Perspectives of Complex Analysis, Differential Geometry and Mathematical Physics: Proceedings of the 5th International Workshop on Complex Structures and ... St. Konstantin, Bulgaria, 3-9 September 2000

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

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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.

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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 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{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 — . . . — 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 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 = — 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

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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