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On the geometry of submanifolds in En2n.

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УЧЕНЫЕ ЗАПИСКИ КАЗАНСКОО ОСУДАСТВЕННОО УНИВЕСИТЕТА
Физико-математические науки
2009
Том 151, кн. 4
UDK 514.76
ON THE GEOMETRY OF SUBMANIFOLDS IN E
n
2n
S. Haroutunian
Abstrat
A speial lass of 2m -dimensional submanifolds in a 2n -dimensional pseudo-Eulidean
spae with metri of signature (n, n) , known as a pseudo-Eulidean Rashevsky spae, is studied.
For suh submanifolds, anonial integrals and parametri equations are found.
Key words: even-dimensional submanifolds, pseudo-Eulidean Rashevsky spae, double
ber bundle, anonial integral, dierential-geometri struture, bration, foliation.
One of the most harateristi features of modern Dierential Geometry is the ative
appliation of its methods in the adjoining elds of the mathematial siene. Essentially
inreased eetiveness of these methods whih aumulated fundamental ahievements,
rst of all from the general algebra and theory of dierential equations, in ombination
with the tendeny to onsider various mathematial objets as dierential-geometri
strutures on orresponding manifolds, has led, on the one hand, to the appearane of
new diretions of the dierential geometri study, and, on the other hand, to a more
fundamental, geometri interpretation of these objets.
The next step is the dierential geometri analysis of these strutures and identiation of their most general harateristi (geometri) properties. Finally, on the last stage
of researh, these properties or their part beome the foundation for generalizations and
new problems in the initial theory.
Moreover, in aordane with [1?, they assume in partiular the exat desription of
the ategory of the strutures under study and also the identiation of the ategory of
algebrai systems neessary for their study.
All above mentioned is true for the geometry of multiple integral depending on
parameters. The study of dierential geometri strutures dened by suh an integral
on the manifold of integration variables and parameters to a ertain extent is similar to
the study of the integral geometry [24?. At the same time the presene of parameters
totally hanges the yle of arising problems and orresponding results. By systemati
study of multiple integrals depending on parameters (in a speial ase when the number
of parameters is equal to the number of variables) and orresponding integral transforms
one an see a good number of interesting geometrial problems onneted with the
desription of invariant properties of suh integrals.
The present artile is devoted to the study of a speial lass of 2m -dimensional submanifolds with struture of double ber bundle in the 2n -dimensional pseudo-Eulidean
n
spae E2n
with metri of index n . We nd multiple integral depending on parameters,
determining the struture of suh a submanifold on the orresponding manifold of integration variables and parameters, also parametri equations of this submanifold.
216
S. HAROUTUNIAN
1. Pseudo-Riemannian Rashevsky spae
In 1925, Russian geometer P.A. Shirokov from Kazan State University introdued [5?
the speial lass of even-dimensional symmetri spaes known as A -spaes or ellipti
A -spaes. In 1933, E. K
ahler [6? studied the same spaes known now as Kahler spaes.
Let us onsider a 2n -dimensional manifold M with loal oordinates x1 , . . . , xn ,
y1 , . . . , yn suh that in all admissible transformations of oordinates two sets of n oordinates are separated: the transformed oordinates x1 , . . . , xn are funtions of x1 , . . . , xn
and the same is true for the seond set of oordinates. Consider a real kern funtion
U (x1 , . . . , xn , y1 , . . . , yn ) and introdue the following values
gji =
?2U
.
?xj ?yi
It is easy to hek that the matrix
G=
0
gij
gji
0
is invariant under all admissible transformations of loal oordinates. This matrix is
nondegenerate and, therefore, its elements introdue a metri on M . In its turn this
metri generates an ane onnetion on M . On this manifold, the bers from dierent
families are omplex onjugate.
The so alled hyperboli ase when both the families of bers are real n -dimensional
manifolds was introdued by P.K. Rashevsky [7?. He studied an invariant salar eld
U (x1 , . . . , xn , y1 , . . . , yn ) with nondegenerate matrix of seond order derivatives:
2
? U
det
6= 0
?xj ?yi
and, using this matrix, introdued a pseudo-Riemannian metri of index n on M and
the orresponding pseudo-Riemannian onnetion. This spae is known as a Rashevsky
pseudo-Riemannian spae. It has the following harateristi properties.
1. The salar eld U (x1 , . . . , xn , y1 , . . . , yn ) generating the struture of a pseudoRiemannian spae on M is determined with arbitrariness
U (xi , yj ) ?? U (xi , yj ) + U1 (xi ) + U2 (yj ).
2. Eah point of M belongs to one and only one ber from eah of the two families
of bers. Fibers from dierent families have intersetion in no more than one point.
3. The bers of both the families are isotropi.
4. The bers of eah family have the property of absolute parallelism (auto parallelism): vetors tangent to bers from one of the families remain tangent to them after
parallel transfer along an arbitrary smooth urve.
It follows from eah of the two latter properties that both the families of bers are
totally geodesi in M .
This spae was studied by P.K. Rashevsky and other researhers as an example of
a pseudo-Riemannian spae only, without any relations to other elds of Mathematis
and Physis.
Later, in 60-th, professor V.V. Vishnevsky from Kazan State University introdued
the third type of A -spaes (paraboli A -spaes) [8? and ompleted the lassiation of
these strutures.
