I N S T I T U T E F O R H I G H E N E R G Y P H Y S I C S

И Ф В Э 8 3 - 1 4 1

О Т Ф

M.V.Saveliev

CLASSIFICATION PROBLEM

FOR EXACTLY INTEGRABLE EMBEDDINGS

OF TWO-DIMENSIONAL MANIFOLDS AND COEFFICIENTS

OF THE THIRD FUNDAMENTAL FORMS

Serpukhov 1983

M.V.Saveliev

CLASSIFICATION PROBLEMFOR EXACTLY INTEGRABLE EMBEDDINGS

OF TWO-DIMENSIONAL MANIFOLDS AND COEFFICIENTSOF THE THIRD FUNDAMENTAL FORMS

M - 24

AbstractSoveliev II.V.

Classification Problem for Exactly Integrable Sibeddings of Two-Dinensional Manifolds andCoefficients of the Third Fundamental Forms. Serpukhov, 1983.

p. 20. (IHBP 83-341)Refs. 20.A method i s proposed for c lass i f i ca t ion of exactly and completely integrable embeddings

of two dimensional manifolds into Riemannian or non-Riemannian enveloping space, which arebased on the algebraic approach''"'8'' to the integration of nonlinear dynamical systems. Herethe grading conditions and spectral structure of the Lax-pair operators taking the values ina graded Lie algebra that pick out the integrable c lass of nonlinear systems are formulatedin terras of a structure of the 3-d fundamental form tensors. Corresponding to every embeddingof three-dimensional subalgebra s ( (2) into a simple finite-dimensional (infinite-dimensionalof f i n i t e growth) Lie algebra £ i s a def inite c lass of exactly (completely) integrable еж-beddings of two dimensional manifold into the corresponding enveloping space supplied withthe structure of £ .

АннотацияСавельев М.В.

Задача классификации точно интегрируемых вложений двумерных многообразий и коэффициенты третьихфундаментальных форм. Серпухов, 1983.

20 стр. (ИФВЭ ОТФ 83-141) .Библиогр. 15.Предложен метод классификации точно и вполне интегрируемых вложении в римановы или веримановы

объемлющие пространства, в основе которого лежит алгебраический п о д х о д ' " 1 " ' к интегрированию нели-нейных динамических систем. При этом градуировочные условия и спектральный состав операторов Лакса.принимающих значения в градуированной алгебре Ли, выделяющие интегрируемые классы двумерных систем,формулируются в терминах структуры тензоров третьих фундаментальных форм. В рамках данного методакаждому вложению трехмерной подалгебры sE(2) в простую конечномерную (бесконечномерную конечногороста) алгебру Ли £ сопоставляется определенные класс точно (вполне) интегрируемых вложений двумер-ного многообразия в соответствующее объемлющее пространство, снабженное структурой £ .

(Институт физики высоких энергий, 1 9 8 3 .

1. A classical problem of geometry is a classification of mani-

folds embedded in this or that fashion into enveloping Riemannian

or non-Riemannian space. According to the embedding theorems of

differential geometry/*"^/ a manifold Vn is defined up to the motion

in the enveloping space VJJ as a whole by its fundamental tensors

satisfying the Gauss, Peterson-Codazzi and Ricci equations (somehow

generalized). Thus, one of the major aspects of the constructive

solution of the problem involved consists in the investigation of

these equations with a view to pick out their exactly integrable

subclasses, classify the latter and find the pertinent solu-

tions in the sense of the Cauchy or Goursat problem. We have to do

the same for subclasses which are completely integrable in the sen-

se of the inverse scattering problem and they contain the appropri-

ate spectrum of soliton-type solutions. Then the characteristics

of enveloping space (particularly its connections at the points

of ^n) enter in the equations for the tensors of V^ as "external"

functions. For this reason it appears to be convenient to consider

the triple of embedded manifolds Vn С VJJ С SJJ for the seix-consis-

tent justification of the problem where the Riemannian curvature

tensor of sjg (or its generalization in non-Riemannian geometry) has

zero components.

Prom the technical point of view the problems encountered are

concerned with the following. The equations under consideration rep-

resent a very complicated system of nonlinear partial differential

equations with the arbitrary choice of- the (pseudo-) normals to the

manifold Vn as well as the parameters on it.(In this connection it is

also important to reformulate the equations in question in terms of

the reduced functions, the latter possessing no arbitrariness menti-

oned above and serving for the gauge invariant quantities.) Apart

from that, even with the adequate methods for their solution, an

approach is to be worked out to identify exactly or completely in-

tegrable subclasses of manifolds.

