En+l which are one-parameter systems of torses. We also characterize ruled hypersurfaces in terms of their second fundamental tensor and show that they can be considered as a special class of the above mentioned hypersurfaces. 2

Hypersurfaces of conullity two in Euclidean space

In this section we consider hypersurfaces of conullity two in Euclidean space with respect to their second fundamental tensor. For a Riemannian manifold (Mn,g) as usual TpMn will stand for the tangent space to Mn at an arbitrary point p e Mn and XMn will denote the Lie algebra of all C°° vector fields on Mn. The Levi-Civita connection of the metric g is denoted by V and the Riemannian curvature tensor R of V is given by the equalities R(X,Y)Z = V . v V y 2 - Vy VXZ - V[X)Y]Z; X, Y, Z e XMn, R(X, Y, Z, U) = g(R(X, Y)Z, U); X, Y,Z,U e XMn. Let (Mn,g,A) be a Riemannian manifold endowed with a twodimensional distribution A. Since our considerations are local, we can assume that there is an orthonormal frame field [W, £} on Mn, which spans A, i.e. A p = span{Wi£}, p 6 Mn. We denote by u> and r; the one-forms corresponding to W and £, respectively: u(X)=g(W,X);

r](X)=g(^X);

X e XMn.

A Riemannian manifold (Mn,g,A) is of conullity two (see Boeckx et al. [1]), if at every point p 6 Mn there exists an orthonormal frame {ei =

137

W) e 2 = £, ^ 3 , . . . , e n } such that i) i?(ei, 62,62,6!) = -J?(e2,ei,e2,ei) = -R(ei,e2,ei,e2) = /2(e2,ei,ei,e 2 ) = fc ^ 0; e

e

e

j5 fei m)

—

(1)

0 otherwise.

Let Ap be the vector space of all bivectors in TpMn, p e Mn. We recall that the dot product (,) in A2 is determined by {X AY,ZAU)

X A Y, Z A U e A2p.

= g{X, Z)g{Y, U) - g{Y, Z)g{X, U);

The linear curvature operator R, acting in A2, is given by the equality X AY, Z AU £ A2p.

(R(X A Y), Z A U) = R{X, Y,Z,U); Proposition 2.1.

A Riemannian manifold (M,g,A)

R(X AY) = -k{u>{X)n(Y)

- LO(Y)T](X)}W

is of conullity two iff

A £;

X A Y e A2p.

Proof. Let {ei = W, e<2 = £, e%,..., en} be an orthonormal basis for the tangent space T p M", satisfying (1). Then R(ejAe.j) = 0, whenever e,Aej _L WA£. This implies that R(W A£) = -kW A£ and

R{XAY)=0;

XAYLWAt,.

Let X A Y be a bivector in Ap\ Using that X A Y ~ (X A Y, W A £)W A £ is a bivector orthogonal to W A £, we find £(x A y) = -fc(x AY,WA

£)W A £.

Hence R(X A Y) = -jfc{w(X)»j(y) - w(F)?y(X)}V1/ A £. The inverse is a straightforward verification.

•

The last equality can be written in a tensor form R(X,Y,Z,U)

= -k{u{X)ii(Y)Lj(Z)i](U) - w(X)V(Y)u(U)v(Z)

+

W(Y)TI(X)U(Z)TI{U) U>(Y)T,(XMU)T,(Z)}.

138 Further we express the second fundamental form of a hypersurface of conullity two in En+l in terms of t h e vector fields W, £ and t h e one-forms LU, VP r o p o s i t i o n 2 . 2 . A hypersurface (Mn,g,A) its second fundamental tensor h has the form

in En+1

is of conullity

two iff

h = Xui 6$ u> + fi(w .JO q + r) $0 LJ) + vr\ & r;, where X, /_i and v are functions

on Mn,

satisfying

the condition

Proof. Let (Mn, g, A) be a hypersurface of conullity two in En+1 fundamental tensor

h(X,Y)

Xv — /x2 ^ 0. with second

X,YeXMn.

= g(AX,Y);

Denote by e*, [i = 1, . . . , n) t h e principal vectors of A: Ae-i = XiCi\

i = l , . . . , n.

Consider the base a A ej (i < j) of A . From the condition i?(e, : A ej) = -A,;Aj e% A ey

i ^ j

and Proposition 2.1 it follows (up to nonieration) t h a t e\ A e^ = W A £, XiX2 = k and A ^ = 0 for all (i, j) ^ (1,2). Hence Ax0 = 0;

x0 _L A p

(2)

and A p , p € M n is an invariant space of A. Then

AW = \W + n£, where Xv — fi2 = k. For an arbitrary vector X in TpMn t h e vector X - UJ(X)W orthogonal t o A p . From (2), taking into account (3), we obtain AX

= XLO{X)W

+ n{w{X)£

+ v(X)W}

+

- r)(X)£ is

vi)(X)Z,

i.e. h = XLO ® to + /j,(u> <8> •)) + 7} <8> ui) + vn ® r/.

T h e inverse is a straightforward verification. R e m a r k 1.

T h e hypersurface Mn is fiat iff Xv - \i2 = k = 0.

•

139 3

Torses in Euclidean space

In this section we shall consider surfaces in Euclidean space En+l endowed with the standard metric g. The Levi-Civita connection of g (the canonical connection) is denoted by V . A hypersurface Mn, which is (lies on) a one-parameter system {En^1(s)}, s £ J of planes of codimension two, defined in an interval J, is said to be a ruled hypersurface. The planes En~1(s) are called generators of Mn. A ruled hypersurface Tn = {En~1(s)}1 ,s e J is said to be developable (a torse), if its unit normal vector field TV is parallel (constant) along any generator En~l(s) of Tn. Torses in En+l were characterized by Ganchev and Mihova [2] in terms of the second fundamental tensor as follows: Lemma 3.1. Let (Mn,g) be a hypersurface in En+l tal tensor h. Then Mn is locally a torse iff h(X,Y)

= kw{X)<j(Y);

X,Y

e

with second fundamenXMn,

where k and u are a function and a unit one-form on Mn, Remark 2. fc = 0.

respectively.

Every hyperplane Mn = En can be regarded as a torse with

Now let T n _ 1 be a torse in a hyperplane En in En+1(n > 3). Such a torse is called a torse of codimension two. The unit vector field ortogonal to the generators of T n _ 1 and its corresponding one-form are denoted by W and u>, respectively. It is clear that the vector field W is determined up to a sign. If / is the unit vector field normal to En and n is the unit vector field normal to T n _ 1 in En, then the pair {l,n} at the points of Tn~l is called the canonical normal frame to T " - 1 . With respect to the canonical normal frame from Lemma 3.1 we have V ^ = 0,

V > = -ku>{x)W;

where k is a function on Tn~l. An arbitrary normal frame {TV, Q to Tn~l frame to Tn~1 are related as follows £ = cos ip I + shupn, where if = Z(n, TV).

xeXTn~\

(4)

and the canonical normal

TV = — simp I + cosipn,

(5)

140

A normal frame {N, £} to Tn * is said to be semi-canonical, if it is constant along the generators of T n _ 1 , i.e. dip(x0) = 0, xo -L W. Hence dip = [AU>, M = W{ip)._ If M™ -1 = T n _ 1 is a torse of codimension two and {iV, £} is a semicanonical normal frame to Tn~1, then (4) and (5) imply VXN

= -k cos ip u(x)W - nui(x)^,

V'x£ = -k s,m
^'

Now we shall characterize torses of codimension two in En+1 which are not planes. Let ( M n _ 1 , g , W) be a surface in Ea+^ of codimension two endowed with a unit vector field W and let u> denote the one-form corresponding to W. The distribution on M n _ 1 orthogonal to W is denoted by Ao, i.e. A 0 ( P ) = { x0 e TpMn~l \Xo±W},

Pe

Mn.

Lemma 3.2. Let (Mn~1 ,g,W) be a surface in En+} of codimension two with normal frame {N, £}. Then Mn~l is locally a torse with serai-canonical frame {N, £} iff y'xN

= -pu(x)W

Vx£ = -qw{x)W

-

fiu(x)£,

+ nu(x)N,

x £ XMn~l

(7)

for some functions p, q, /j, which satisfy the conditions p2 + q2>0,

/z = W ( a r c t a n - j .

(8)

Proof. I. Let M n _ 1 be a torse T n _ 1 with semi-canonical normal frame {N, £}. Denote p = k cos ip and q = ksinip. Then the equations (6) imply (7). On the other hand the equalities p2 + q2 = A;2, - = tan ip and /x = W(tp) imply (8). P II. Let R' — 0 be the curvature tensor of the canonical connection V'. Calculating R'(x,y)N and R'(x,y)(, from (7), we find {fiduj(x, y) - (to A dfi)(x, y)}£ + {pdu(x, y) - (w A dp)(x, y)}W {ndu(x, j | ) - ( w A d/z)(x, y)}A' - {g<Mx, y) - (^ A dq-)(x, y)}W + ^ ( : r ) V ; ^ - qu(y)YxW = 0.

(9)

141

From (9) it follows that for every xo € AQ we have pdu(x0, W) + dp(x0) = 0, qdu>(xo, W) +dq(x0) = 0. Eliminating duj(xo,W) from these equations, we obtain d(^\(x0)=0

{ord(Z\{x0)=0),

XOGAQ.

(10)

Let us denote tp — arctan - . Then the condition (10) implies dtp(xo) = 0. P Using the equality of (8) we find dip — W(ip)ui

— pmj.

Now the condition (7) takes the form W'XN = -pu)(x)W Vj

- dp(x)(„

= -qcv(x)W + d
XMn~x.

(11)

Setting I — cos tp £ — sin tp N,

n = sinip£ + cosipN,

(12)

we find Vxl = 0,

V'xn = -kw(x)W;

x e XMn~\

(13)

where k = p cos ip + q sin
with normal frame

{N,£}.

[(14) }

142

Proof. I. Let Mn

x

be a plane En V ^ = 0,

l

with canonical normal frame {l,n}, i.e.

V > = 0;

xeXE71-1.

(15)

Using (5), for an arbitrary normal frame {N, £} of En~l V'XN =-dtp(x)£,

we find n l

V;r«e = d
(16)

Setting a = dip, we obtain (14). II. Let M " ^ 1 be a surface in En+1 with normal frame {N, £} such that (14) holds good. The integrability conditions of (14) imply da = 0. Then locally there exists a function p> such that a = dip. Defining / and n by (12), from (14) we obtain (15). Hence Mn~l is locally a plane. • 4

Hypersurfaces of conullity two and one-parameter systems of torses

Let Mn be a hypersurface of conullity two in En+l tensor h =

with second fundamental

® LO + n{uj r] + r\ <£) u) + vi] ® r/,

\UJ

Xv — /j,2 ^ 0.

Taking into account (17), from the Codazzi equation (V*/i)(y, Z) = (VyA)(I, Z);

X,Y,Ze

XMn

we obtain the following Integrability conditions for a hypersurface of conullity two: l)VXt^ 2)VXl,W

=

1(x0)W;

-y(x0)Z; vd\{x0) - fjdnjxp) u(X + v) 3 g(VwW,x0) = : 5 + 1 jlM; Z Xu - /J, Xu - \lz Xdii(xo) - ndX(x0) X2+2fi2-Xu 4) g(Vw^x0) = 7(-x-o); 1-—? A,-M2 5)

=

fl(V^,x0)

=^ ) - A ^ ( x o )

6)
Ay

2

_

7) {(A - v) + An }g(VwW,C) +(A-i/)dA(£);

+

V+_^7(x.o); XT^T(-TO);

= 2/«f/z(0 - (A - */)d//(W0 - 2/idi/(W)

(17)

143 8) {(A - vf + 4n2}g(V&

W) = (A - v)dn{i) + 2ndii(W) - (A -

v)dv{W)

-2/zdA(£), where x0 € A 0 and 7(.x0) = fifCV^, W). Let (Mn,g, W, £) be a Riemannian manifold endowed with an orthonormal frame field {W,£}. We denote by A 0 the distribution on Mn, orthogonal to W and £, i.e. A 0 (p) = { x-o G TpAfn | xo J- W, x0 1 £ },

P

eM"

and by A the distribution on Mn, orthogonal to £, i.e. A(p) = { x e TpMn | x ± £ },

P G M".

Now we shall prove the main theorem in our paper T h e o r e m 4 . 1 . Let {Mn,g,W,£) be a hypersurface of conullity two in E"-+1 with second fundamental tensor (17). Then Mn is locally a one-parameter system of torses iff

i)\ + 0; ii) the distribution A is involutive; in) 7 = 0; iv) \i = —W I( aarctan rctan^). Proof. I. Let Mn = {Tn~l(s)}, s 6 J be a one-parameter system of torses. We denote by {W, £} the orthonormal frame field on Mn such that £ is orthogonal to the torses Tn~1(s) and PV is orthogonal to the generators of Tn~l(s). If N is the unit normal to Mn, then {JV, £} is a normal frame field to each torse

Tn-\s).

From (4) and (5) we find

V^£ = -k sin (pw(x) W + dip(x) N; If h(X,Y) of AT", then

x <= A.

^ '

= #(.4X, Y); X, F G AfA/™ is the second fundamental tensor Ar =

\LO(X)W

+ dip(x)£;

A = A; cosy,

x e A.

(19)

Since Mn is not fiat, then k ^ 0 and cosy ^ 0. Hence A ^ 0. Since Tn~1(s) are the integral submanifolds of the distribution A of ?/, then A is involutive.

144

From (19) it follows that Axo = d(p(xo)£. Since Mn is of conullity two, then dif(xo) = 0, i.e. dip = W((f)uj = pio.

Now the second equality of (18) implies V^n£ = 0. Hence the one-form 7 on Ao is zero. Further let { e i , . . . ,e n _2} be an orthonormal basis of Ao at p G Mn. Then from the formula n-2

divC = Y,9^e,Z,et)

+ g(Vwt,W)

(20)

we find div£ =

fl(VH^,W0.

Now the second equality of (18) implies div£ = — k sirup. Taking into account that A = k cos
—. Hence p =

— W I arctan —-— II. Let Mn be tensor (17) and the every point p G Mn of A containing p. of surfaces Sn~1(s) conditions for Mn,

a hypersurface of conullity two with second fundamental conditions i) -iv) hold good. Since A is involutive, then for there exists a unique maximal integral submanifold S " - 1 Thus Mn = { 5 n _ 1 ( s ) } , s G J is a one-parameter system of codimension two. Taking into account the integrability we get VXiN

VWN

= 0,

V;.,^ = 0;

= -XW - fi£,

.xoGAo,

Vw£ = div iW + pN.

Let x G A. Then x - w(x)W G A 0 and V'XN = -Xw{x) W - fiw(x) £, V ^ = div &{x) W + fiu{x) N. Denoting p = X and q = — div£, from the condition iv) we get p = W ( a r c t a n - 1. Applying Lemma 3.2, we obtain that Sn~1(s) are (lie on) V PJ torses Tn~1(s). Hence Mn is locally a one-parameter system of torses of codimension two. •

145

With similar arguments we obtain T h e o r e m 4.2. Let (Mn,g,W,t;) be a hypersurface in En+1 with second n fundamental tensor h. Then M is locally a ruled hypersurface iff i) h = n(u> <8> r] + r\ u>) + vq <£> 77; ii) 7 = 0; in) div£ = 0. Proof. Let Mn = {En~1(s)}, s e J be a ruled hypersurface and £ be the unit vector field, orthogonal to the generators En~l(s). If N is the unit normal to Mn, then {N, £} is an orthonormal frame field, orthogonal to each generator En~[(s). From (16) we have V'XN =-a(x)Z;

x ± £.

(21)

Denote by u> the unit one-form, orthogonal to 77, such that a = fiw,

fi=\\a\\.