Quite similar to the geometrical problem discussed is the fully

solved problem of the classification of simple Lie algebras and the

description of all their semisimple subalgebras. Thus it is natural

to expect the application of the algebraic notions and methods in

this or that form to be adequate to the solution of the embeddingproblem of differential geometry. Recently it was established'0»''that the treatment of the classification problem of simple Lie al-gebras £ and the description of the embeddings of three-dimensionalsubalgebras in £ are closely linked with the problem of picking outexactly and completely integrable two-dimensional nonlinear systems.It is this relationship that serves as a hint to the possibility ofa constructive solution of the differential geometry problem con-cerned in ref.'S/. (The concept in question in fact goes back to thepioneering idea conceived by S.Lie со the effect that the continuoustransformation groups have a decisive significance for the integ-ration of partial differential equations and mean the same for thelatter as the Galois groups for the algebraic equations.)

Witb>the algebraic approach/6>8/ to the construction of exactlyand completely integrable nonlinear systems the starting point is a

Lie algebra £ and the choice of some of its grading, £ = 2 Ф £_ ,

£,.„• da = dim£a<°° (for infinite-dimensional algebrasM + =°°). These notions are introduced with the realization of theLax-type representation

[d/dz+ + A+, d/dz_+A_] =0 (1.1)

Here the Lax-pair operators take the values in £ , A+(z+, z_) t£ ,

and have a definite spectral structure, namely, A+ i 2 ® £ + , m +<M +.

At the same time differential geometry starts with the Gauss, Peter-son-Codazzi and Ricci equations themselves. Here the characteristicsof VJJ (to be more exact, the structure of a Lie group with which VJJis supplied) play the role of £ introduction, while the choice ofgrading and spectral structure of the Lax-pair provide the condi-tions for the components of the fundamental tensors of VQ ' . Hereto reveal the geometrical implication of these conditions is ofcrucial importance, so are the notions of grading and spectral stru-cture. This is what this paper is eonoerned with. As will be seenbelow, the appropriate geometrical objects are the third fundamentalforms whose matrix structure adequately expresses the required al-gebraic conditions.

2. In this section following the classical monograph by Eisen-hart' ' we briefly give the necessary information from the theoryof Kiemannian manifolds as well as certain definitions of non-Rie-mannian geometry'^»**'.

Consider Riemannian space V N of an arbitrary dimension N with

coordinates yA and metric G^g(y) = Gg^ (which is generally indefi-nite) consistent with the symmetric connection run. An embedding

of Riemannian manifold V with coordinates x1 into V

N is determined

by an analytical dependence yA = f

A(x*) with matrix df^/dx*^^ of

rank n.Here*'

A

where n are components of the vector fields in VJJ normal to Vn at

its points, g .(x) is the metric in V ; the numbers ea are equal to

+1 or -1. Then V_ is described by its fundamental tensors g..(x) ="n ,ab.у its fundamental tensors g..Iba ...

JJ.I. а __^ ob ,.-„__ It= q

a

±, t*

u(x) = -t£a with q±j and t°

D being the com-

ponents of the second quadratic forms and torsion vectors, respec-

tively. These functions satisfy the Gauss, Peterson-Codazzi and

Ricci equations,

Here КД т у т

, is the Riemannian curvature tensor for the metric G.B

defined at the points of V ; semicolon denotes the covariant diffe-

rentiation with respect to the metric gj.. The equations represent

the integrability conditions of the Gauss-Weingarten derivative for-

mulae for the vectors у . and n" in the moving frame,

A _ a A ^A В С

(2 3)A a jk A . ba A pA с В

na - i

= "^iiS У к

+ ек*ч

ni, "

Гпг-У ч

пяa

» •*• 1

J fK b z b ВС ,1 a.

(In what follows the factors e will be omitted for the sake of

brevity.) It is obvious that the corresponding equations for mani-

folds supplied with a structure of an arbitrary Lie group (not only

orthogonal) can be written down analogously.

In the classical theory of surfaces apart from the first and se-

cond fundamental quadratic forms

Q2

5 "

GA B

D y A

^FTOB now on the a i a u t l o n convention i s used tor a l l types of indices: i ,J,k,f (u,v.p.r)*1 , . . . , n; A,B,C,D( A, B.C, »)«=],. . . .N; «,b,o,d=«+l,.. .,H; E , A , 0 , ¥ • « + ] , . . .,lf+p7 в S ,

S> 1 Я + >0

the third fundamental form

< = GA]K'

Dnb - <<***

Ьц - tJ

CtJ

b)dxW. (2.4)

is sometimes applied. Similar to the two forms mentioned above theother one also allows the direct geometrical interpretation andplays an important role in what follows.

Equations (2.2) and (2.3) are very much simplified for the embed-dings of the manifold V

n into a flat space SJJ, N>n+p (p is a class

index of Vn) . They can be completely formulated in terms of V

n and

naturally do not contain the terms with the Riemannian tensor andthe connection of V

N. For this reason it often appears to be conve-

nient to consider the triple of spaces Vn С V

N С S^

+p (with

subsequent reduction) for the investigation of properties of theembeddings of V

n into Riemannian enveloping space V^ (not necessa-

rily flat) and for the solution of their classification problem*^.The calculations foi the investigation of the embeddings (VgC Vjy,in particular) are provided by the following formulae on the baseof the triple V

n С V

NC S

N + p. Let us introduce the repere E^ (y) in

an enveloping space VN,

V ^8 - 8

t e (2.5)

and thereby construct the "spin connections" у = - У

A A

***'*** +Г*В*

11Л-ТР

В** = О. (2.6)

It is easy to show that the Riemannian tensor in V™ is expressedvia these connections

Then t h e embeddings of V i n t o VN a l low t h e following r e d u c t i o n

^ J B = УГ *f*f- </***? + *?* a5b' < 2 ' 8 )

which is equivalent to the derivative formulae (2.3) for V СУ„ dueto (2.6). Here y^ v(x) and e^(*) are analogous to y^ and E"» quan-t i t ies , respectively, defined in V .

ПIn the case of non-Riemannian spaces V с V the derivative for-A

n " Дmulae for the vectors у . and pseudo-normals n* are essentially

,2 a

complicated as well as the integrability equations resulting fromthem. The latter generalize eqs. (2.2). They contain three typesof new tensors'2'

a a A B * „ i A i B . b A b B , „ „..

* Apparently in algebraic terms the transition from V to S,, is equivalent to a contrac-tion of the corresponding algebra.

я

and are expressed byA

a A - A B Cy

V L y y

А „ЗА В

Ъ А -А С В ( 2 Л 0 )

na,i

where the symbol "; " denotes the covariant differentiation withrespect to the connection l&. in V ; L~~ is the connection in V».,both being asymmetrical, generally speaking. When the determinantof <of,- is not; equal to zero and the pseudo^normals can be chosenin such a way that £ai = 0, the tensors g

1J (nondegenerated with

det £i Ф 0) and g., . are defined from the relations g^ou, = £*, ,ak !J JK ак

sk*

I n t h i s context the tensors g

i . and g

l J

are used only for raising and lowering the indices of the tensor quan-

tities of Vn. Here a variant with L ^ = L ^ - L ^ + L ^ I ^ -

— £ — A= 0 represents an analogue of a flat enveloping space of

Riemannian geometry. In what follows we shall call, in brief, thespaces V»T with zero RADrT4 flat. In terms of the tensor G*

B and con-nections Lgp and LgQ realizing the covariant differentiation of co-

and contravariant tensors, respectively, the Weyl and Schouten gene-ralizations (see, e.g.'2»**/) of Riemannian spaces can be expressedby the introduction of three tensors in V N

GA!J в GAB, Li, - l£B s (An, 4c -

LBC = G AC' (2

-П)

,G (3)

G ВС **В (2

ВС ВС ВС (а!>

ВСequal to zero in the Riemannian case. Here, defining in V

N the repere

and spin connections by formulae (2.5) and (2.6), one gets convin-ced that the symmetrical parts of these connections are expressedby the formula

4W ?

(2Л2)

3. We shall use the algebraic method of constructive investiga-tion of the two-dimensional exactly and completely integrable nonli-near systems' » ' for the study of the embedding problem. The methodis based on the representation theory of Lie algebras or superalgeb-ras £ and the Lax-type representation (1.1) in two-dimensional space(z+, z_) realized by the operators A + taking the values in the sub-

m+ -

SDaces 2 ® £ of a graded algebra £ . In a specific case when A.о +a

+

— л ~~

take the values in the local part £ = £ _ 3 e £ o

ф £ + 3 of a finite-di-

mensional Lie algebra £(m + = 1) supplied with a grading consistent

with the integral embeddings of three-dimensional subalgebra A^=

= I H,J+; [H, J+] = + 2J+, [J+l J_] = Hi in £ there emerges the

following system of equations' ':

dig"1 dg/dz_)/dz+ = [J_, g"3J+g]-. (3.1)

Here g is an element of the complex hull of the Lie group GQ withthe Lie algebra £ o. The system allows an explicit solution of theGoursat problem characterized by 2d^ arbitrary functions Ф

+ (z

+)

and <A_a(

z_). l<a ^ dj.