Then (21) implies Ax = tiLo{x)£; x _L £.

(22)

Since AW = /z£ and span {W, £} is an eigen space of A, then Let X e XMn.

Using that the vector X - r;(X)£ ± £, from (22) we find AX = IJW(X)Z

+ m(X)W

+ vri(X)£,

h = n(u (8> 77 + 77
I'TJ

$5 ?/•

Under the assumption A = 0 the integrability conditions 1) and 4) imply dr}(x0,W) = -7(x- 0 ),

.x'o £ A 0 .

Since the distribution A of 77 is involutive, then 7(0:0) = 0, x$ G Ao- On the other hand the integral submanifolds of A are totally geodesic. This condition implies that g(V\vW,£) = 0. Then from (20) it follows that div£ = 0. II. Let Mn be a hypersurface in En+X such that the conditions i)-iii) hold good. From the integrability conditions 2) and 4) under the assumptions A = 0 and 7 = 0 we find dri(x0,yo) = 0,

dr]{xo: W) = 0,

x0, y0 G A 0 .

Hence the distribution A is integrable and Mn = {S" 1_1 (.s)}, s e J is a one-parameter system of the integral submanifolds Sn~1(s) of A.

146 Taking into account that 7(.x'o) = 0, xo G Ao and the equality (20), we find X7'XN = -M*)Z, V'x£ = div&{x)W

+ ficu(x)N;

x e A.

Now the condition iii) and Lemma 3.3 imply that Sn~l(s) are (lie on) planes En~1(s). Hence Mn is locally a part of a ruled hypersurface. • References 1. E. Boeckx, L. Vanhecke and O. Kowalski, World Scientific, Singapore, 1996. 2. G. Ganchev and V. Mihova, C. R. Acad. Bulg. Sci. 50(1997), 17. 3. K. Nomizu, Tohoku Math. J. 20(1968), 41. 4. K. Sekigawa, Tensor, N. S. 25(1972), 133. 5. Z. Szabo, J. Differential Geometry 17(1982), 531. 6. Z. Szabo, Acta Sci. Math. 47(1984), 321. 7. H. Takagi, Tohoku Math. J. 24(1972), 105.

147 REAL H Y P E R S U R F A C E S OF A KAEHLER MANIFOLD (THE SIXTEEN CLASSES)

GEORGI GANCHEV Bulgatrian

Academy of Sciences, Institute of Mathematics and Acad. G. Bonchev Str., bl. 8, 1113, Sofia, Bulgaria E-mail: ganchev(§math. has. bg

Informatics,

MILEN HRISTOV University 1 Teodosiy

of Veliko Tirnovo, Faculty of Pedagogics, Mathematics and Informatics, Tirnovski Str, 5000, Veliko Tirnovo, Bulgaria E-mail: milenjhQyahoo.com

In this paper we consider real hypersurfaces of a Kaehler manifold with respect to the induced almost contact metric structure and obtain four basic classes of such hypersurfaces. These four basic classes generate sixteen classes of real hypersurfaces of a Kaehler manifold. We find some geometric conditions under which real hypersurfaces of a Kaehler manifold belong to some of the sixteen classes.

1

Introduction

An almost contact manifold M(ip,£,r)) is a (2n + l)-dimensional manifold M endowed with an almost contact structure (ip,£,r]) consisting of a tensor field (f of type (1,1), a vector field £ and a one-form r/, satisfying the conditions
r;(0 = l,

(1)

where Id is the identity on the tangent space TpM, p G M. As a consequence of (1), we have v(0 =0'

T)0(p = 0,

rauk
A Riemmannian metric g on A/(i^, £,?/) is compatible with the given almost contact structure if g(
X,YeTpM.

(2)

In that case {
g(X,0=v(X),

XeTpM.

The Levi-Civita connection of the metric g is denoted by V.

148

Every fiber TPM of the tangent bundle TM is a (2n + l)-dimensional vector space with structure {ipp, £ p , i]p, gp). The decomposition TPM = {Dp = Kerr? p }©span {£p} is orthogonal and U(n) x 1-invariant. The 27i-dimensional vector space (Dp, P e M} — the vertical distribution. Operators: h:TM - • D,h = - { ? } , « = f/8^; Tensor fields: $ : $(X,Y) = g(X, £ — the generalized Nijenhuis tensor field of type (1,2), where [p,
f* P O = g^Ffa,

ipehX),

w(X) =

Y),Z)\

F($,t,X),

where { e i , . . . , C2n+i} is an orthonormal basis of TPM, p e M. These fundamental objects satisfy the following well-known conditions [e.g. Ganchev and Alexiev 5]: Lemma 1. Let M2n+1(tp,£,r],g) be an almost contact metric manifold. Then: a) F(X,Y,Z)=g((Vx{X, tpY, Z) + 2F(Z, Y, yX) +2r1(X)(VYv)Z-2dr)(Y,Z); e) 2dV(H,X) = (£&){£, X) = -u(
- (Vzr})
149

Let (M2n,g, J) be an almost Hermitian manifold with metric g and almost complex structure J. Studying the vector space of all tensors of type (0,3), which have the same symmetries as the covariant derivative F(X, Y, Z) = g((VxJ)Y,

Z),

X, Y,Z e TpM2n, p e M2n

of the almost complex structure J, Gray and Hervella [7] obtained a geometricdecomposition of this space into four basic subspaces. Thus there arise four basic classes of almost Hermitian manifolds, wich generate the sixteen classes of almost Hermitian manifolds. Following a similar scheme Alexiev and Ganchev [1] studied the covariant derivative X,Y,Z e TpM'2n+1

F(X, Y, Z) = -g((Vx
of the structural tensor if of an almost contact metric manifold M2n+1(if,£,ri, g) and obtained 12 basic classes Wi (i = 1, . . . ,12) of almost contact metric manifolds. If hF(X, y, Z) = F(hX, hY, hZ), P{X,Y)

= F(hX,hY,0, q(X,Y)=F(£,hX,hY), then we have the following

Wi

W2

Ws

Characterization conditions for the twelve basic classes of almost contact metric manifolds f F(X, y Z) = V(X){V(YMZ) - r,{Z)«>(Y)}, \p = q = hF = 0; . f F(X, Y, Z) = ^{9(X, \ q = hF = 0;

.JF(X,Y,Z)

=

Y)V(Z)

- g(X, Z)r,(Y)},

^{*(X,Y)V(Z)-*{X,Z)V(Y),

\ q = hF = 0;

w,

(F(X,Y,Z)=dr}(
w^

(F(X,Y,Z) \

Wa

q

= hF =

=

1

-{{L^)^X,Y)V{Z)

-

(Li9)(vX,Z)n(Y)},

-

{Lzg){
f*(Z)=0;

iF(X,Y,Z) = \{{L^g){ifX,Z)n{Y) \ q = hF = 0;

150

W7

f F(X, Y, Z) = dr](
w8

( F(X,Y,Z) = ri(X){q(Y,Z) \ p = hF = 0; F(X,Y,Z)

12

Y)r,(Z), +

2r)(Z)u,(Y)},

^-±-^{9(X,Y)f(Z)-g(X,Z)f(Y) +*{X,Y)f(
W,10 W n

- 2V(Y)CJ(Z)

=

Wo

- dr,^X,

f F(X, Y, Z) = F{
(F(X,Y,Z) = - \ p = q = 0;

-F(Y,X,Z),

. f F ( X , Y, Z) + F(Y, Z, X) + F(Z, X, Y) = 0, ' \ p = g = 0,

It was shown by Chinea and Gonzalez [4] t h a t in the case n = 2, the classes W\Q and W u are empty and in the case n = 1 the classes W4, W5, WT, WQ, WQ, WIQ, WU, W\I are empty. These 12 basic classes (n > 3) generate further 2 1 2 classes of almost contact metric manifolds. Many earlier studied classes take place in the above classification scheme. For example: T h e class of cosymplectic manifolds is characterized by the condition F = 0. It is denoted by W0 and W0 e W,., i = 1, . . . , 1.2. T h e class of almost cosymplectic manifolds is characterized by the conditions d $ = dr\ = 0 and it is exactly the class WQ © W\ \. T h e class of a-Sasakian manifolds is characterized by the conditions N = 0, dr\ = a $ , a / 0 and it is exactly the class W2. T h e class of quasi-Sasakian manifolds is characterized by the conditions N = d$ = 0 and it is exactly the class W 2 © W 4 . T h e class of normal almost contact, metric manifolds is characterized by the condition N — 0 an it is exactly the class W-2 © W-i © W 4 © W 5 © Wg ©

WW.

T h e class of a-contact metric manifolds is characterized by t h e condition dq = a $ , a / 0 and it is exactly the class W2 © W 3 © W4 © W 5 © W 6 © W 8 © W 9 © W 1 0 © W u © W i 2 ; Characterisations of the clsses W,;, i = 1, . . . , 12, in termes of the essential complex components of the fundamental tensors are given by Hristov [8].

151

2

The sixteen possible classes of real hypersurfaces of a Kaehler manifold

Let M (J, G) be an almost Hermitian manifold with almost complex structure J and Riemmannian metric G: J2 = —Id, G(JX,JY) = G{X,Y), 2n+2 X,Ye TM . If M2n+1 is a hypersurface in M with a unit normal vector field N, then there arises naturally an almost contact metric structure (
g = G\M,

(p =

J-ri®N, 2n+l

r){X)=g(£,X),

XeXM

.

Let V and V be the Levi-Chivita connections on M2n+1 and M" , respectively. If h(X,Y) = g(AX,Y), X, Y e XM2n+l is the second fundamental tensor of M2n+i, then the Gauss and Weingarten formulas VXY

= VXY + h(X, Y)N,

V'XN = -AX;

X, Y e

XM2n+1

imply immediately (VxV)Y = (V.r^)y, (VXJ)Y = {Vxtp)Y + {h{X,tpY) + (Vxl)Y}N - r,(Y)AX + h(X,Y)£, F'(X, Y, Z) = F(X, Y, Z) + v(Z)h(X, Y) - r,(Y)h(X, Z), where F' = - V ' $ , F = -V<E>, and $ is the fundamental 2-form for the structure {(p,£,ri,g): $(X,Y) = g(X,
T,(Y)AX-h{X,Y)Z,

= F(X,
-h{X,
2dr](X, Y) = h(Y,
2dTi(
(Ceg)(X, Y) = -{h(Y, ipX) + h(X,
?)(Y)h(ipX,S),

(4)

d $ = 0, N(X, Y) = [
152

F(X, Y, Z) = V(Y)h(X, Z) - ri(Z)h(X, Y), h{X, Y) = -F(X, Y, 0 = -p(X, Y), h(£,Z) = F{U,Z)=u{Z) = {C^g){VZ,0 = -2dr,(
(5) (6) (7)

The next theorem is the main result in the paper. It describes the sixteen classes of real hypersurfaces of a Kaehler manifold in terms of their second fundamental tensor. T h e o r e m 2. Let M (J,G),(n > 2) be a Kaehler manifold. 2n+1 hypersurface M ((f,£t,r),g) of M is in the class Wi

Any real

®W2®W4®WQ.

The four basic classes W\, W2, W4 and WQ generate sixteen classes of real hypersurfaces, which are characterized in terms of their second fundamental tensor h as follows: i) W0: (W0eWi,i

= 1,2,4,6) h(X,Y)

=

v(r,®r,)(X,Y);

ii) Wx: h{X,Y)

= (r)®w + uj®ri +

vri®r)){X,Yy,

iii) W2: h(X,Y)

= -^Qg(hX,hY)

+v(r)®ri)(X,Y),

tr/i^O; iv) W4: h(X, Y) = tjp-gihX,

hY) ~ d-q{
/I

= 0;

v) Wa: h(X,Y)

= ±(Czg)(
v(V®t])(X,Y);

vi) Wi ®W2: h{X, Y) = ~^g{hX,

hY)

+ (I1®U

+ W®T) + VT)® T]){X,

Y);

153

vii) Wx ®W4: h(X: Y) = f~~9(hX, hY) - dr)( rf)(X, Y); viii) Wi © W6 : h(X, Y) = UC(:g)((pX, hY) + (JJ u + LO r? + 1/17 ® »y)(X, y ) ; ix) ^

e

^ /i(X, y ) = -dri{vX, hY) + v{j) 77) (X, Y);

x) w2ew6./t(x,y) = -M 5 (/ l X ,/ l y) + i(£5ff)(^x,/iy) + «/(r?® »?)(*. n xi) W 4 © W 6 :

/i(X, Y) = ?!&-g(hX, hY) - dr,{
+

v(ri®T,)(X,Y);

xii) Wi © I Y 2 © W 4 : h{X, Y) = -dr)(ipX, hY) + (77 u> + u 77 + ur) ® rj)(X, Y); xiii) Wi

®W2®W6: h(X,Y) = ±(Csg)(ipX,hY) -

Mg(hX,hY)

+ (77 (8) w + w (8» 7] + vi] ® i))(X, Y); xiv) Wi

®W4®W6: h(X, Y) = l-{Ci9){yX, hY) - dj){
hY)

+ (r)®uj + uj®Ti + vri
±(Cig)(vX,hY)-dT1(^X,hY) + (?? W + U) <S> 7? + l/T) (g> T))(X,

Y).

154

Proof. For any vectors X, Y e TpM2n+l, orthogonal decomposition X = x + V(X)Z,

p e M2n+1 we use the unique

Y = y + V(Y)S,

x, yeD

and compute h(X,Y)

=

h(x,y)+r](X)h(ty)+v(Y)h(x,0+h(^Ov(X)V(Y).

We denote v = h(£,£) and using (7) we find h(X, Y) = vv(X)V(Y)

+ LJ(X)T,(Y)

+ v(X)u,(Y)

+ h(x, y).

(8)

Further we denote h0(X,Y)

=

vV(X)r,(Y),

IH{X,Y)=U;(X)TI(Y)+V(X)OJ(Y).

By using the orthogonal irreducible decomposition of the symmetric: tensors over the Hermitian vector space (D,ip.g), we have h(x,y) =

f

-^9{x,y)

+ \{h{x,y)

+ 2^(x'^)

+ h{
^9(*,v)

-h(px>vy)}-

Putting h2{X,Y) hi(X,Y)

=

= \{h{x,y) hG(X,Y)

^Qg{x,y),

+ h{
= ~{h(x,y)

-

~9(x,y),

h(px,tpy)},

and taking into account (3), (4), we obtain h2(X,Y) h4(X,Y)

=

^-g(hX,hY),

= -dV(
=

+

^Qg(hX,hY),

±(£sg)(
Hence the second fundamental tensor h of a real hypersurface M2n+i of has the form a Kaehler manifold M h = ho + hi + h-2 + h.\ + fiQ.

155 T h e equalities (5) and (6) imply t h a t the tensors p, q and hF satisfy the following conditions: p{X, Y) = F(hX,

hY, 0 = -h{hX,

hY) = p(Y,

X),

q(X, Y) = F{Z, hX, hY) = 0, hF = 0. Now the characterizing conditions for the twelve basic classes of almost contact metric manifolds imply t h a t any real hypersurface of a Kaehler manifold belongs to the class Wi

®W2iBW4®W6.

In view of the characterizing conditions for the twelve basic classes, and taking into account the equalities (5), (6), (7), we obtain characterizations for each of t h e sixteen classes of real hypersurfaces as follows: W 0 : h = h0, W 0 G W , i = 1, 2, 4, 6, Wf. h = hi, WitBWf h = hi + hj, Wi © Wj © Wk: h = hi + hj + hk, i, J, & € {1, 2, 4 , 6 } , Wi 8 W 2 © W 4 © W 6 : h = hi + h2 + h4 + h6. This completes the proof. D R e m a r k . If n = 1, then the class W\ = 0 . In this case there arise 8 classes of real hypersurfaces of a 4-dimensional Kaehler manifold. 3

Examples

In this section we find conditions under which real hypersurfaces of a Kaehler manifold ( M

, J, G) are in some of the sixteen classes.