The present approach to the completely integrable systems is con-

nected with a realization of the Lax operators in the corresponding

subspaces of infinite-dimensional Lie algebra £ of finite growth' 1 4Л

Here the existence of finite-dimensional representations of £ pro-vides the possibility of nontrivial introduction of a spectral para-meter and construction of the soliton-type solutions by the inversescattering method (see, i.e./-10') or by the invariant method sugges-ted inref. / U / . In this situation the choice of a definite spectral

~ m± ~structure of the operators A. in £ , A + f 2

a £ is equivalent in_ — о +^

a certain sense to their realization, on the elements of the corres-

ponding finite-dimensional Lie algebra £,which form a degenerate(finite-dimensional) representation of the generators of £ +

0 < a <; m+.Note representation (1.1) is form-invariant with respect

to the gauge transformation

К - g ;1^ +

g;1ag

o/az+ (3.2)^ ; a go/

л -*which conserves the spectral structure of the operators A at g e G

Q

due to the condition [£ . £ 1 c£ o. ~

The algebraic approach is quite applicable to the study of theembedding problem specifically for two-dimensional manifolds of dif-ferential geometry because up to now it failed to be extended tothe systems in spaces with more than two dimensions. (Nor couldthe approach mentioned be extended to other methods for integration

""Aof nonlinear equations). Here if the tensor L^— of the enveloping

space Vjy is non-zero one has to consider as an intermediate stepthe triple of spaces V 2 С V N

С sfM with zero components of the curva-

ture tensor of the space Щ^. That is why in the future we shall con-

fine ourselves to the case of a flat enveloping space V"N referring

the reader to ref./9/ for the details of the modifications of reaso-

ning concerning the embeddings into non-flat space VJJ.

The applied representation (1.1) is the synonym, for the- zero cur-

vature of enveloping space. Here the operators &+ are the reduced

connections of type (2.8) conjugated with the elements М д в of the

Л D A (В., г.

orthogonal algebra in Riemannian case, A+ = У

f+Y

D Tl Д $, where

differentiation is performed over the variables z^ = l/2(xj + ix2).

Therefore for the effective application of the algebraic approach

in the investigation of the erabeddings of two-dimensional manifolds

V2

C S supplied with the structure of Lie algebra £ the indicial

system of equations (2.2) with n = 2 and R A B C D = 0 (or itsgeneralization in non-Riemannian case) should be reformula-ted in the form of representation (1.1) with the helpof the explicit realization of the operators A + inthe graded Lie algebra £ • Here the aforementioned arbitrari-ness in the choice of (pseudo-) normals to V 2 and the parameters onit is clearly related to transformation (3.2). The selection ofvarious types of exactly integrable embeddings is realized in theframework of the algebraic approach by choosing the spectral struc-ture of the operators A + in the algebra £ , whose grading is consis-tent with the integral embeddings of subalgebra Aj in £ . As weshall see, this leads to the conditions for the coefficient func-tions at the elements of £ in A+. The latter define (up to thetransformations of the Weyl group of £) the structure of the 3-dfundamental forms. Evidently it is necessary to realize the opera-tors A+ by the Cartan elements and the root vectors of £ rather thanwith the help of the tensor basis like M A to provide the possi-bility to consider the embeddings of manifolds supplied with thestructure of various Lie algebras £ . In the next Section we illust-rate the afore-said by the example of the embeddings V,С S

N, i.e.

for £ E 0(N).Another types of the simple Lie algebras can be considered inde-

pendently or by the reduction over the simple roots of D or B^.forexample, D^ -» B3 -> G2. For these algebras, as well as

I74=773 пЗ

= Я1

for the series Вд and Dn,there is a complete Dynkin's classification

of the embeddings of three-dimensional subalgebras.The study of the

embeddings of non-Riemannian manifolds follows the completely analo-

gical scheme and demands the use of noncompact real forms.In particu-

lar, non-Riemannian generalization with non-zero tensor G£ can be

obtained the basis of the algebra SL(N, E ) , with the symmetri-cal parts ui the connections %p ® (2.12) being conjugated to the

symmetrical tangent matrices of this algebra.

4. Consider an embedding of two-dimensional Riemannian manifoldV 2 into the flat Euclidean enveloping space S™ and realize its zerocurvature representation (1.1) with the help of the operators X+ == A + (z+, z_)M д jg taking the values in the orthogonal Lie aTgeb-

ra of rank [N/2] with the elements M д в = -M jg д satisfying the com-

mutation relations

[ MA < B '

MC D

] -

SA«C

MHD

+ SB C

MA C -^BC^AD -

gAD

MiBC.

( 4 Л )

(Note the consideration of pseudo-Euclidean enveloping spaces is

carried out quite analogically. The only difference being that the

Kronecker symbols in (4.1) are supplied with the factors e

A. The

latter express the pseudo-Euclidean character of the metric of SN

and, correspondingly, the pseudo-orthogonality of the algebra.)

It would be convenient to come over to the complex structures

A IBin order to formulate the functions A

+ (z

+, z_) in terms of the

coefficients of the fundamental forms of a manifold V2 depending

on the parameters x^ = i ~-*z + (-i)J~-*z_. Then using relations

(4.1) one can easily verify by a direct check that the substitution

of the operators A+ = A

+ ™ А. в

w i th

in representation (1.1) leads to the Gauss, Peterson-Codazzi and

Ricci eqs. (2.2) for an embedding V2C S

N < Here the d/dx±+l/2y£4b =

= д.+ у. Hi realizes the covariant differentiation of the corres-1 1

12 д A iB

ponding tensor quantities. (Note, the operator д/ду +1/2 у. М * _,

plays the same role for the embeddings VN C S

N + as the previous exp-

ression for V2C Sj.).

The operators A can be equivalently expanded over the elementsEAB _ _

EBA 1

of Lie algebra satisfying commutation relations (4.1), where thesymbols <5 |g are replaced by

AB _ ij А В ab А В

лRewrite the expression for the operators A+ in terms of the Car

tan-Weyl invariant root basis. Due to the orthogonal algebras of

even (N = 2£) and odd (N = 20+1) dimensions corresponding to diffe-rent classical series (D« and Bn, respectively) consider firstthe even case following the notations of ref .'^' . The series D^are described by the extended Dynkin diagram

10

"2 " 3

-М £_2

Here -M = -(»7i+ 2 2

i (£ >4). (4 .3)

"£-з "£-2 ^ ^"I

по + по ) i s t h e minimal root, тг are

the simpls roots. The Cartan generators пд and the root vectors X~

corresponding to the simple roots satisfy the commutation relations

[V / , 6

] =°'

[V

XP = ±

k| a 4 '

[X«'

X^] m*afi

ha' <

4'

4)

with к =k being the Cartan matrix of £ = Dp . Denote the rootvectors of all positive and negative roots of the algebra via Xy -,1 Л < t(l-l). Consider in the root space (R the canonical basis

(€a), where na = ед - *

a+1, 1 4 a< ^-1, я--=

е

г + €

£т and all positive

roots are given by formulae _

e - e o = 2 n , 1 <a<$4 i;£ -2

s« = 2 n + n , 1y= a ' l

£ - 1 ;

In these notations the operators A have the form

A £ a _ "ЯA^ = 2 U+ h + 2 (4 .5)

where the trace is calculated over the indices i,j = 1,2 of the

matricesX

= e

/ 8 (4 .6)

/S

2a-l,2fi-l 2oг т

.2a, 2/8-v =

o-l 2y+l,2y+2 21,U* = - i [ 2 2 t , + У L ] , 14a < ? -2 ;

_ y = ! — J.

Ua= i[2t+

£-1 2(£-y)+l,2(£-y)+2

- l

-1/2

11

„,, a

A Aa

The functions r+ = -r are

|o,

г+Ъ = q+je

jM , 4

b = t f , p = 2,2; 3«a,b«2e, 1< A^ 2? (4.7)

Here the 3-d fundamental forms (2,4) are expressed via functions

(4.7) or via matrices (4.6) as

%, в2 = +, a = 2a + i - 2 , b=2/3 +j-2; i,j = 1,2.

To single out the exactly integrable subclasses of system (2.2) re-

sulting from the substitution of operators (4.5) in representation

(1.1) consider, in accordance with the general construction of

ref .'**', the gradings of the algebra Dp that are consistent with

the integral embeddings of three-dimensional subalgebra. Here the

Cartan element H of an embedding is, in fact, the grading operator,

[H, £а] = 2a f

a. Since the Dynkin's procedure'

13/ for the descrip-

tion of all types of embeddings is uniform for all finite-dimensio-

nal simple Lie algebras £ , we remind briefly the main steps of theprocedure for an arbitrary £ .