E x a m p l e s o f real h y p e r s u r f a c e s i n t h e c l a s s Wi © W 6 Let Af 2 " + 1 = {Qn(s)},s G / be a one-parameter family of complex hypersurfaces Qn(s) in M , defined in the interval / . If N is the unit normal to M 2 n + 1 , then £ = ~JN is orthogonal to Qn(s) and Qn{s) are integral submanifolds of -q. In this case h(hXJiY) = —h(
hY) + (?/ cg> UJ + UJ r] + vq r))(X,

It follows from Theorem 1 t h a t

Y),

156

Any real hypersurface in a Kaehler manifold, which is a one-parameter family of complex hypersurfaces is in the class W\ ® W§. Now let Qn(s) be totally geodesic complex hypersurfaces, i.e. Qn(s) are complex "hyperplanes". Then M"n+i is a "ruled" hypersurface, whose generators are complex "hyperplanes". In this case h(hX,hY) = 0 and h = r\ <8> to + to
It follows from Theorem 1 that Any real hypersurface of a Kaehler manifold, which is a one-parameter family of totally geodesic complex hypersurfaces, is in the class W\. The one-parameter family of complex hypersurfaces M2n+1 = {Qn{s)}, s £ I is called regular [Ganchev and Mihova 6] if the tangent space to any Qn(s) is an invariant space of the second fundamental tensor of M2n+X. Let M2n+1 = {Qn(s)}, s
= hiCtg)(
+

v{7)®T,)(X,Y).

It follows from Theorem 1 that Any real hypersurface of a Kaehler manifold, which is a regular oneparameter family of complex hypersurfaces, is in the class WQ . Examples of real hypersurfaces in the class Wi © W\. 2n

lanifoli with Kaehler metric g, locally defined Let M (J,g) be a Kaehler manifold on a neighbourhood (U; z — {za; z13}), by d2h 9a0

~

dzadz^

where h = /i(2 a ,z^) is a positive function on U. In U x C = {(za = xa iya; w = u + iv)} the real hypersurface M 2n+1

.

r

=

h(z\^

^ .,n; 7T _ _ _ ; Tn) W uJ _ 1 =

0

is considered. Blair [3] defined an almost contact structure on M by d £=v ^ du

u

d —, dv

1 f Ohih , . ?? = tor = - ——ax ' 2\dy«

dlnh \ vdu - udv —dy H 2^ =—. 8xa J J u + v2

157

d We rectify £s = — by the solution dt t = arcsin

u =+ C Vu 2 + v2

of the linear nonhomogenious partial differential equation dt

dt

,

_.

,

Then 1 / dhih , „ Putting ACT = d2/dx°2

+ d2/dya2

dlnh , „ \

we compute

n

$ = -2 ^

1

(ACT/i) da;CT A dy°,

n

dr; = i09r = - - ] T (ACTln/i) dxCT A dy°.

(7=1

These formulas imply: The real hypersurface M2n+i belongs to the class W^ © W.%. The real hypersurface M 2 n + 1 (n > 2) is in the class W4 iff tr*dr; = 0, i.e. iff ln/i is a harmonic function. The real hypersurface M 2 n + 1 is in the class W2 (drj = a $ , a ^ 0), iff the function h satisfies the following equalities dh \ a^;

2 +

/ dh N 2 U r J =/l(l-4a/l)A
<x = l , 2 , . . . , n .

References 1. V. Alexiev and G. Ganchev, Math, and Educ. in Math., Proc. of the XV Spring Conf. of UBM, Sunny Beach, 155(1986). 2. D. Blair, Lecture Notes in Math. 509, (Springer-Verlag, Berlin, 1976). 3. , Bull, de la Soc. Math, de Belgique, XXXV(1983), 25. 4. D. Chinea and C. Gonzalez, Ann. Mat. Pura Appl. 156(1990), 15. 5. G. Ganchev and V. Alexiev, Math, and Educ. in Math., Proc. of the XV Spring Conf. of UBM, Sunny Beach, 186 (1986). 6. G. Ganchev and V. Mihova, Complex Geometry and Vector Fields, eds. S. Dimiev and K. Sekigawa (World Scientific, Singapore, 2001). 7. A. Gray and L. Hervella, Ann. di Mat. pura ed appl. (IV), CXXIII(1980), 35.

158

8. M. Hristov, Aspects of Complex analysis, Differential Geometry and Mathematical Physics, eds. S. Dimiev and K. Sekigawa (World Scientific, Singapore, 1999). 9. Y. Tashiro, Tohoku Math. J. 15(1963), 62. 10. K. Yano and M. Kon, Progress in Mathematics, 30 (Birkhauser: Boston, Basel, Stuttgart, 1983).

159 ALMOST CONTACT B-METRIC H Y P E R S U R F A C E S OF K A E H L E R I A N MANIFOLDS W I T H B-METRIC MANCHO MANEV Faculty of Mathematics and Informatics. University of Plovdiv, 236 Bulgaria blvd., Plovdiv 4^04, Bulgaria E-mail: mmanevQpu.acad.bg, [email protected] In this paper, we construct two types of almost contact B-metric hypersurfaces of an almost complex manifold with B-metric and we characterize subclasses of these hypersurfaces of a Kaehlerian B-metric manifold with respect to the second fundamental tensor.

Introduction The geometry of almost complex B-metric manifolds is determined by the action of the almost complex structure as an antiisometry in each tangent fibre. The basic classes of these even dimensional manifolds are given in [1]. The special class Wo in this classification is the class of the Kaehlerian manifolds with B-metric, where the almost complex structure is parallel with respect to the Levi-Civita connection of the B-metric. This class is contained in each other class. Examples of Wo-manifolds are considered in [2], [3], [4]. The geometry of the almost contact B-metric manifolds is a natural extension of the geometry of the almost complex manifolds with B-metric to the odd dimensional case. A classification of the almost contact manifolds with B-metric is given in [5]. There are studied some examples of manifolds belonging to the basic classes in [5], [6], [7]. In this paper, we construct two types hypersurfaces of an almost complex manifold with B-metric, which are equipped with almost contact B-metric structures. We determine the class of these almost contact B-metric hypersurfaces of a Wo-nranifold and we characterize its important subclasses with respect to the second fundamental tensor.

1

Preliminaries

Let (M',J,gr) be a 2n'-dimensional almost complex manifold with B-metric, i.e. J is an almost complex structure and g' is a metric on M' such that: J2X = -X,

g'(JX, JY) = -g'(X, Y).

(1)

160

for all vector fields X, Y on M' . The associated metric g' of the manifold is given by g'(X, Y) = g'(X, JY). Both metrics are necessarily of signature (n',n'). The Levi-Civita connection of g' will be denoted by V . The tensor field F' of type (0,3) on M' is defined by F'(X, Y, Z) = g'((V'x J)Y, Z) for arbitrary X, Y, Z £ X(M') — the Lie algebra of the differentiable vector fields on A/'. This tensor has the following symmetries: F'(X, Y, Z) = F'(X, Z, Y),

F'(X, JY, JZ) = F'(X, Y, Z).

If \e,i\ (i — 1, 2, . . . , 2n') is an arbitrary basis of TP>M' at an arbitrary point p' in M', and g'lJ are the components of the inverse matrix of g', then the Lie form 9' associated with the tensor F' is defined by 0'(x)=g'iiF'(ei,ej,x) for an arbitrary vector x G TP'AI', p' e M'. A classification with three basic classes of the almost complex manifolds with B-metric with respect to F' is given in [1]. Further, we shall consider only the class Wo: F' = 0 of the Kaehierian manifolds with B-metric belonging to each of the basic classes. Let (M,ip,£,7],g) be a (2n + 1)—dimensional almost contact manifold with B-metric, [5], i.e. (y, £,?/) is an almost contact structure determined by a tensor field

= -X + V(XK,

7?(0 = 1,

(2)

and in addition this almost contact manifold (M,(p,£,r)) admits a metric g such that g(
(3)

where X, Y are arbitrary differentiable vector fields on M, i.e. X, Y 6 3t(M). It follows immediately 770^ = 0,

^ = 0,

v(X)=g(X,0,

g(
(4)

Moreover the endomorphism ip has rank 2n. The associated metric g given by g(X,Y) = g(X,LpY) + r](X)r](Y) is a B-metric, too. Both metrics g and g are indefinite of signature (n,n + 1). Further, X, Y, Z will stand for arbitrary differentiable vector fields on M and x, y, z — arbitrary vectors in tangential space TPM to M at an arbitrary point p in M •

161

Let V be the Levi-Civita connection of the metric g. The tensor F of type (0,3) on M is defined by F(x, y, z) = g((Vx
9* (•) = g1^ F{eu^ev

•),

w(.) = F(£,£, •)•

A classification of the almost contact manifolds with B-metric is given in [5], where eleven basic classes T{ are defined We use that characterization conditions changing the definitions of the classes FQ, FT, F$ and Fg with equivalent conditions for computing reasons. Let us denote f(x,y) := F(
FA • F(x,y,z)

=——

{g(px,
F5 : F(x,y,z)

=

F6 • F(x,y,z)

= f(x,y)r){z)

f)*(C\

f(x,y)

= f(y,x),

^-{g(x,py)r](z)+g(x,pz)r](y)}; +

f(x,z)r)(y),

f(
F7 : F(x, y, z) = f(x, y)v(z) + f(x, z)rj(y),

f(x,y) = -f(y,x),

f(yx,
0 ( 0 = 0* ( 0 = 0 ; (5)

F8 : F(x, y, z) = f(x, y)v(z) + f(x, z)r)(y), f(x, y) = f(y, x), Fg : F(x,y,z)

= f(x,y)r)(z)

f(x, y) = -f{y, x), F\\ : F(x,y,z)

f(px, ipy) = f{x, •
f{x,z)r){y), py) = f(x, y);

= r){x){r](y)io(z) + r/(z)w(y)}.

The classes F% © Fj, etc., are defined in a natural way by the conditions of the basic classes. There exists 2 1 1 classes of almost contact B-metric manifolds. The special class FQ: F = 0 is contained in each of the defined classes.

162

2

Time-like hypersurfaces of an almost complex manifold with B-metric

It is known, [9, 10] that every differentiable orientiable hypersurface of an almost complex manifold has an almost contact structure. In [5] it is shown that on every real nonisitropic hypersurface of R 2ra+2 as complex Riemannian manifold with a canonical complex manifold and B-metric there is arised an almost contact structure with B-metric. In a similar way we construct a hypersurface of an almost complex manifold with B-metric. Let ( M ' , J ,
= tant.

We define the structural vector field £ on M by the equalities: Z = \.N + fi.JN, Then we have we receive

£,= s'mt.N + cost.JN,

g'(S,N)=0.

J£ = —cost.N+

sint.JN,

whence

JN =

£-tai\t.N, J£ = tan*.£ N. cos t cos t From the last equality it is clear that J£ is transverse to M. The transform vector field JX of X has a tangent component to M denoted by ipX and a component with respect to J£ denoted by T](X)J£, i.e. it is valid the unique decomposition JX = ipX + r](X)J£, where r/ is a differentiable 1-forni on M. The last decomposition in tangent and normal components takes the following shape JX=
+ ta,nt.i)(X)£,

—V(X)N. (6) cost By such a way we define the structural (l,l)-tensor tp and the 1-form r\ in TpM at an arbitrary point p <E M. The restriction of g' on M we denote by g. Then, because of (1), we get immediately (2)-(4). Thus, we obtain that ((p, £, 7j, g) is an almost contact B-metric structure on the hypersurface M. So, we give the following Definition 2.1. The hypersurface M of an almost complex manifold with B-metric (M',J,g'), determined by the condition the normal unit N to be

163

time-like regarding g', equipped with the almost contact B-metric structure if := J + cost.g'(-, JN){cost.N € := s'mt.N + cost.JN,

-

sint.JN},

r] := cost.g'(-, JN),

g:=g'\M,

where t := arctan {g'(N, JN)} for t
- g(AX, Y)N,

VXN

= -AX,

(7)

where A is the second fundamental tensor of M corresponding to N . Using (6) and (7) we compute (VXJ)Y and (VXJ)N, whence we obtain F'{X,Y,Z)

=

F(X,Y,Z) + f<mt{F(X, tpY, Ov(Z) + F(X,
£)T,(Y)}

^-t{g(AX,Y)V(Z)+g(AX,Z)r)(Y)}

+

-L-dt{X)r,{Y)ri{Z), cos^ t F'(X, Y, N) = g(AX, VY) + tan t.g(AX, Y) 1

cost F'(X,N,

{VxvW

+ tan t j ~dt(X)

N) = — \—dt{X) cost [cost

+

+ r/(AX) \

V(Y),

2V(AX)\ J

In case when (Af, J, g1) is a Kaehlerian manifold with B-metric there are valid the following conditions for (M,ip,£,r),g): (Vx¥)Y

= r){Y){smt.tpAX

-

cost.ri{AX)£}

+ {sint.g{AX,ipY) (Vxv)Y

= -smt.{g(AX,Y)

-cost.g{AX,Y)}£, -

V(AX)T)(Y)}

- cos t.g(AX,
= sint.{T]{AX)ri{Y) - cost.g((ip o A-

V(AX)

= -^-tdt(X),

-

r^AY^X)}

Ao tp)X, Y), (8)

164

V x £ = - sin t{AX - ii(AX)£} - cos t.ipAX. F{X, Y, Z) = smt{g(AX,

ipZ)r,(Y)}

- cost{g{AX, Y)n(Z) + g(AX, Z)r,(Y) -2T,(AX)T}(Y)V(Z)}

(9)

(10)

,

6{Z) = {siat.tr{A o
v(M)]}v{Z),

9*(Z) = {- smt[trA - V(A£)] - cost.tr(A o
--{dt(
From (8), (9) and because of the symmetry of A regarding g, we obtain the general shape of the second fundamental tensor of the hypersurface of first type and the corresponding traces:

+ cos t{V^, X)^} (11)

ti-A = - ^ 1 _ CoS t.6(£) - sint.0*(O tr(Aoip)

= sin £.
-cost.0*(£).

Substituting AX in (10) we get the general shape of F and its associated 1-forms on the hypersurface of first type are F(X, Y, Z) = F(X, Y, Ov(Z) + F(X, £,

o(Z) = e(0v(z),

Z)V(Y),

e*(Z) = e*(cHZ)-

Hence, according to (5), it is valid the following Theorem 2.1. Every hypersurface of first type is mi almost contact Bmetric manifold belongs to the class T\ © T$ © T§@TT® T% © Tg © T\\. It is clear that 2 7 subclasses of hypersurfaces under consideration are possible but some subclasses are restricted to J-QNow, we will give the characteristics of some subclasses by the second fundamental tensor of the submanifold. Theorem 2.2.