Each embedding is unambiguously defined (up to equivalence) bythe decomposition structure of the Cartan element H of the subal-

lgebra A- over the generators h of the algebra £ , H = 1 С h ,

I = rank £.In their turn the unknown structure constants Ca of theembedding are determined by the following scheme. Consider the ex-tended Dynkin's diagram for £ and select its subset P^s^ = fp-,, ..., ps;

l ! of any roots corresponding to some semisimple Lie subalgebraof rank 2g^E of the initial algebra £ . Then the unknown ele-

± ±ment H = 1 С h

n is found from the condition [H, X ]= + 2X

<z=l a Pa Pa - Pa

Here the embedding of A., in £ .where [H, X~] = + 2X^, l<a^ I ,plays a peculiar role. The corresponding 3d-subalgebra is calledthe principal 3d-subalgebra. Really, the description of all embed-dings A^cf is reduced to the consideration of the principal 3d-

subalgebras in all algebras £*-s' . (The exceptions are [ ] embed-

dings for the series D. and [ ] for E» * , I ' - 6,7,8,) Therefore

the structure of H for other embeddings (non-principal) is estab-lished from this condition written in the form

12

+ + — -[Н, Х~] = +2Х~, 1^ а £ s^e ; [Н, Х+]=+2Хм (4.9)

for l ^ s ^ £ and from the subsequent reduction. Formula (4.9)leads to

• - H ( S ) - Лл

+ S d]g >/d|f,where d are the decomposition coefficients of the root Л over the

I Asimple roots, Л = 2 d^r . Thus, taking into account the relation

a=1 a a

± I A A ±

[H, Хд] =±2( 2 d - d v )Хд , we come to the following structure

of the local part £ of the algebra £ with the grading H :: J(4.11,)

Here i t seems natural that the case s=0 corresponds to the canonical grading (the principal subalgebra Aj of £ ), i . e .

£o = {ha, 3 <C a4l 1 , £+ 1 = { Х д» 1 « а « П . (4.11

For the case of the series Dp under consideration formulae(4.11^) lead to

; XM;

±1 "у " - -в е_2 t

2 «r/+ 2 »

y=a ' y=(-—a

H a $ min(s, t-s-1)},

jo, s = i, t-i, г

. s * i, 2

<:s <£-2

or

£ = j h , l«y«£ ; f XT ; fgX1 , l«a<min(s, 2 -в)},У s

ГЧ ea+*£-a+l+ + + (4.122)

2

As can be easily seen after excluding the corresponding set ofpoints in scheme (4.3), some of these gradings are evidently equi-valent.

Retain in expansion (4.5) only the terms with the elements ofthe subspaces £ e £ + J and £ e £ . (for the operators A and A_,respectively).Then taking into account formulas (4.12) we obtain therequired relations for the quantities r ^ from (4.7). Let us enume-rate all nonequivalent possibilities.For the case of even Ve wehave

«/8 _ R a /3 Ra/3 _+i2 - R +i2 5

a ?_i 5 /3£ R+22 ~ 0>

2 ^s^ [---]:

R+12

According to the definition, the functions Ra\. are constructed

from R ^ J via the formula

*.. (4.14)

14

To consider the odd values of £ one has to make the replacement

R+jl~ R+J2 i n <4 Л З ) a n d

<4-

14>-

The solution of the same problem for the orthogonal algebrasof odd dimensions is obtained by equating the simple roots тт. andTT in the root system of D. . This procedure corresponds to

the transition from (4.3) to the w-system for the series Bg,

-M

t

-Ms-»- -2 21 fii /8=2*18

or. in terms of the coefficient functions, to the equalities

r+ ~ ' = 0 .

Thus, is accordance with expression (4.8), the classificationof exactly integrable embeddings of V

2 into S^ is defined by the

structure of the 3-d fundamental forms with nonzero coefficientfunctions (4.13) and (4.14). It is via nonzero elements of the mat-rix Qo'

3 that system (3.1) describing these embeddings can be formu-

lated. For this to be accomplished, one has to reduce expansion(4.5) to the forn/

8 /

A+ = g~

1J

+g, A*_ = J_ + g-

3<9g/(5z_ (4.16)

by expressing the group parameters of the element g e Go through the

coefficients at the differentials (2.4). Here the form-invarianceof (1.1) with respect to transformation (3.2) with g e G is applied.

Let us illustrate the general conclusions by the examples of non-trivial embeddings of V

2 into the enveloping spaces Sg and S

5 rela-

ted to the algebras Bj(^A,) and B~ and then trace the reduction tothe case of the enveloping space supplied with the structure of thealgebra G .

Вд: For this case formula (4.5) is reduced to the form

A*. = u-H + f^X* + frX", (4.17)

— ± +

-j+ + 12

where f+ = q

+.e

J , ±z = q

+.e , u~ = i/2y . (Here and in what fol-

lows the inessential normalizing factors at the root vectors will

be omitted for the sake of brevity,) The condition that the coef-

ficients at X~ in A+ and at X

+ in A_ be equal to zero, i.e. f* =

= f~ = 0 , leads to

ki -1Hence, qik£ = qil-det(q e). In the invariant form this conditionis written as

or Q4 = expo (dxJ) . (4.18)

15

Putting (4.17) with f* = f~ = 0 into representation (1.3), we come

to the system of equations

u,+

z_ - u~

z+ = f+ Г, (in f+)

fs_ = -2u", (fn O

> z += 2ut

As a consequence of the latter the function p satisfies the Liouvil-

le equation p = 2 exp p . With the conformally-flat metric,

_2 i

z+

z-

gk£ = ^ \pexPp» ^ = const, condition (4.18) in terms of thetensor Ы = b* = q

n tg J takes the form b.b. = X S Then it fol-

lows that the exactly integrable embedding described by the Liouvil-

le equation corresponds either to the minimal surface with 2 Ы = 0

or to the constant curvature surface with Ы = AS1?- ^

For the Tchebycheff metric the constant curvature surface is

well known to be described by the (completely integrable) sine-Gor-

don equation p z z

= 2exp p-2exp(-p). To obtain this equation in

the framework of the given construction (see Section I) one has to

take into account all terms in (4.17) imposing on them the condi-

tions f+f^ = flf+ = 1. The latter correspond to the canonical gra-

ding of the infinite-dimensional Lie algebra A-j of f inite growth'

in its degenerated representation (with зГ- = X , XZ = X+: Л-, =

<w 1 Z A

= -h2 = H) and they are equivalent to

5 i 2(expp-l)] or (4.19)

Q3 = expp(dx3)2 +(expp-3)(dx2)2.

л

В„: Here the operators A+ have the form

+ + ± + + - ± + ± - ± +A j_ = 1ГТП,- + Urst ln + I , - | A i "*" 1 -]A^ + I . «Ar t "f" 1 O"O "*" ^ l O ^ I O "*"

J J[ & & ТА J ^^ J Л "г<5 ^ — Ля Л AZ A £t

f f-12X12 + f122X122 + f-122X122' ( 4 - 2 0 )

where+ з 4 i - +• 3 4 i +

f~ = (q . + iq .)e , T~ = (q . - iq .)e ;

f , =(q . - i q 4 )e^ , Г „=(q . + iq .)e ;122 +j + j -122 J;J _+j

fj = t» + it«. Q = tf - it? ;

f1 - a 5 e j ~ f1 - a 5 e J + - u 1 - -2У3 2

*12 " q + j e ' -32 " q + i e ' u l " 2У+ >l+3 ' -32 ^±3

The indices at the root vectors denote the number of the correspon-

ding simple roots я, and n over which the given root is expanded.

For the case under consideration we obtain two nonequivalent gra-

dings consistent with the integral embeddings of A in Bo. The lo-

3 л

16

cal parts for these gradings are written (up to the Weyl transfor-mations) as

+ +

= i h r h2; x2, x~2\, £ + 1 ={jq\ jq2, x122" V ^ ' " J 2 '

In the former case corresponding to the exactly integrable sys-

2 -2tem (the Toda lattice for Bo)

2

, = к a exp p

n, к = I

_ «P /3 1-1 2

the conditions for the spectral structure of A+

f122

= f-122

= f12

= f-12

= f+l

= f+2

= f-l

= f-2

lead to

3 k£ 3 . 4 k£ 44 e * = 5 e x p ' = q g q

In the latter case corresponding to the string-type exactly integ-rable system^-15'

sh[(Pl - P2)/2]

3 ' Z + Z ~ 3 ch38-P] - p 2 ) / 2 ] ' Z + * Z -

sh[(p3 - P2) /2]

2 > Z + Z - 2 ch3[(p3 - p 2 ) / 2 ] ' + ' ~

j 2 z 2_ 3 2 , Z J , Z + = 0,

the conditions for the spectral structure of A +

f-3 ~ f+l = f-12 = f+12 = f-122 = f+122 = °

lead tok4a (4-23)Here the functions P/a\(z , z_) are related to the solutions of the

string-type equations given above.