I) The following classes of hypersurfaces of first type are

165

characterized in terms of the second fundamental tensor A by the conditions: 2 cost f4 : A= -—y-^-n ® £ - ^-{smt.p 2 cos t 2n

- cost.tp2};

T*> : A = - r - ^ U ( g > £ + — — {cost.p + sint.p 2 }; 2 cos t 2n F6: Aow = <noA, trA

— =tv{Aoif) 2 cost

=0;

Fix •• A = ^[•n®(,-cost{i}<%>n + u>®Z} 2 cost — sin t{rj ® = g{-,Q); F4®Fb:A=

-§^-n

® £. - ^~ {[sin t.0(£) - cos t.6* (£)]
£> COS b

£iTh

-[cos*.0(O+sint.0*(O]¥> 2 }; Ti © T% : A o ip = ip o A, sintltrATi © J"u : A = -~^-V 2cost

> + cost.tr {A o tp) = 0;

® £ + cost I - ^ V [ 2n

-r/ft-w®0 J

- sint < ——
£ I ; Jr5®Jr6:Ao
= ipoA,

H © J"n : A = 2cost

cost \itr. trA-

-•- \ - sin t.tr {Aotp) = 0;

»£ + cost<^ — ^ V - ?y ft - u; &> O [ 2ra J

+ sin t •{ -~-tp2 2n

- ri ® ipQ. - {u> o ip) a, £

... 4, ^ © 7 1 1 : ^„2„ 0 ^„ 0„ ^ = ^ 0„ 4, 0_ ^ 2, tr

^— ( 0 = tr M o w ) = 0; 2 cost

166

FA ®F5 ®FU : A =

-PQ-V®Z

2 cost

•cost -^ -—V 2 '-! [ 2n

—
sint -J -~P [ 2n

^ ^ 2?i

2

+ i ) ® ^ + (woi^)®q;

FA © F6 © Fix : ^ o A o ^ ^ o i o i^2 sin* < J t r A -

^

[• + cost.tr (A o
F5 © -^6 © ^11 :

\

2

- ^ 1 I - s i n t . t r (A o f) = 0; v

2costJ

^'

2

F4 © F5 © F6 © ^"n : ^ o A o ¥ J = ( ^ o i o ¥ j ; II) yl hypersurface of first type can't belong to the classes F7, Fg, F$, to their direct sums or to the direct sums of someone of them with someone of F4, F5, or F6. Proof. Having in mind (5), the covariant derivative of £ can be expressed explicite in the subclasses FQ, F4, F5, F\\, J-4 ® F5, F4 © F u , .F5 ©^"n and J-4 © ^5 © Fu- Then A takes the corresponding shape for considered classes. Now, let us consider the classes Fe, FT, .Fg, Fg and their direct sums. The conditions F(X,Y,() = F ( y , X , 0 and F(pX,pY,£) = -F{X,Y,£) imply individually A o p = f o A, and the commutation of A and p implies the properties F(X,Y,£) = F(Y,X,0 = -F(pX,pY,£)Besides, each of properties F{X, Y, £) = -F(Y, X, £) and F ( ^ X , <^y 0 = F(X, Y,£) follows to A o f = —p o A, and the anticommutation of A and y; implies the properties F(X,Y,£) = -F{Y,X,£) = F(pX,pY,£). Obviously, if we suppose that a hypersurface of first type belongs to Fr or to F%, w e obtain that it is only an Fo-manifold. The conditions for F on JFg-manifold imply the anticommutation of A and p. In other hand the 1-form r; is clossed on .Fg-manifold as well as on a manifold in F4, F$ and FQ- Taking into account (8) and (9) we obtain A o if = p o A. Hence a hypersurface of first type can't be in the class Fg without FQ. It is easy to get the propositions about Fa, F4 (B F$ (9 F$ © F\\ and the remaining classes in I).

167

The part II) of the theorem is received immediately from the explanations above. • We shall give geometric interpretation of some of these classes. We recall, if A = 0, tr A = 0 or A = XI, then the corresponding hypersurface is totally geodesical, minimal or umbilical, respectively. Having in mind (11) and (5), we get Proposition 2.3. (i) The totally geodesical hypersurfaces of first type form a subclass of J-Q with the condition t = const; (ii) The minimal hypersurfaces of first type form a subclass of the general class of hypersurfaces with the condition cost.6(E) +smt.0*(£) H

= 0.

(iii) The umbilical hypersurfaces of first type form a subclass of T\ © T$ with the condition sin £.#(£) - cosf.0*(£) = 0. There can't exist umbilical hypersurfaces of first type, in J-A or T*,. The umbilical T^-hypersurfaces of first type are totally geodesical. Remark. If M' = R2n+2, then it is received the example in [5] of the timelike unit hypersphere of the class T± © J~5, which is an umbilic hypersurface ofR2n+2. 3

Isotropic hypersurfaces regarding the associated metric of an almost complex manifold with B-metric

In [5] it is defined an real isotropic hypersurface of the associated metric in R 2 n + 2 , considered as a complex Riemannian manifold with a canonical complex structure and B-metric. In a similar way we introduce an isotropic hypersurface regarding the associated metric of an almost complex manifold with B-metric. Let (M',J,g'), d i m M = 2n + 2, be an almost complex manifold with B-metric. We determine a (2n + l)-dimensional differentiable hypersurface M embedding in M' by the condition AT. g'(Z, Z) = 0 for a vector field Z on M'. It is clear that Z and its transform vector field JZ by J are orthogonal with respect to the B-metric g', i.e. g'(Z, JZ) = 0. At every point we put g'(Z, Z) = cosh t, t > 0 for the sake of the impossibility Z to be a main isotropic direction and in view of definiteness. We can choose the time-like unit normal N =

:—JZ, i.e. g'(N,N) coshi — 1. Hence, JN is a space-like unit tangent vector field on M.

=

168

We determine the structural vector field £ on M by

£ = -JN =

-\-Z.

cosh i Then the vector field J<J coincides with N. Thus, in this case the introducing of the structural (l,l)-tensor if and 1-form 7) is made by the unique orthogonal decomposition Jx = fx + i](x)N.

(12)

The restriction of g' on M we denote by g. Then, ascertaining (2)~(4), we obtain that (
£ := -JN,

,, := -g'(-, JN),

g := g'\M

will be called a hypersurface of second type of (M', J,g'). Now, we shall study a classification of these manifolds with respect to the second fundamental tensor A of the hypersurface. The formulas of Gauss and Weingarten are VXY

= VXY - g{AX, Y)N,

VXN = -AX.

Taking into account (12) and (13) we compute (VXJ)Y Then we get

(13) and

(VXJ)N.

F'(X, Y, Z) = F(X, Y, Z) - g(AX, Y)rj(Z) - g(AX, Z)rj(Y), F'(X,Y,N)

= -(Vxv)Y+g(AX,fY),

F'(X,N,N)

=

-2V(AX).

In case when (M',J,g') is a Kaehlerian manifold with B-metric, the left hand sites of the last equalities are vanished. Then we obtain the following conditions for (M,tf,£,r),g): ( V A ^ ) F = V(Y)AX

(Vxv)Y

+ g(AX,

Y)£,

= g(AX,
dr,(X,Y)=g((<poA-A0
Vx€ =
(14)

169 F(X, Y, Z) = g(AX, Y)r,(Z) + g{AX, Z)r,(Y), 6(Z) =tr A.r)(Z),

9*{Z) = tv{A o ip)r)(Z),

(15)

<J(Z)=0.

Because of (14) the second fundamental tensor on the hypersurface of second type is AX = — on the hypersurface of second type we receive F(X, Y, Z) = F(X, Y, Ov(Z) + F(X, £,

9 = 9(Ov,

Z)V(Y),

e* = e*{Ov, w = o.

Having in mind the classification (5) and the results above, we ascertain the truthfulness of the following Theorem 3.1. Every hypersurface of second type is an almost contact Bmetric manifold belongs to the class T^ 0 T$ © T§ © T%. Obviously, 16 subclasses of hypersurfaces of second type are possible. Now, we shall write up some of these classes with respect to A. Theorem 3.2. Some classes of the hypersurfaces of second type are characterized by the second fundamental tensor A as follows: To : A = 0; 2n T •A T5.A=-^-^;

°*^

T6 : A o if = ip o A, A£ = 0, tr A = tr (A o ip) = 0; F$ : A o ip = -

--^{9(0^

+ P(Z)
r

F4®J 6:Ao
= tpoA,

A£, = 0, tr (A o p) = 0;

T5®J76:Aop

= lpoA,

At, = 0, trA = 0;

T& © T8 : AS, = 0, tr A = tr (A o ip) = 0; Fi © Tf, © Tz : A o ip = ip o A, A£, = 0;

170

We give a geometrical interpretation of some of these classes according to the results of the last theorem by the following Proposition 3.3. the To-manifolds;

(i) The totally geodesical hypersurfaces of second type are

(ii) The minimal hypersurfaces of second type are the J^®TQ®

J-s-manifolds;

(iii) Every umbilical liypersurface of second type is totally geodesical hypersurface, i.e. it belongs to the class JT0 . Remark. If M' = R 2ra + 2 , then it is obtained the example in [5] of the isotropic hypersphere of g' belonging to JF5 and it is a minimal liypersurface oiM.2n+2.

References 1. G. Ganchev and A. Borisov, Note on the almost complex manifolds with Norden metric, C. R. Acad. Bulg. Sri. 39(1986), no. 5, 31-34. 2. A.P. Norden, On a class of four-dimensional A-spaces (in Russian), Izv. Vyssh. Uchebn. Zaved. Mat. 17(1960), no. 4, 145-157. 3. , On the connection's structure of the manifolds of lines in a, nonEuclidean space, (in Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 17(1960), no. 4, 145-157. 4. G. Ganchev, K. Gribachev and V. Mihova, Holomorphic hypersurfaces of Kaehlerian manifolds with Norden metric, Plovdiv Univ. Sci. Works •--Math. 23(1985), no. 2, 221-236. 5. G. Ganchev, V. Mihova and K. Gribachev, Almost contact manifolds with B-metric, Math. Balk. 7(1993), no. 3-4, 261-276. 6. M. Manev, Examples of alm.ost contact manifolds with B-metric, Plovdiv Univ. Sci. Works — Math. 32(1995), no. 3, 61-66. 7. G. Nakovaand K. Gribachev, Submanifolds of some almost contact manifolds with B-metric with codimension two, II, Math. Balk. 12(1998). no. 1-2, 93-108. 8. D.E. Blair, Contact manifolds in Riemannian geometry, Lect. Notes in Math. 509 (Springer-Verlag, Berlin, 1976). 9. Y. Tashiro, On contact structure of hypersurfaces in complex manifolds, I, Tohoku Math. J. 15(1963), 62-78. 10. S. Kobayashi and K. Nomizu, Foundations of differential geometry, II (Intersci. Publ., New York, 1969).

171 PROJECTIVE FORMALISM A N D SOME METHODS FROM A L G E B R A I C G E O M E T R Y IN T H E T H E O R Y OF GRAVITATION

B.G. D I M I T R O V Bogoliubov Laboratory for Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141 980, Russia, E-mail: [email protected] The purpose of this paper will be to propose the implementation of some methods from algebraic geometry in the theory of gravitation, and more especially in the variational formalism. It has been assumed that the metric tensor depends on two vector fields, defined on a manifold, and also that the gravitational Lagrangian depends on the metric tensor and its first and second differentials (instead on the partial or covariant derivatives, as usually assumed). Under these assumptions, it has been shown that the first variation of the gravitational Lagrangian can be represented as a non-homogeneous polynomial in respect to the variables, defined in terms of the variated or differentiated vector fields. Therefore, the solution of the variational problem is found to be equivalent to finding all the variables (elements of an algebraic variety), which satisfy the algebraic equation.

1

Introduction and statement of the investigated problem

Variational approach is an essential and powerful method [1, 2] in the theory of gravitation, not only for obtaining the Einstein equations, but also the equations of motion in the case of some space-time decomposition, for example the (3 + 1) splitting [3] of space-time. Presently, another space-time splitting(4 + 1) is frequently used within the framework of the Kaluza-Klein theories, called Randall-Sundrum models [4, 5], where the main idea is that the fourdimensional Universe may have appeared as a result of a compactification of a five-dimensional one with a line element, given by ds2 = e-'lkr'r'> • if" • dx'1 • dxv + rc2 • dxl

(1)

where rc is a compactification radius, rfLV is the ordinary Minkowski metric, £5 C [0,7r] is a periodic cooddinate and fw are four dimensional indices. In both cases, the gravitational Lagrangian is of the form L = -sf^g-

R = Hgij^ij.k^ij.u),

(2)

where ; may denote either a partial or a covariant derivative as a result of the standard scalar curvature representation. Let us decompose the metric tensor

172

gij according to the known formulae 9ij =Pij

-Ui-Uj,

(3)

where pi3 is the projective tensor (satisfying the projective relation p% • p{ = Pi 7^ H )i ui a r e the (covariant) components of the vector field u with a lenght e = Ui-u'. If formulae (3) is applied in both cases, a substantial difference will be noted. In the ADM (3 + 1) approach, the vector field components are identified with some components of g^ so that p\ = Sj and the projective tensor components turn out to coincide with the three-dimensional components g\, , defined on the three-dimensional submanifold, i.e. pij = g^ . In a (4 + 1) space-time splitting, or in some other kind of decomposition, for example, warped compactifications [6] to four dimensional Minkowski space on sevendimensional manifolds, this will be of course no longer the case. The already known nice geometrical meaning of the embedded space will not be valid, and one will have to deal with some kind of a multidimensional projective formalism and a decomposition gij =>Pij + htj,

(4)

where space-time is decomposed into a p-dimensional subspace and orthogonal to it (n — p)-dimensional space [7]. If we restrict ourselves only to the gravitational part in the action (it is present in all mentioned cases in its standard form), then the combined system of equations of motion for p^ and for Ui (or respectively, for h^) have to be solved. Another example in the same spirit is from relativistic hydrodynamics, where the vector field u will be the tangent vector, defined at each point of the trajectory of motion of matter. However, there is also another "alternative", and it shall be investigated in the present paper. Namely, in view of the expression (3), let us simply assume that g^ depends on two vector fields u = u(xi,X2, • • • ,xn) and v = v(xi,x2,...,xn): 9ij(xi,x2,..-,xn)

=gij(~u,~v).

(5)

The left-hand side suggests that g^ may be regarded (for each pair of i and j) as some hypersurfaces on the n-dimensional manifold, but they may also be unterpreted as defined on a two-dimensional manifold, represented by the vector fields u and v. In terms of the differentials du and dv, definite differentialgeometric characteristics may be assigned, such as the first and the second quadratic form [8]. This is the reason why in the present paper the choice of the variables willl be related with differential quantities.

173

2

First and Second Differentials and Variations

In general, the differential of a vector is not necessarily a vector. Here it shall be assumed that du and dv are defined in the corresponding tangent spaces Tu and Tv of the vector fields u and v. The first differential d§ij(u,v) will be given by the expression

<*<„..) = g ^

+

^.^

d u +

&tdv

(for brevity further the indices k will be omitted).

(6)

In the case when

—-j-fx?,... ,x") and —-¥-(x?,... , x°) form a basis in the tangent space to ouk ovk the hypersurface gij(xi,x-2, • • • ,xn), d u and dv may also be interpreted as the linear coordinates of the tangent vector. It can easily be proved that in a curved space the differentials of a vector, having vector transformation properties are only the covariant differentials. The kind of the differential is of no importance for the presented formalism in this paper. Also, instead of u and v one may write down some other (tensor) variable, related for example to a fc-dimensional hypersurface, embedded in the n-dimensional spacetime. Of more importance is the expression for the second differential

**<-.*> - ^ • «•")' + * • t • t ^

+

S 1 •

The expressions for the first and second variations of gv,(u, v) have the same structure. Note that

„ ' J • (du) 2 is the concise notation for ouz

8

^f • (du) 2 EE -^L• du' • duk + pH- • (du*)2, (8) 2 v l k v du ' du du dul2 v ' ' and also it has been assumed that S2u = d2u = 52v = d2v = 0. If du = a,i-dx%, then we will have d2u = a4 • d2xi + dai A dx{ = at • d2xi + (^

- ^ - J • dxl • dxk = 0.

(9)

Provided the Poincare's theorem is fulfilled and d2xl = 0 (i.e. dx% = const. ~ d x is a coordinate line), one would have d2u = 0 only if dai

da^ _

n

/..•,.

dxk dxl for every pair of i and k. This means that du is either an exact differential (a,i = const.), or that the vector field d u has zero-helicity components at [rota = 0, eq. (10)] in regard to the chozen basic vector field components dxl.