G2: The operators A + realizing the embedding of two-dimensional

surface into the flat 1зрасе supplied with the structure of the

37

group G_ can be constructed with the corresponding operators for

VOC S

7 by equating the roots in /г-system of B

o (n = я- ). This pro-

£ * о о Лcedure is equivalent to the conditions

"*'7; a » 3,5; 0= +;

7.eJ(-^ = tf - tf - i*(tf

+ tf); tf + t?

6 =

and leads to the following expression

i± = -

i yf h, - ±<

y« + 2 t

3 4) h 2 + (q

3_ .

^ .,e xf1 H + t t^ + t lOtt? - t«

4 t f - t f + i*(tf +t«)]z«12.

(Here we use the same notations as in formula (4.20) for the indi-

ces of the simple roots of G^) The subsequent singling out of

exactly integrable embeddings follows completely the general scheme

described above.

5. Summing up briefly the main conclusions. A method is sugges-

ted for the classification of exactly and completely integrable em-

beddings of manifolds V into Riemannian or non-Riemannian envelo-

ping spaces VJT. However, it appears constructive but for two-dimen-

sional manifolds (n=2) due to the absence of effective methods for

the integration of nonlinear systems in spaces of more than two di-

mensions (n> 2). The classification is based on the algebraic

approach to the integration of nonlinear dynamical systems'®»**' as-

sociated via the Lax-type representation (zero curvature of envelo-

ping space) with the Lie algebras £ supplied with a grading con-sistent with embeddings of three-dimensional subalgebra s£(2) in£.Here the coefficient functions of the expansion of the Lax opera-tors over the elements of £ are expressed in a certain way via thetensors of the principal fundamental forms. Therefore, the choiceof the grading of £ and that of the spectral structure of the ope-rators result in the Gauss, Peterson-Codazzi and Ricci equations(in the integrable sector involved). They also result in the requi-red classification of manifolds and consequently establish the geo-metrical meaning of the latter in terms of the structure of the 3-dfundamental forms.

Specially for the embeddings of two-dimensional Riemannian sur-faces V 2 into the flat enveloping space Sj we present Table (4.13)

18

which expresses the conditions on the coefficients defining the 3-dfundamental forms. The latter correspond to every nonequivalent em-bedding of s£(2) into the orthogonal-type algebra of rank [N/2].

Apparently, as is seen from the examples presented at the endof the previous section, the requirements on these coefficients canbe clearly and directly expressed through the eigen values of thematrix of the 3-d forms. For this purpose the gauge arbitrariness(3.2) is to be effectively applied.

Finishing up my paper I should like to express acknowledgementto my worthy colleagues profs. A.T.Fomenko,A.N.Leznov and Yu.I.Ma-nin for their helpful discussions.

19

REFERENCES

3. Eisenhart L.P. - Riemannian Geometry. Princeton, Uhiv. Press.,

1926.

2. Eisenhart L.P. - Non-Rieraannian Geometry.Amer.Math. Soc. Collo-

quium Publ., vol VIII, 1927.

3. Kagan V.F. - Riemann's Geometric Ideas and Their Modern Deve-

lopment, Moscow, 3933 . (In Russian).

4. Rashewsky P.K. - Riemannian Geometry and the Tensor Analysis,

Nauka, Moscow, 1964. (In Russian); Dubrovin B.A., Novikov S.P.,

Fomenko A.T. - Modern Geometry, Nauka, Moscow, 1979.(In Russian);

KobayasM S. ,Nomizu K.- Foundations of Differential Geometry,

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Phys., 51, 10-18, 1982 (in Russian).

8. Leznov A.N., Saveliev M.V. - Comm. Math. Phys., 89, 59-75, 3983.

9. Gabeskiria M.A., Saveliev M.V. Preprint IHEP 83-58, Serpukhov,

1983.

10. Manakov S.V., Novikov S.P., Pitaevsky L.P., Zakharov V.E. - The

Soliton Theory: Inverse Scattering Method. Nauka, Moscow,

1980.(In Russian). Faddeev L.D. - Sovrem. Prob. Mat., 3,

93, 3974. (In Russian).

11. Leznov A.N. Preprint IHEP 83-60, Serpukhov, 1983.

32. Bourbaki N. Groupes et algebres de Lie. Paris, Hermann, 3968.

13. Dynkin E.B. - Mat. Sb., 30, 349-462, 1952 (in Russian); Trudi

MMO 1, 3952. (In Russian).

34. Kac. V.G. - Math. USSR - Izv 2, 1271-1311, 1968. (in Russian).

15. Barbashov B.M., Nesterenko V.V., Cherviakov A.M. - Comm. Math.

Phys., £4, 471-486, 1982.

Received 11,July, 3983.

20

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