174 Note t h a t if r o t a = 0, but dx1 are not basic vectors, one would have d 2 u / 0. From a mathematical point of view, models with d2xl ^ 0, when the scalarproduct (ei,dx^ = ff 7^ Sf have been considered in [9]. 3

F o r m u l a t i o n of t h e V a r i a t i o n a l P r o b l e m in t h e C a s e of Different O p e r a t o r s of V a r i a t i o n a n d D i f f e r e n t i a t i o n

T h e gravitational part of the action, which will be investigated, is of t h e type L(gij1dgij,d2gij)dnx.

S

[ID

T h e first v a r i a t i o n of t h e a c t i o n (provided the volume element is not being varied) is SS :

/"*W

dL dgtj

dL • lJ

:odgij

d(dgij)

->

d(d2gij)

-Sd'gij

dnx.

(12)

T h e operators 6 and d are defined in one and the same way, but here they are distinguished. In the spirit of C a r t a n ' s works [10], they may correspond to variations and differentiations along different paths (not necessarily u and v). For example, one of the operators may be defined on a submanifold. To find the equations of motion for u and the divergent terms, one would need to interchange the places of the operators d and 5. For t h a t purpose, the corresponding expressions for [S,d]gij and (Sd2 — dS2) gij have t o be found. T h e first one is [6, d] gi:j ( u , v ) = 8dgvj (u, v)

d5g(u,v)

= ^ [ < J , d ] u + (ii <->«), (13)

where (u <-» v) means the same expression with interchanged u and v. As for the s e c o n d expression, it shall be presented in the following compact and symmetric form (6d2 - dS2) 0 y ( u , v ) = Qij = 5 Q y ( d u , d v ) - d Q 0 ( 5 u , « v ) ,

(14)

where Q i j ( d u , d v ) is the quadratic form in respect to d u and d v from eq. (7) , 2 „ 9^f t.j ^ dp 9d iu jd. v. +, - d'29,j (dv) ou av ovz (15) Qij(du:6v) is the same form, but with <Su and Sv. By means of the last three expressions, expression (12) for the variation of the action can be represented in the following form Qij ( d u , d v ) = d2gij (u, v)

6S

I

d2

9ijiA^2 (du) du'2

-5 gij + Aij + 5Bij + dd Sgt.

dnx = 0,

(16)

175

where 5LW

dL

dL

dgij

d(dgij)

'

+ Sd

dL ' d{d2gtJ)

(17)

This expression will represent the equation of motion for u, provided also that the variations at the endpoints vanish and SBij = 0. A\j is a term, which appears in (16) due to the assumption 5 ^ d and will disappear when S = d Aij(u,v)

dL Qij (du,dv) d{d2glj) d dgl0 d2L 9i: dv dud(dgij) du

= -6

+d

dL d(d?9ij)

d2L dvd(dgij)

Qtj(du,5v)

(duSv — Sudv).

(18)

Bjj is the expression B,

dL :dg,j d(dgi

dL \d(d?gi.

Sg.K

dL d(cPgi_

j(du, dv),

(19)

dL -KjjSudv. d(d2gK

(20)

and Cij is equal to Cj

dL 2 J gij d(d29l.

dL Qij(du,dv) d(d29lj)

=2

In (20) Kij is given by Ki.

d29i:j __ dgjj dgi:j

dudv

du

(21)

dv

Note that if the variations at the initial and final endpoints of the chosen curves disappear and also 5 = d, then the equation Kjj = 0 (taking into account second variation of the metric tensor 52gij) appears as a necessary and sufficient condition for the fulfillment of the equations of motion (17). 4

First Variation of the Lagrangian as a Third-Rank Polynomial

From the representation (16) of the variated Lagrangian, let us express all variations and differentiations in terms of the vector fields u and v. The following notations are introduced X\ = 5ul Y( = du

1

X\ = dv1 Y^ = dv*

Z\ = 5du% T\ = d5u

l

Z\ = Sdv T.j = dSv1

(22)

176

After some transformations, an expression for the first variation of the Lagrangian will be obtained: SL = [P^X,

+ P?X2 + Q\ZX +

Q\Z2}

+ P^Y1X2 + PV2X2Y2 + P%Y2XX + Q2J\X2 +

+ {p^XxYi + [P^X,X2YX

+ PIXXX2Y2

+ PgXMYz

Q2T2XX)

+ F%X2Y{Y2}

+ { P%X, Y2 + P^X, Y22 + P?X2Y2 + P£X2Y22 }=0.

(23)

If we assume (just from a general consideration and not from a concrete example) that the coefficient functions P™, P " , . . . , P " , P£', Qi, Q2 are independent of the variables Xi, X2, ..., T\, T2 and depend only on u and v, then (23) will represent a non-homogeneous polynomial of third degree (the highest degree in the polynomial) and it is linear in respect to the variables Xi, X2, Z\, Z2, quadratic in respect to X\, X2, Yi, Y2, T\, T2 and cubic only in respect to X\, X2, Y\, Y2. Note that the polynomial structure is rather specific, since Z\, Z2 enter only the linear terms and T\, T2 only the quadratic terms (and in combination only with X\, X2 and not Y\, Y2). In (23), Pi, . . . , P-j, Qi, Q2 are functions of u and v , which are of the following form PXUE

-^3

dv

dL d(dgi: PI

pu -M3

dL dgij dgtj du dgi3 du

du

dL d{d2gi:j) du d2L d2gtj 2 dud(d gij) du'2

p« _ ^'2 —

dL d2gtj d(dgij) du2

dL dgr [d(dg7j) dv

P,u

dL d'2gV] d(dgl3 dudv' d2L dudi&gijY d2L d29lJ dvd(d2gij) du'2

(24) (25) (26)

Pf, . . . , P!j are the same expressions, but with u and v interchanged. It should be noted, however, that depending on the concrete Lagrangian, powers in du, dv, 8u, 5v may come also from the coefficient functions, therefore changing the highest degree of the polynomial. Further the problem shall be formulated in terms of a third-rank polynomial, keeping in mind that the degree of the polynomial does not change the formulation, but only the technical methods for solving the algebraic equation.

177 5

Third-Rank Polynomials — Formulation of the Problem from an Algebro-Geometric Point of View

Let us formulate the investigated problem from the point of view of algebraic geometry, using the well-known approaches and terminology in [11-13]. For a clear and illustrated with many examples exposition of the subject, one may see also [14]. Let the defined in (22) set of variables (X\,Xj, Yf, Y\,Z[,Z\, T\, Tl2) (i = 1, 2, . . . , n) belong to the algebraic variety X C An(k), where An(k) is the n-dimensional affine space, defined over the field k = k[Xi,X2,Yi,Y2, Zi, Z2,Ti,T2] of the functions in 8-variables. The coefficient functions P " , Pf, P j \ P2", . . . , Pf, P?,Qi,Q2 are defined not on the same field k, but on the manifold M. In fact, An(k) is the cartesian product of n-tuples of k. Since all the the components of the metric tensor and of the vector fields are also defined on M D {x\,x^,...,x°), for each point (x\,x2), • • • ,x^) a mapping ip : M —> X between the elements of the manifold and the elements of the algebraic variety is also defined t

P\xlix2i

• • • 'Xn)

=

( - ^ U ^ l ) • • • ^Xn)-<^-2\Xl^

• • • 'X'n)i

0

...,Ti(x 1,...,x°n),T2i(x0l,...,x°n)).

(27)

l

Now let R[XlX^Yl,Y2\ZlZ 2,TiTl} denotes the ring of all polynomials / i , /2, • •., fn, • • •, defined on the points X of the algebraic, variety C An{k) and belonging to the ideal V(J, k), such as V(J,k)

= {XcAn(k)

: (f1(X),f2(X),...,fn(X),...)

= 0}.

(28)

Therefore, from (27) and (28) it is easily seen that the following sequence of mappings is defined M^X—>V(J,k).

(29)

The considered in this paper problem can be defined in the following way: Proposition 1. The variational problem 5L = 0 is equivalent to finding all the elements X\, Xl2, Y{, Y2\ Z\, Z\, T ^ T j of the algebraic variety X, which satisfy an algebraic equation f(X) = 0, defined on the elements of the variety and with a finite number of coefficient functions Pf, Pj", P] 4 , P | , . . . , Pf, Pf, Qi, Q2 — functions of the metric tensor and the two chozen vector fields. The algebraic equation belongs to the ideal I = (Pf, P>, P2,P%,..., P?,P?,QUQ2) C R[X[,Xl Y{, Y2\ Zi,Zi, T{, r2<], where R is the ring of all the polynomials, defined on X.

178

If found and considered as functions, defined on the manifold M, the elements of the algebraic variety X are no longer independent, but will represent a system of partial differential equations in respect to du (or 5u, for example). Further, if du is known, then again the new system of partial differential equations (this time in respect to u) will give an expression for u (if v is assumed to be known), or it will give a relation between the two vector fields (if u and v are not known). Furthermore, the obtained relation between the vector fields from the variational principle SL = 0 might be used in the determination of the equation of motion for u. In an algebraic language, the simultaneous investigation of the variation equation SL = 0 and the equation of motion means that the intersection "varieties" of the two algebraic surfaces (defined by the corresponding algebraic equations) should be found. Acknowledgment. The author is grateful to Dr. S. Manoff from the Institute for Nuclear Research and Nuclear Energy of the Bulgarian Academy of Sciences for some interesting discussions, as well as to Dr. L. Alexandrov (Laboratory for Theoretical Physics, JINR, Dubna) for his interest towards this work and encouragement. References 1. L.D. Landau, E.M. Lifshitz, A Course of Theoretical Physics. Field Theory, vol.11, "Nauka" Publ. House, Moscow, 1988. 2. B.A. Dubrovin, S.P. Novikov, A.T. Fomenko, Contemporary Geometry. Methods and Applications, vol. I and II, Fourth edition,, URSS Publ. House, Moscow 1998. 3. R. Arnowitt, S. Deser, C.W. Misner, The Dynamics of General Relativity, in Gravitation: An Introduction to Current Research, ed. by L. Witten, (John Wiley & Sons Inc., New York, London, 1962). 4. L. Randall, R. Sundrum, A Large Mass Hierarchy from, a Small Extra Dimension, hep-th/9905221, Phys. Rev. lett. 83(1999), 3370-3373. 5. , An Alternative to Compactification , hep-th/9906064, Phys. Rev. lett. 83(1999), 4690-4693. 6. K. Becker, M. Becker, Compactifying Ad-theory to Four Dimensions, hepth/0010282, JHEP 0011(2000), 029. 7. E. Zafiris, Kinematical Approach to Brane Worldsheet Deformations in Spacetime, Annals of Physics 264(1996), 75. 8. S.P. Finikov, A Course of Differential Geometry, Moscow, 1952. 9. S. Manoff, Spaces with Contravariant and Covariant Affine Connections and Metrics, Phys. of Part, and Atomic Nuclei, [Russian Edit.: 30(5),

179

(1999), 1211-1269], [Engl.Edit.: 30(5), 1999, 527-549]. 10. E. Cartan, Riemannian Geometry in the Orthogonal Basis, Moscow, Moscow State University, 1960 (in Russian); also E. Cartan, Lecons sur la geometrie des espaces de Riemann, Paris, 1928. 11. M. Reed, Algebraic Geometry for All. 12. W. Fulton, Algebraic Curves. An Introduction to Algebraic Geometry, New York, Amsterdam, W. A. Benjamin Inc., 1969. 13. D. Mumford, Algebraic Geometry. Complex Projective Varieties, NewYork-Berlin-Heidelberg, Springer-Verlag, 1976. 14. D. Cox, J. Little, D. O'Shea, Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, Second Edition, Springer-Verlag, 1998.

180 G E O M E T R Y OF MANIFOLDS A N D D A R K

MATTER

I V A N G O E B. P E S T O V Joint Institute for Nuclear Research, Dubna, Moscow Region,Ut19S0 RUSSIA E-mail: pestov(§thsunl .jinr.ru It is shown that the theory of dark matter can be derived from the first principles. Particles representing a new form of matter gravitate but do not interact electromagnetically, strongly and weakly with the known elementary particles. Physics of these particles is defined by the Planck scales.

1

Introduction

The problem of invisible mass [1], [2] is acknowledged to be among the greatest puzzles of modern cosmology and theory of elementary particles. The most direct evidence for the existence of large quantities of dark matter in the Universe comes from the astronomical observation of the motion of visible matter in galaxies [3]. One does neither know the identity of the dark matter nor whether there is one or more type of its structure elements. The most commonly discussed theoretical elementary particle candidates are a massive neutrino, a sypersymmetric neutralino and the axiom It is self-evident that existence of such particles is a great question. So, at present time there is a good probability that the set of elementary particles is by no means limited by those particles, the evidence for whose is proved by experience. Moreover, we are evidently free to look for more deep reasons for the existence of new particles unusual in many respects. Here the theory of structure elements of the so-called dark matter is derived from the first principles and all properties of these particles are listed at the end. 2

Geometrical framework

According to the modern standpoint a fundamental physical theory is the one that possesses a mathematical representation whose elements are smooth manifold and geometrical objects defined on this manifold. Most physicists today consider a theory be fundamental only if it does make explicit use of this concept. It is thought that curvature of the manifold itself provides an explanation of gravity. Within the manifold, further structure are definedvector fields, connexions, particle path and so forth- and these are taken to account for the behavior of physical world. This picture is so generally

181

accepted and it is based on such a long history of physical research, that there is no reason to question it. Geometry on manifold M with a local coordinate system xl, i = 0, 1, 2, 3, is defined by the metric gtj and linear (affine) connexion rJjk. Tensor field gij is symmetric, gtj — gji, but linear connexion TJik is nonsymmetric with respect to the covariant indices, T3ik ^ Y3ki in the general case and in any way does not link with the metric g. Really, these notions define, on a manifold M, different geometric operations. Namely, a metric on a manifold defines at every point the scalar product of vectors from the tangent space and linear connection gives the translation along any path on M. 3

Symmetry

Symmetry on a manifold can be introduced as follows. Consider the Einstein law of gravity Rjj = 0. Let g^ be a solution of the equations Rij = 0 and P{x) are four functions defined by the demand that all expressions in the formula gij(x) =

ft(x)fJ(x)gki(f(x)),

where /f(x) = difk(x), have the meaning. It can be shown that gij(x) is a symmetric tensor and a new solution of the Einstein eqs. Rij = 0. The point transformation xl => p{x) is a local diffeomorphism. All such transformations form a group of local diffeomorphisms. As it follows from the above statement, the group thus defined is a symmetry group of gravitation interactions. It should be noted that it is the most general group of coordinate transformations. Another very general group of transformations on M can be defined as follows. Consider the most general linear transformation of vector fields V1 =

S)Vj,

where V1 and V1 are components of vector fields, Sj are components of a tensor field of the type (1,1), det(5'j) ^ 0. Of two tensor fields S* and T\ of the type (1,1) a tensor field Pj = SkTj of the type (1,1) may be constructed, called their product. With the operation of multiplication thus defined, the set of tensor fields of the type (1,1) with a nonzero determinant forms a group, denoted by Gi. This is a most general gauge group on a manifold. It can be shown that the diffeomorphism group is the group of external automorphisms of the gauge group, i.e. the gauge group is invariant under the transformations of the group Diff (M). Thus, we have a nontrivial unification of these symmetries.

182

The tensor fields can transform under the gauge transformations in different ways. We will say that a tensor field T of the type (m,n) has the gauge type (p, q) if under the transformations of the gauge group there is the correspondence S---STS-l---S~\,

T = P

Q

where 0 < p < m, 0 < q < n, and S~x is the transformation inverse to S, S~x = (T*), S\T^ = Sj. From the equations Rij = 0 which express the Einstein law of gravity one can find that the Einstein gravitational potentials gtj have the gauge type (0,0). 4

Main Conjecture

Now, we have the possibility to put forward the idea that the group of gauge symmetry G, defines properties of a new form of matter called the dark matter. From the theory of linear vector spaces and for the reasons of symmetry and simplicity it follows that there is a single q u a n t i t y which can be put in correspondence with hypothetical particles of this matter. This quantity is a tensor field of the type (1,1), ^ J , and the gauge type (1,1). Since under the action of the gauge group a tensor field \I> is transformed as follows * = StfS" 1 , then scalars ** = T r * and ^)^\ = T r ( * * ) are evidently invariants of the gauge group G,. It is known from the theory of linear operators that there also exist other invariants, but in what follows we will only use the invariant Tr(#tf). To derive the nontrivial gauge invariant equations for \I/, first of all investigate the link between the linear connection T°ik and gauge group G,. To this end consider the properties of covariant derivative with respect to linear connection F^fc from the standpoint of gauge symmetry in question. The covariant derivative of ^ with respect to the affine connection F7; = (T^fc) can be written in the form

The use of the matrix notation is rather evident and does not require special explanations. Let Ti = (Tjk) is another affine connection on M and V, denotes the covariant derivative with respect to this connection. Then

183 where STi .= Ti — Ti. instead of $ , we get

Substituting, into the above relation, V> l = S^S

l

V,,* = ^ ( V , * ) ^ - 1 + [STi - S V j S - 1 , * ] . From this it follows t h a t Vi* = 5(Vi*)5_1

(1)

if STi = S V i S 1 - 1 , or t h a t is t h e same,

r j = r i + 5V i s- 1 .

(2)

Let ( B y f ) = BVJ = diTj - djTi + [Ti, Tj] be the Riemann tensor of the affine connection I \ , then from (2) it follows that Bij = SBijS

,

(3)

where B^ is the Riemann tensor of the connection f,;. Thus, from (2) and (3) it follows t h a t in the framework of the gauge group Gi one can consider the affine connection T3ik as the gauge field and the tensor B^ as t h e strength tensor of this field. According t o (1) a tensor field V i * has the same gauge type as * , but this is not true for the second covariant derivative of * or Btj. For this reason it is necessary to introduce the important notion of the gauge covariant derivative, which does not change the gauge type of the quantity in question. Let T be a tensor field (tensor density) of the gauge t y p e (1,1), then by definition DiT =

diT+[Ti,T]

is the gauge covariant derivative. For example, for t h e Riemann tensor we have DiBjk

= diBjk

+

\ri,Bjk].

As it must be, for t h e field ^ t h e gauge covariant derivative coincides with the standard covariant derivative, D^ = V;\I'. In the general case the operator Di is not general covariant, since D{F will not always be a tensor field together with T. However, the commutator [Di, Dj] is always general covariant, because [Di,Dj]T=[Bij,T].

184

From this we get the important relation for the Riemann tensor [DuD^Bk^lBi^Bki].

(4)

Thus, the gauge symmetry shows that not only the Einstein gravitational potentials g^ but also the tensor field * and gauge field I \ are to be treated as primary fields tightly connected with the symmetry and geometry on the manifold. Before writing the simplest gauge invariant equations, that express the law of interaction of these fields, it must be noted that all the three fields have geometrical interpretation. For the field g^ it is well known, so we dwell only on the geometrical interpretation of the fields $ and I\ considered as a whole. Let 5Vl = dV + Tljk±iVkdt be an infinitesimal change of the vector field V on a curve 7(f) and SJ = Sj + ^dt be an infinitesimal gauge transformation. We assume that at every moment of time t an infinitesimal change of a vector V along a curve j(t) is equal to an infinitesimal linear transformation of the vector field V induced by the gauge group, SVl = t j F ' d t . Thus, we obtain a system of ordinary linear homogeneous differential equations for V'(t) dV dt

, dx\,. dt

Jh

,

,

J

from which it follows that a composite geometrical object (\I>, I\) defines the general law of translations of a vector field along the given curve j(t). 5

Field Equations

Now we shall assume that the field * has the gauge type (1,1), under the action of the gauge group the T-field is transformed by the law (2) and construct a Gi-invariant theory of the interaction of those fields. The simplest gauge invariant Lagrangian of the field * has the form L* = - | ' I V ( A * £ ' ' * - m 2 * * ) ,

(5)

where m is a constant, Di = gl^Dj. From (5) by the variation with respect to $ we obtain the following equations Dt(^\g~\D^)

+ m2^\g~\^ = 0,

(6)

where \g\ is the absolute value of the determinant of the matrix (gij). When deriving (6) one should take into account that Ti(Di^) = 9,(Tr\I>). In accordance with (6) one can consider m as the mass of a particle defined by the

185

field * . The simplest Lagrangian of the gauge field T is a direct consequence of (3) L r = -^Ti-(B i j -B i J),

(7)

where Blj = glkgjlBki. Varying the Lagrangian L = L* + Lr with respect to T with the help of the relation SBtJ — DjSTj - Dj8Yl we obtain the following equations of the gauge field V Diiy/WB*)

= s/\g\J^

(8)

the right hand side of which contains the tensor field of the third rank ./' = [*,£**].

(9) 1

This field obviously has the gauge type (1,1). The tensor current J has to satisfy the equation

Dt(V\g\Jl) = o,

(io)

%:

as in accordance with (4), DiDj(y/\g\B ') = 0. From (6) and (9) it follows that J1 really satisfies equation (10) and thus the system of equations (6), (8) is consistent. Varying the Lagrangian L = L<j, + Lr with respect to g1^ we obtain the so-called metric tensor of energy-momentum of the considered system of interacting fields TV] = Tr (£><*£>,-*) + TV (BlkB3k)

+ 9ijL,

(11)

kl

where Bj ' = Bjig . If the fields \P and T satisfy equations (6) and (8), then one can show that the metric tensor of the energy-momentum satisfies the well-known equations Tij,,. = 0, where the semicolon denotes as usual the covariant derivative with respect to the Levi-Civita connection belonging to the field g^j jk J

=

2glt(dj9kl

+ dk9jl

~

dl9jk

^

It is evident that the metric tensor energy-momentum is gauge invariant. Now we can write down the full action for the fields gy, \t, F ^3

S = ~^

f R^/\i\d4x

- ^ f Tr ( D i * D 4 * - m 2 t t # ) VTsfd4*

-^JTv(BtjB^)^\g~\d4x,

(12)

186

where R is the scalar curvature, G is the Newton gravitational constant and h is the Planck constant. From the geometrical interpretation of the fields * and r it follows that they have the dimension sm~l. As all coordinates can be considered to have the dimension sm, the action S has a correct dimension. It is necessary to substantiate only why we have introduced the Planck constant h into the full action S and not, say, the constant of interaction e with the gauge field T, similar to the electric charge of the electron e. Consider the infinitesimal transformations of the diffeomorphism group, xl = x1 + Kl(x)dt. If under such transformations the gravitational potentials gij do not vary, i.e. the vector field Kl(x) satisfies the Killing equations K%gji

+ gudjK* + gjAK* = KH + Khj = 0,

then the vector field P1 = T^Kj will satisfy the equation Plj = 0. Integrating this equation we obtain the conservation law. The Killing equations impose severe constraints on the gravitational potentials. Thus, the Killing equations are completely integrable if the tensor of curvature of the metric g^ satisfies the equations: R ,

.

Ran = rrzKgikgji - gngjk)In the general case the Killing equations have no solutions at all and there are no conservation laws. This result allows us to understand the absence of the constant of interaction of matter fields with the gravitational field similar to the electric charge e and why the Newton gravitational constant G does not enter into the equations of matter fields. It is impossible to switch on or switch off the gravitational field. It does not admit the existence of the gravitational screen, the "gravitational charge" does not exist as an invariant fundamental notion. Now consider infinitesimal gauge transformations S'j = S^+iYjdt. If under such transformations the gauge field I \ does not vary, i.e. the tensor field fi*satisfies the equations

ami + r^fi'fr - fifr|-fr = v , ^ = o, then the vector field Ql = Tr( Jlfl), where J ? is a tensor current (9), will satisfy the equation Ql.t = 0. Integrating this equation we obtain the conservation law as usual. The equation VjfijJ. = 0, like the Killing equations, imposes severe constraints, in the given case, on the gauge field IV Thus, the equation Vjftj[, = 0 is completely integrable, if the strength tensor Bjj satisfies the equations B^ = 1/45^™^. So, in the considered case, generally speaking, there are no conservation laws too. From this fact it is natural to conclude,

187

that there is no special coupling constant (like the electric charge) of the gauge field T with the field of matter * . The gauge field T is impossible to screen. In the known sense, the Planck constant h is analogous to the Newton gravitational constant G. The Planck constant h characterizes the intensity of interactions of the field * with the gauge field T. Interactions in question has a pure quantum nature. From this consideration one can conclude that nonabelian gauge fields and gravity are not in the scope of perturbation theory and we need new approaches here. 6

General Physical Interpretation

Varying the full action (12) with respect to g%i we derive the Einstein equations

where I = \JhG/ci is the Planck length and 7Vj is the metric tensor of energymomentum (11). Thus, it is shown that the problem formulated above has the solution. A tensor field of the second rank \P' describes unknown gravitating particles which, as it was shown earlier, are a single source of the gauge field F that is known as the linear (affine) connection in geometry. As it is known, the affine connection has always played a fundamental role in all attempts to develop General Relativity from the very start of its creation [4], [5], [6]. The conclusion drawn here that the affine connection has a conserved energy-momentum tensor and therefore may be a source of gravitational field radically changes the view on that object. In this connection it should be emphasized that equations (6) and (8) can be considered on the background of the Minkowski spacetiine. It is enough to suppose that <^ is the Minkowski metric. It is not difficult to show that for the field * there is no nontrivial gauge invariant equations of the first order. Let S be an element of the gauge group; obviously, (-S) also belongs to that group. Further, under the gauge transformation ^ = S^S^1 the same transformation of the field \I/ corresponds to different elements S and (—S) of the gauge group. We have remarked these properties because they characterize the bose particles. The equations of motion for a test particle in external gravitational and gauge fields have the form d2xi ds2 where Wy = Ti(Bij)

f i 1 dx* dxk \jkj ds ds

h mc

t J

dxi _ ds

= diQj - djQz and Qt — T r ^ ) = Ykik. An additional

188

gauge force acting on a test particle has the quantum nature and the same form that the Lorentz force. However, the new force has one essential feature which can be seen from a rather general consideration. Taking the trace of equations (8) and taking into account that in accordance with (9), Tr (,/*) = 0, one obtains that if T satisfies equations (6) and (8), then o ^ will satisfy the equations a; y ;i = 0. These equations are the same as the Maxwell equations without sources. From this one can conclude that the new gauge invariant force connected with the gauge field F can neither be central nor Coulomb.

7

Summary

In accordance with the theory developed above, the main properties of particles of dark matter are as follows: 1. These particles practically do not interact with the known elementary particles. Really, it is impossible to construct the Lagrangian of interaction of the field ^ with other known fields being invariant with respect to the transformations of the gauge group in question. 2. However, these particles, as it is established, interact gravitationally. 3. The spin of particles equals 1. To see this, one can remind that any real matrix has the canonical (Jordan) form [7]. In one case, for example, * — diag(Ai, A2, A3, A4), where A^ are roots of the characteristic equation I* - XE\ = 0. Besides, it should be noted that the expansion * = * l/4Tr(^)E + l / 4 T r ( * ) £ ' = * 0 + l / 4 T r ( * ) £ is gauge invariant. Thus, the field \&o has three degrees of freedom and we have deal with the Bose particles. 4. The mass of the particles is a single parameter of the theory. However, under the natural condition m = mp = \JhcjG we shall obtain the theory free from parameters completely. 5. We have not deep reasons to introduce complex value wave function and hence electric charge of particles equal zero. So, actually all is known about unknown particles. However, experience remains a sole criterion of the validity of a mathematical construction for the physics. Last but not least. If the suggested physical interpretation of the theory is true, then the evidences of the existence of the dark matter, obtained from the astronomical observations, show that the macroscopic quantum effects can appear not only on the scale of solid (superconductivity) but also on the scale of the galaxies.

189

References 1. W.C. Saslay, Gravitational physics of stellar and galactic systems, Cambridge University Press, USA, 1987. 2. D. Pavon and W. Zimdahl, Dark matter and Dissipation, Phys. Lett. A179(1993). 3. Particle Data Group, Phys. Lett. B239(1990) III. 3. 4. A.S. Eddington, Mathematical Theory Relativity, Cambridge University Press, 1923. 5. E. Schrodinger, Space-Time Structure, Cambridge University Press, 1950. 6. A. Einstein, Relativistic Theory of the Nonsymmetric Field, The Meaning of Relativity, Princeton, 1955. 7. B.F. Schutz, Geometrical Methods of Mathematical Physics, Cambridge University Press, 1982.

190 L A G R A N G I A N FLUID M E C H A N I C S S. MANOFF Bulgarian Academy of Sciences, Institute for Nuclear Research and Nuclear Energy, Department of Theoretical Physics, Blvd. Tzarigradsko Chaussee 72, 1784 ~ Sofia, Bulgaria E-mail: [email protected] The method of Lagrangians with covariant derivative (MLCD) is applied to a special type of Lagrangian density depending on scalar and vector fields as well as on their first covariant derivatives. The corresponding Euler-Lagrange's equations and energy-momentum tensors are found on the basis of the covariant Noether's identities.

1

Introduction

In the recent years it has been shown that every classical field theory [1], [2] could be considered as a theory of a continuous media with its kinematic characteristics. On the other side, "perfect fluids are equivalent to elastic media when the later are homogeneous and isotropic" [3]. The theory of fluids is usually based on a state equation and on a variational principle [4], [5], [6] with a given Lagrangian density depending on the variables constructing the equation of state (rest mass density, entropy, enthalpy, temperature, pressure etc.) and the 4-velocity of the points in the fluid. There are two canonical types of description of a fluid: the Eulerian picture and the Lagrangian picture. In the Eulerian picture, a point of the fluid and its velocity are identified with the state and the velocity of an observer moving with it. This means that every point in the fluid has a velocity which coincides with the velocity of an observer moving with it. In the Lagrangian picture, the observer has different velocity from the velocity of a point of the fluid observed by him. He observes the projections of the velocities of the points of the fluids from its own frame of reference. This means that an observer does not move with the points of the fluid but measures their velocities projecting them on its own velocity arid standpoint. In other words, the Eulerian picture is related to the description of a fluid by an observer moving locally with its points, and the Lagrangian picture is related toN the description of a fluid by an observer not moving locally with its points. In the Eulerian picture the velocity of the points (the velocity of an observer) in a fluid appears as a field variable in the Lagrangian density describing the fluid. In the Lagrangian picture the

191

velocity of an observer could be considered as an element of the projection formalism [called (n + 1) — 1-formalism, dim M = n, where M is a differential manifold with dimension' n, considered as a model of the space (n = 3) or the space-time (n = 4)]. The Lagrangian invariant L by the use of which a Lagrangian density L is defined as L = ^—dg • L, where dg = det( dx? + dx^ 05 dxl) is the metric tensor field in M determining the possibility for introducing the square of the line element ds2 = g{d,d) = gTj • dx{ • dxj with d = dxl • <9, e T(M) := UXTX(M) [TX(M) is the tangent space to the point x £ M, := means by definition, i, j , k, • • • = 1, 2, . . . , n], interpreted as the infinitesimal distance between two points: point x with co-ordinates xl and point x with co-ordinates xl + dx1. The differentiable manifold M is considered here as a space with contravariant and covariant affine connections (whose components differ not only by sign) and metrics, i.e. M = (Ln, g)-space [7], [8]. In (Ln,g)spaces with n = 4 the Lagrangian invariant L could be interpreted as the pressure p in a material system. This interpretation is based on the fact that for n = 4, the invariant [force] L has the same dimension as the pressure p : \p] = -, — r = 1X1 of a [2-surfacej physical system, described by its Lagrangian density L, i.e. L = \J—dg • p. It allows us to consider L as the pressure of a fluid. On this basis, we could formulate the following hypothesis: Hypothesis. The pressure p of a dynamical system could be identified with the Lagrangian invariant L of this system, i.e. p := L(gij,gij.k,gij.,k.l,VAB:VAB;i,VABj;j),

(1)

where g^ are the components of the metric tensor field of the space (or spacetime), where the system exists, and VAB are components of tensor fields describing the state of this system. gij-tk, gij-,k;i are the first and second covariant derivatives of the metric tensor g with respect to a covariant affine connection P and VAs-,i, VAB,i;j are the first and second covariant derivatives of the tensor fields V G ®fc;(Af) with respect to a contravariant affine connection T and a covariant affine connection P. The hypothesis could be considered from two different points of view: 1. If the pressure in a dynamical system is given, i.e. if a state equation of the type P •= P(gij, gij;k, gij;k;l,VA

B,VA

B;i,VA

B;i;j)

is given, then p could be identified with the Lagrangian invariant L of the system. On this basis a Lagrangian theory of a system with a given pressure p could be worked out.

192 2. If a Lagrangian invariant L of a Lagrangian density L = \J—dg • L is given, i.e. if L : = L(gij,gij;k,gij.k;i,V

B,V

BA,V

B-UJ)

is given, then L could be identified with t h e pressure p of the dynamical system. On these grounds a Lagrangian theory of fluids could be worked out with a given pressure p. In this case L := p. Therefore, we can distinguish two cases: Case 1. L := p with given p € Cr{M). Case 2. p := L with given L G Cr(M). In the present paper we consider a Lagrangian fluid mechanics by the use of t h e method of Lagrangians with covariant derivatives of t h e type (1). T h e Lagrangian invariant L :— p (considered as the pressure p) could depend on the velocity vector field u G T(M) of the points in the fluid, on other vector fields £ <E T(M) orthogonal (or not orthogonal) to u [g(u,£) = 0 or g(u,£) ^ 0], on scalar fields f^ = fN(xk) {N = 1, 2, . . . , m G N ) describing t h e s t a t e of t h e fluid as well as on t h e (first) covariant derivatives of t h e vector fields u and £. For a given in its explicit form Lagrangian density L and its Lagrangian invariant L respectively, we find the covariant Euler-Lagrange's equations and the corresponding energy-momentum tensors. 2

L a g r a n g i a n d e n s i t y a n d L a g r a n g i a n invariant

Let us consider a Lagrangian density of the type L = ^—dg grangian invariant L in the form: L:=p:=po

+ a0- p-e + bo • g(Vu(p

• u),u)

+ CQ • g(Vu(p

• L with La-

•0,0

(2)

h

+ /o • g(Vu{p

• 0 . « ) + o •
M0 • p

+ fcl • y O r + P l ( / j V , / / V , I , W * ) .

T h e quantities p0, oo, bo, c 0 , / o , ho, «o, ko, mo. Mo, and ki are constants, k, m, r are real numbers, JM := fN(xk) G Cr(M) are real functions identified as thermodynamical and kinematical variables, N G N . T h e function p = p{xk) is an invariant function with respect to the co-ordinates in M. T h e vector field u = ul • <9, G T(M) is a contravariant non-isotropic (non-null) vector field with g(u, u) := e : ^ 0. T h e vector field £ G T(M) is a contravariant vector field with # ( 0 0 :7^ 0 m t n e cases when KQ ^ 0, m 0 ^ 0, k0 ^ 0, M 0 ^ 0, /c.(a:fe) ^ 0 and fl(00 = 0 or 9 ( 0 0 = 0 if «o = m0 = k0 = M0 = 0.

193

The constants ao, bo, Co, /o, ho, «o, ko, and k\ could also be considered as Lagrangian multipliers to the corresponding constraints of 1. kind: ao : p- e = 0,

(3)

fro : g(Vu(p-u),u)

=0,

co : « 7 ( V u ( p - O , O = 0 , /o:5(Vu(p-0,«)=0, ho • # ( V „ ( p - u ) , 0 = ° ,

M0 • p u

' [
0,

Depending on the considered case the corresponding constants could be chosen to be or not to be equal to zero. 2.1

Representation of the Lagrangian invariant L in a useful for variations form

For finding out the Euler-Lagrange's equations one needs to represent the Lagrangian invariant L in a form, suitable for the application of the method of Lagrangians with covariant derivatives [9]. For this reason the pressure p = L could be written in the form + ki- pr + p- f + (up) -b,

P = Po+Pi{fN,ufN)

(4)

where f :=a0-e

+ bo- g(a, u) + c0 • ff(Vu£, 0 + /o • 0(V U £, u) + h0 • g(a, 0 Mo

TOQ

*°'y/MT)

+

°>MM5(00]™'

b : = b0 • e + c0 • g(£, 0 + (/o + ho) • I, a : = Vuu = u'-j • u^ = a1 • di,

I := g(u,£).

In a co-ordinate basis / , b, a, and I have the form: / = m • Wo • uk • ul + b0 • u fe ; m • urn • ul + c0 • £k.m • um • Zl + fo^k..m-um-ul

+

h0-uk,m-um-e]

, . (5)

194

b = gM • [b0 • uk • ul + co • Cfc • £' + (/o + ho) • uk • £']. Therefore, we can consider p, / , and b as functions of the field variables /JV, p, u, £, and g as well as of their corresponding first covariant derivatives. 3

Euler-Lagrange's equations for the variables on which the pressure p depends

We can apply now the method of Lagrangians with covariant derivatives to the explicit form of the pressure p and find the Euler-Lagrange's equations for the variables /JV, p, u, £, and g. After long (but not so complicated computations) the Euler-Lagrange's equations follow in the form: 1. Euler-Lagrange's equations for the thermodynamical functions /jy: dpi dfN

( dpi \ KdfNjJ.i

dpi +
(7)

where Qi = Tkik-l-9iri-gki.,i k

l

l

Tkl = g?-Tkt ,

+ glk-g?;i, l

Tkl = r\k~T ki,

^

(8)

=l t + / t

2. Euler-Lagrange's equation for the function p: ub=k1-r-pr~1+f

+ (q-6u)-b,

(9)

where ub = uk • b,k,

q = qi-ul,

Su = ul-ti = ul-k • gk.

(10)

3. Euler-Lagrange's equations for the contravariant vector field u: (ho-fo)-e.,k-uk=gv-[b-(\ogp),

'J

p UU,dmJ

fj

M0 + <^ 2 • [a0 - k0 • k • fc+1 [S(u,«)] •[
M

+ b0-[q-

8u + u(log p)] > - u%

+ {h0-(q-Su)+f0-[u(logp)}}-C

(11)

+ 9kl • [(bo • ul + h0 • £') • (ufc;, - 4 k

Jl

+ (co • e + /o • «') • Z ;j • 9 ]

m

•«'")•
195

-fl°'-0JE i m -« m -(&o-« f e + / * • £ * ) , where yjk;m

:

= ^j

• fk

• 9nl;m,

(12)

u(\og p) = W? • ( l o g p ) , , .

4. Euler-Lagrange's equations for the contravariant vector field £ (/o - h0) • a1 = [/o • (q - 8u) + h0 • u(logp)] • u1 + I Co • [q - Su + u(\og p)} m

o

K0 •

. „ ,

3/1 • + 2 •fco• ™ •

[»&01

M0 b(u, «)]*•[
ff7fcim-«mm-(co-r + /o-« fc )

r,4J . „

.'J . / i ^ ' •
(13)

• « • m • (co • £' + /o •«').

5. Euler-Lagrange's equations for the covariant metric tensor g: 2 P

b0 • (up) + p • ( a 0 - fco • fc

c0 • (wp) - p

KO

2

?™o

[s(£,0]

3/2

+ ko • m

M0 Mo fe

[s(«.u)] -[s(£.0] m+1

+ - • p • [bo • (a* • v? + a? • b%) + h0 • (a* • & + a? • C)

j

m

(14)

j

m

+ co • (?;m • e + i -,m • o • u + /o • (e ; m • « + e ; m • «*) • u ] + i - M - ( / o + ^o)-K-^ + ^ - r ) | 2

4 • ui • uj + B • C • e + I • p • Cij + i • (up) • Dlj

where Mo \9(u,u)]M-\9(Z,0Y 'Ko m0 , , M0 B = c0 • (up) ~~ p • +fco• rn •i+i 3/2 2 [s(^)] ' ^ '"* '\9(u,u)]k-m,Z)]mA 1 j j 1 j J &' = bo • (a • u + a • b*) + h0 • (a • £ + a • C) (15) ^4 = bo • (up) + p • ( a0 - k0 • k

+ co • (C:m • ¥ + ?im •€)• um + fo • (C-,m • uj + e,m • u') • um,

DI> = (fo + ho)-(u*-e +

uj-e).

196

The Euler-Lagrange's equations for the different variables are worth to be investigated in details in general as well as for every special case with a subset of constant different from zero. 3.1

Conditions for the pressure p which follow from the Euler-Lagrange's equations for the covariant metric field g

The Euler-Lagrange's equations (ELEs) for the metric field g lay down conditions to the form of the pressure p and its dependence on the other variables. The Euler-Lagrange's equations for g could be written in the general form as

After contracting with g^ and summarizing over i and j we obtain the condition dp dgij

,n 2

2 dp n dgtj

On the other side, if we contract the ELEs for p with £j = g3rn, • £ m or with Uj — gjn • un we obtain the following relations respectively: {A\ - gi) -e = 0,

(A'k - gl) • uk = 0,

(18)

where A\ = -2--^--g3m-f\-h\ P dgij

(19)

fmk are components of the contraction tensor Sr [7] and fmk • fml — gkSince 2

P

d

P ogi3

i

corn

it follows that Aik = g\ and therefore, the relations for C and ul are identically fulfilled. The only condition remaining for p follows in the form \po+Pi+h-Pr

n+2 3

m

°

(21) , ii, ,

i t\ i

+ P- 2 Ko • —?=== VgiU) + (k + m + 1) • Ko

M

°

[g(u,u)}^m,0Y

197

In the special case, when n = 4, we have P= 3 • (Po+Pi +h + P-

-2

• n0 •

• pr)

VWT)

+ 3- • (k + m + 1) • fco

[ff(«,«)]fc-^,0]rn (22)

By the use of the method of Lagrangians with covariant derivatives we can also find the corresponding energy-momentum tensors. 4

Energy-momentum tensors for a fluid with pressure p

The energy-momentum tensors for the given Lagrangian density L could be found by the use of the method of Lagrangians with covariant derivatives on the basis of the covariant Noether's identities [9] Fj + 0/;j = 0

first Noether's identity,

8i — sTiJ = Qt

(23)

second Noether's identity.

One has to distinguish three types of energy-momentum tensors: (a) generalized canonical energy-momentum tensor Oi ; (b) symmetric energymomentum tensor of Belinfante S T / , and (c) variational energy-momentum tensor of Euler-Lagrange Qi . All three energy-momentum tensors obey the second Noether's identity. After long computations we can find the energymomentum tensors. 4-1

Generalized canonical energy-momentum

tensor

mera The generalized energy-momentum tensor 0, could be obtained in the form dpi Q

,

fN,i + b-

p,i-u3

OfN,j

+ P-9ln-

K&O -Un + h0- C) • «';i • Uj + (CO • C + / o • « " ) • ?

;i

'"'I

k

+ 9jk • {(/o - ho) • (Zp + p-SZ) • ui • u - (/„ - ho) • (up + p-5u) • ? • uk + p-(ho-

/ o ) • [(? • ak + b> • uk) - (vP -dk + uk- d ' ) ]

+ P • (9ln;m + 9rn ' 9l,m)

+ \-(c0-C

+ h-un)-

+ g* • {uk .?»+um-Zk)-

• [(<*> • Un + h0 • C)

[gkl • («' -C~um-

' ( ^

' Uk • Um - glm

?)

glm • (u> • £fc + uk • £>')]]} - p.gj,

(24) • U* • Uk)

198

where ak = uk,t • ul,

Su = um,m, mm ikk

6£ = $'.„ kk

A

rk

£p = PiJ • ?

(25)

fk

k

v = e,m • u ,d = u ;l • ?,f = e-,n • r . 4-2

(26)

Symmetric energy-momentum tensor of Belinfante

The symmetric energy-momentum tensor of Belinfante sTi3 could be obtained in the form sTf = 9ik • {lbo -(up + p- Su) + (/ 0 - ho) -{£p + p- SO] • u> • uk + Co • (up + p • Su) • £j • £fc + h0 • (up + p- Su) • {uj • £k + uk • £j) + p-[b0- (uj • ak + uk • a3) + c0 • {& • bk

+£k-V)

+ /o • (uj • dk + uk • dj) + h0 • (? • ak + £k • a3) + h0 • (uJ -bk + uk -h3)-

ho • (uJ • dk + uk • dj)}

+ P • (9ln;m + 9nr • gl.m) ' [(k

+ gjl •uk-um-gml

-uj

' « " + /*> • C)

(27)

' (
-uk)

+ fo- un) • [gkl • (u3 • C1 + um • e)

+ \-(co-C

+ g* • (uk • r + um • e) - glm • w -e + uk- e m - P • g\. 4-3

Variational energy-momentum tensor of Euler-Lagrange

The variational energy-momentum tensor of Euler-Lagrange Q i could be obtained in the form 3 Ql

dpi =

du^-uJ+b'p>*'ul Kb0 • «" + h0 • e) • < i ' U3 + (C0 -C+f0-

+ p-9lnb

J

Un) • ^

k

3

,3}

•W

k

~ 9JU • i o -(up + p- Su) • u • u + co • (up + p- Su) • £ • £ + fo-(up + p- Su) • £j • uk + hQ • (up + p • Su) • u3 • ik j

k

k

j

k

k

+ p-[bo- (a • u + a • u ) + co • (V • £, + b • &) + /o • (V • uk + ak • £J) + h0 • (aj -£k + bk- u>)} • {(b0 • Un + h0 • C)

+ P • (9ln;m + 9rn • 9l;m)

+ (co-C +

n

kl

m

fo-u )-g -u -e\.

' 5 * ' ' « J ' ' «™

(28)

199 From the explicit form of the energy-momentum tensors and the second Noether's identity the relation dpi dul

, uJ

dpi djN.j

fN,i-

(29)

follows. 5

Special cases

T h e general form of the Lagrangian density L could be specialized for different from zero constants po, O,Q, bo, Co, / o , ho, KO, ko, mo, Mo, and fci. If only po, ao and ho are different from zero constants, then t h e corresponding Euler-Lagrange's equations and energy-momentum tensors describe a fluid with points moving on auto-parallel lines [10]. All more general cases are also worth t o be investigated. This will be the task of another paper. 6

Conclusion

In the present paper Lagrangian theory for fluids over (Ln, g)-spaces is worked out on the basis of the method of Lagrangiaus with covariant derivatives. A concrete Lagrangian density is proposed. T h e Euler-Lagrange's equations and t h e energy-momentum tensors are found. T h e y could be used for considering the motion of fluids and their kinematic characteristics. It is shown t h a t t h e description of fluids on the basis of the identification of their pressure with a Lagrangian invariant could simplify many problems in the fluids mechanics. On the other side, every classical field theory over spaces with affine connections and metrics could be considered as a concrete Lagrangian theory of a fluid with given pressure.

References 1. S. Manoff, Invariant projections of energy-momentum tensors for field theories-, in spaces with affine connection and metric, J. Math. Phys. 32(1991) 3, 728-734. 2. S. Manoff, R. Lazov, Invariant projections and covariant divergency of the energy-momentum tensors, in Aspects of Complex Analysis, Differential Geometry and Mathematical Physics, Eds. S. Dimiev, K. Sekigava, (World Scientific, Singapore 1999), pp. 289-314 [Extended version: Eprint (1999) gr-qc/99 07 085].

200

3. J.D. Brown, D. Marolf, Relativistic material reference system, Phys. Rev. D 53(1996) 4, 1835-1844. 4. A.H. Taub, General relativistic variational principle for perfect fluids, Phys. Rev. 94(1954) 6, 1468-1470. 5. B.F. Schutz Jr., Perfect fluids in General Relativity: Velocity potentials and a variational principle, Phys. Rev. D 2(1970) 12, 2762-2773. 6. B.F. Schutz, R. Sorkin, Variational aspect of relativistic field theories, with application of perfect fluids, Ann. of Phys. 107(1977) 1-2, 1-43. 7. S. Manoff, Spaces with contravariant and covariant affine connections and metrics, Physics of elementary particles and atomic nucleus (Physics of Particles and Nuclei) [Russian Edition: 30(1999) 5, 1211-1269], [English Edition: 30(1999) 5, 527-549). 8. , Lagrangian theory of tensor fields over spaces with contravariant and covariant affine connections and metrics and its applications to Einstein's theory of gravitation in V4-spac.es, Acta Appl. Math. 55(1999) 1, 51-125. 9. , Lagrangian formalism for tensor fields, in Topics in Complex Analysis, Differential Geometry and Mathematical Physics, eds. S. Dimiev, K. Sekigawa (World Sci. Publ., Singapore, 1997), pp. 177-218 10. , Auto-parallel equation as Euler-Lagrange's equation over spaces with affine connections and metrics, Gen. Rel. and Grav. 32(2000) 8, 1559-1582.

201

T R A N S F O R M A T I O N OF C O N N E C T E D N E S S E S GEORGIZLATANOV University of Plovdiv "Paissii Hilendarski", Faculty of Mathematics and Informatics, 24 Tsar Assen, 4000 Plovdiv, Bulgaria E-mail: [email protected] Let n independent fields of directions vl (a = 1, 2, . . . , n) and an affine connecta

edness 1F with coefficients 1Ttk be given in the differential manifold Xn. The net (D, v,..., v) defines the symmetric tensor a;,* = Y ] " _ , ViVs, where v„lvi- = 61.. 1 I n ' '—'a-\ K It is proved that for any net (v. v,. . . , v) £ Xn exists only one affine connectedness 1 2

n

2

F such that the mixed covariant derivative of Ojs with respect to iT and 2 F determinates a Weyl's connectedness with a fundamental tensor ais, a complementary vector Wfc = — -VilVfrVt, where 1 V/ t u* = d^-v' + 1V., vs. It is established that n

a

a

fv

^ a

3

F is a mean connectedness of 2 T and the mutual connectedness of 1 F . The transformation defined by !T and 2 T is called (v)-transformation and 1 T, 2 T are called (v)-corresponding connectednesses. Some properties of (v)transformation and transformation between a couple of mutual connectednesses as well as properties of some special nets in them are found.

1

Preliminary

Let diffrent affine connectednesses "F and 'T be given in the differential manifold Xn. The transfer from one to the other can be consider as a transformation of the connectednesses or as a transformation of the law of the parallelly translation. Denote the coefficients of both connectednesses by Tj 0 . and hT^. Then for the covariant derivative of an arbitrary field of directions vl in aT we have a i i a i a k k ka where

v v =d v + v v , "v^-'v^-^/, a

TL = T L - lrks

(2)

is the affine strain tensor. Let n independent fields of directions vl be given in Xn and let after renormalization they be transformed by the law v* = A - V , a

where A is a function of the point.

a

(3)

(i)

202

The reciprocal vectors u, are defined by the conditions

v^i=St

v^s = 6l.

n

a

(4)

Obviously Vi = Xvi.

(5)

The fields of directions vl (a = 1, 2, . . . , n) define a net (u, v,..., v) in Xn. a

1 2

n

Here as in the papers [3], [4] can be written the following derivative equations a

Vkvi

= aTkvi,

a

a

VkVi = -aTkVi.

a a

(6)

a

The net (v, v,..., v) is called geodesic if any its field of directions vl 1 2

n

a

(a = 1, 2, . . . , n) is geodesic (see i.e. [2], p. 315). The net (v, v,..., v) is called Chebyshevian of the second kind if any area 12

n

of elements (v,v,..., 1 2

v , v ,... ,v) is parallelly translated along the lines n

a —1 a + 1

(v) (a = 1,2, . . . , n ) [1]. a

As in [3], [4] we can prove that: The net (v,v,... ,v) e Xn is geodesic in the affine connectedness a r if 1 2

71

and only if a

Tkvk

= 0 for any a^a.

(7)

a a

The net (v, v,..., 12

v) G Xn is Chebishevian of the sec:ond kind in the affine n

connectedness aT if and only if a© k , Tkv = 0 for any a ^ a . (8) a © Note that in (8) and in the sequel summation over encircled indeces is not made. According to [3] the net (v, v,... ,v) G Xn is called generalized metrically 12

n

Chebyshevian in the affine coimectedness T if and only if a V[svk] = 0 for any a.

(9)

Let us consider the following tensors n

ais = y^Vii's,

n

a

ts

= V^ vlvs.

(10)

203

Because of (3), (5) after renormalization these tensors are transformed by the law ai3 = \-2ais,

ais = A'V S .

(11)

From (4), (10) follow diSvlvs = 1, alsViVs = 1; aisvlvs

= 0, a'sViVs = 0, for any a ^ /?;

(12)

a /3

a a

ais aik = <5sfc.

(13)

According to ([2], p. 138) for the mixed covariant derivative of the tensor a, s we have 3

V f c a l s = dkats - ^aps

- 2Tpksaip.

(14)

Since the tensor a,iS is symmetric and nondegenerated, then aj S determinates the connectedness 3 F (see i.e. [2], p. 132) such that 3

r ^ = -ark(di(irj

+ djOir - dratj)

- -ark(sViarj

(15)

+ 3 Vjair - 3 Wraij).

The tensors of the curvatures 1Rksi™ and aRksi™, the tensor of the torsion Slks of the connectedness XF satisfy the equality (see i.e. [2], p. 130) a

Rkai™ = lRksi™ + 21V[kaT$

+ 2aT(kn/p/aT?i

+ 2SpksaT™.

(16)

According to (see i.e. [2], p. 152) the condition 3

Vfcais = 2cokais

(17)

defines Weyl's connectedness with a fundamental tensor diS and a complementary covector ujk. After renormalization of ais by the formula (11) u)k is transformed by the law &k = wfc + dklnX.

(18)

The connectednesses °F and 1T which coefficients satisfy the condition (see i.e. [2], p. 127) a

r L = lT\k

(19)

are called mutual. According to ([2], p. 127) for the mutual connectednesses are fulfilled a a

ns

r L = lns + 2SL= arks - J rL = 25*fc.

(20) (21)

204

Then the connection (16) for 1Rksi* and aRksil- can be written in the form a

= lRks-

Rksil

- ^V[kSi}l

+ ASZS\m.

(22)

The connectedness T , which coefficients satisfy the condition (see i.e. [2], p. 127) c

rL = ^(1rL + °rL)

(23)

is called a mean connectedness of lT and aY. Obviously the mean connectedness of two mutual connectednesses is symmetric. 2

(v)-Corresponding Affine Connectedness

Theorem 1. / / an affine connectedness lT is given in Xn, then for any net (v,v,... ,v) G Xn there exists only one affine connectedness 2 F, such that the 12

n

mixed covariant derivative (14) to satisfy the condition (17), where tok = --lTk. n a

(24)

Proof. From (2), (14) and equality 1 Vkais = dkaiS — lTpkiaps — lYpksa,iP we find 3

Vkazs

= lVkais

- 2Ti°saip.

(25)

With the help of (6) we establish the following representation of uk uk = --%lVkvj. n a After the renormalization (3), (5) for tbk we find tik =

-IA^V^A"

n - LUk + dkln\.

(26)

V) = --Xvii-dkX^v* + X~uVkvl) a n a a

(27)

Hence 3 F, generated by the fundamental tensor (10) and the complementary covector (24) is Weyl's connectedness. • According to (13), (17), (25) we obtain 2

T£ = 'V^sa™

- 2uk8?,

(28)

2 1 Tl Tk s

1

which means that the affine strain tensor is defined uniquely by T^J and the tensors ais, ojk, connected with the net (u, v,..., v). So from (28) we find 1 2

the only possible connectedness 2 ^Ts = ^Z + lVkalsaim

n

- 2uk5?.

(29)

205

Using (14), (15) we establish Remark. of 1T.

3

r is the mean connectedness of 2 r and the mutual connectedness

Definition 1. Any transformation between the affine connectednesses XF, 2 r , generated by the net (v, v,..., v) and satisfying the condition (29) will be 1 2

n

called (v)-transformation. In this case the couple of the connectednesses 1F and 2 r will be called (v)-corresponding and 3 r-associated with the net. Theorem 2. / / two affine connectednesses, belonging to Xn are (v)corresponding then the tensors of their curvatures and the coefficients from the derivative equations (6) satisfy the equalities: 2

Tk = -xTk a

for any

a ± 0,

(30)

0 2

Tk = lTk,

(31)

2

Rsk,l = lRskil. Proof. Because of (10), (4), (6) we can write X

Vkats = ^ 2 E lTkZ{.j;s)l

(32)

aimvsvm

7=1 a

= vW.

a 0

0

Then from (1), (4), (28), (33), we find 2Tk

- lTk

a

a

l

Vkalsamv4m

- 2cokdlnvsvm

(33)

a 0

=

2

lTsvsvm

= - E lTk(5°08Z + P08aa) - 2uk5'i =

0

=

a

-1Tk-

l

Tk - 2wk6£ or 2

Tk = ~lTk~2ujk80a. a

(34)

0

When a ^ (3 from (34) it follows (30). When a = /? from (24), (34) it follows (31). a

Taking into account (10), (6) and (33) we obtain 1Vfcai.9aiS = —2X7V a

Then with the help of (28), (24) we establish 2T§S = 0. So from (16) immediately it follows (32). . •

206

Corollary 1. Let rF and 2T be (v)- corresponding affine connectednesses, belonging to Xn and 3 T be the associated affine connectedness with the net (v,v,... ,v). The following conditions are equivalent: 1 2

n

(i) The connectedness XT is equiaffine; (ii) The connectedness 2T is equiaffine; (Hi) The connectedness 3T is Riemannian. From Theorem 2, (7), (8) it follows Corollary 2.

The net (v, v,..., v) € Xn is geodesic in the one of the couple n

1 2

of the (v)-corresponding affine connectednesses, belonging to Xn if and only\ if this net is Chebyshevian of the second kind in the other connectedness. 3

Mutual Connectedness

Let J T and 4 r be mutual connectednesses and 5 T be their mean connectedness. From (1), (6), (23) we obtain 5

Tk = -CTk+4Tk), a

Z

(4Tfc - lTkW a

a

(35) a

OL

= 4Tlmvm.

(36)

a

a

0

a

After contraction of (36) consecutively by u*, v, and taking into account (4) we find

4k - lTk = 4 T ^ < = 2SlmkvmL a

ct

a 4

Tk -lTk <x

Theorem 3. 4

a

(37)

a

= 4Tki = 2S\k.

(38)

a

The coefficients from the derivative equations (6) satisfy the 4

a

ot

a

conditions T\k Ts\ — 1T\kTs\

= 0.

Proof. Applying the integrability condition for (6), (a = 1, 4) we establish a

Rksmivm

- 2SJS°V m t; i = 2aV[k°Ts]vm

+ 2aT[k<-Ts]v>

After contraction by v^ in view of (4), we obtain

a = 1,4.

(39)

207 4

Rksj-1Rkai}

+ 2S%a(4Tp + 1Tp) a

a"

a

(40)

a

=2( 4 V l f c 4 r s ] - ^ [ / t 1 ^ ] ) + 2 4 T [ f c 4 s ] - 2 1 T [ , 1 T s ] . According to (1), (21), (38) it follows 4Vk4Ts a

25L 4 T S , from where we obtain 4Vk4Ts a

- lVk4Ts a

= ^k4Ts

a

= -4TlksATs

+ 2SL 4 T,: = ^ . ( ' T , + ' n

a

2SPS) + 2S« s ( 1 T i + 2S£) = ' V i ' T , - 2 1 V*S? p + 2S< s 1 T i + 4Sj,S£ or 4

V[k4Ts]

1

-

= a

V[k4Ts] = - 2 1 V [ , ^ ] p + 2Si,1T, + 4SJ 8 S£

a

(41)

Now from (38), (40), (41) it follows 4

RkJ

- lRkJ

= ASl.sSppi - 4 1 V [ i .^ ] l + 24T[k4Ts] - 21T[k1Ts].

So from (22), (42) it follows the validity of the Theorem 3.

(42) •

Since S%mk is an antisymmetric tensor, then from (37) we obtain 4

Tkvk

- lTkvh

a a

= 2S'mkvmvkv,

a a

a

= 0.

(43)

a

From (7), (8), (23), (43) it follows: Proposition 1. The folloing conditions are equivalent: (i) The net (v, v,..., v) G Xn is geodesic in the connectedness J T; 12

(ii) The net (v,v,..., 1 2

n

v) G Xn is geodesic in the connectedness 4T\ n

(Hi) The net (v, v,..., v) € Xn is geodesic in the connectedness 5T. 12

n

From (23), (8) it follows Proposition 2.

/ / the net (v,v,..., 12

v) € Xn is Chebishevian of the second n

kind in two of the connectednesses "T (a = 1, 4, 5), then it is Chebyshevian of the second kind in the third connectedness, too. Theorem 4. / / the net (v, v,..., v) € Xn is Chebishevian of the second kind 12

n

in two of the connectednesses aT (a = 1, 4, 5), then the tensors of the torsion

208

Sf.s and the curvatures 4i?fcsm?, 1Rksml- satisfy the conditions S& = 0, 4

(44)

x

Rksil

= iW-

(45)

Proof. Let us accept that the net (v, v,..., v) e Xn is Chebyshevian of the 12

n

second kind in the connectednesses XT and 4 I \ Then using (8), (37) we obtain ® ® @ TkVk - TkVk = 2S1 kvmvkVk = 0, from where it follows the equality a @

a @

a

@

SnA'k

= ^m,

(46)

a ® where A are arbitrary functions of the point.

a

According to (4), (46) 5^ f c = Xvm. Henc

k

e S

mk

But vm is an arbitrary covector.

= 0.

•

Because of (22), (44) it follows (45). Theorem 5.

/ / the net (v, v,..., v) G Xn is generalized metrically Chebishe12

n

vian in two of the connectednesses aT (a = 1, 4, 5), then these connectednesses are symmetric. Proof. It is easy to check that if the net is generalized metrically Chebyshevian in two of the connectednesses T (a = 1, 4, 5), then it is generalized metrically Chebyshevian and in the third connectedness. From (1) when a = 4 and (9) we find 4T^k\-, = 0. Applying this condition in (21) we obtain 5[" = 0, from where it follows that lT and 4 r are symmetric connectednesses. • References 1. 2. 3. 4.

A. Liber, Proceedings of Sem. Vect. and Tens. Anal. 2(1984), 185-199. A. Norden, Affinely Connected Space, Monographes, Moscow, 1976. G. Zlatanov and B. Tsareva, Journal of Geometry 55(1996), 192-201. G. Zlatanov, Journal of Geometry 38(1995), 234-245.

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