In terms of a o-basis of linear dierential forms ? 1 , . . . , ? n , ?1 , . . . , ?n adapted to
the struture of 2n -dimensional Rashevsky spae, the struture equations of this spae
N
ON THE GEOMETRY OF SUBMANIFOLDS IN E2N
217
an be presented in the form [9?
I
d? I = ?K
? ?K ,
d?I = ??IK ? ?K ,
I, K, P, T = 1, . . . , n
(1.1)
I
P
IT
d?K
= ?PI ? ?K
+ RKP
? P ? ?T ,
IT
where RKP
are the nonzero omponents of the urvature tensor. The metri of this
spae is generated by the nondegenerate bilinear losed form [9?
d? = ? I ? ?I .
(1.2)
It is known [9? that an integral of the form ?? 1 ? . . . ? ? n indues a struture of
a 2n -dimensional Rashevsky spae on the manifold M of integration variables and
parameters under natural ondition of nondegeneray for the matrix of seond order
derivatives of the funtion ln ? .
Rashevsky spae an also be onsidered as a double bered manifold with two families of n -dimensional transverse geodesi bers. It is a generalization of the ross produt
of two manifolds: a (2n + s) -dimensional smooth manifold M is said to be a double
ber bundle if two smooth mappings
?i : M ?? Mi ,
i = 1, 2,
from M onto n - and n + s -dimensional smooth manifolds M1 and M2 are given, the
bers, i. e., full preimages of points from M1 and M2 under the mappings ?1 and
?2 respetively are smooth n + s - and n -dimensional submanifolds, and the tangent
spaes to the bers of the bundles ?1 and ?2 at an arbitrary point have only trivial
intersetion:
TP ?1?1 (x) ? TP ?2?1 (y) = p, ?1 (p) = x, ?2 (p) = y, x ? M1 , y ? M2 .
Therefore, the tangent spae of M at an arbitrary point is a diret sum of n + s and n -dimensional subspaes. The ase of Rashevsky spaes orresponds to s = 0 .
If the urvature tensor of suh a spae is trivial, we have a pseudo-Eulidean Rashevsky spae whih, in terms of a o-frame of prinipal exterior linear dierential forms
n
? 1 , . . . , ? n , ?1 , . . . , ?n adapted to the struture of a double ber bundle on E2n
and
n
dened on the prinipal ber bundle of tangent frames on E2n , an be presented by
the following struture equations [9?
I
d? I = ?K
? ?K ,
d?I = ??IK ? ?K ,
I
d?K
=
?PI
?
I, K, P = 1, . . . , n
(1.3)
P
?K
,
I
n
are dened on the manifold T 2 E2n
of seond order
where the seondary forms ?K
n
tangent frames assoiated to E2n and do not depend on the prinipal forms. There is
a natural onnetion between suh spaes and the Fourier transform [9?: this integral
n
transform invariantly indues a struture of E2n
on the double bered manifold of
integration variables and parameters M .
An (n + s) -tuple integral depending on n parameters is said to be a anonial integral of a dierential-geometri struture on a 2n + s -dimensional manifold M if this
integral generates the struture on M . Suppose that n -tuple integral depending on n
parameters generates a struture of a Rashevsky 2n -dimensional spae on a manifold
218
S. HAROUTUNIAN
of integration variables and parameters. An n -tuple integral depending on n parameters
onstruted on parameters of integration generates the same struture of a Rashevsky
2n -dimensional spae on the manifold of integration variables and parameters M if and
only if M is an Einstein spae (Rii tensor is proportional to the metri tensor with
onstant oeient). This is the geometrial meaning of the invertibility of the orresponding integral transform. This means that the ategory of Einstein 2n -dimensional
spaes with metri of signature (n, n) is the most general one for the onstrution of
invertible integral transforms. One of the most important problems here is ndind of
an integral transform generating the struture of a given Rashevsky (Einstein) spae
on M .
2. 2m -dimensional submanifolds with struture
of double ber bundle in a pseudo-Eulidean spae En2n
n
The neessity to onsider submanifolds of the pseudo-Eulidean spaes E2n
and
orresponding anonial integrals follows from the problem of nding anonial integrals
of Rashevsky (Einstein) spaes beause in the speial ase when the spae under study is
pseudo-Eulidean (the urvature is equal to zero) the orresponding anonial integral
oinides with the lassial Fourier transform. Taking into aount that an integral
generates the orresponding dierential-geometri struture in an invariant way and
that the geometry of a pseudo-Riemannian spae, in general, is dened by its urvature
tensor, it is natural to suppose that the anonial integral of a Rashevsky (Einstein)
spae is related to the urvature tensor of this spae in a speial way.
Let us onsider 2m -dimensional submanifold M with struture of a double ber
n
bundle in an 2n -dimensional pseudo-Eulidean spae E2n
with metri of index n
(pseudo-Eulidean Rashevsky spae) when the dimension n satises the ondition
2m > n .
Suppose that, in terms of a o-basis of linear dierential forms ? 1 , ? 2 , . . . , ? n , ?1 ,
n
?2 , . . . , ?n adapted to the struture of a pseudo-Eulidean Rashevsky spae E2n
, the
struture equations of the spae are represented in the form (1.3) and that a 2m dimensional submanifold M is dened by the equations
? m+i = ?2m?n+i ,
?m+i = ? i ,
i = 1, . . . n ? m.
(2.1)
Let us note that relations (2.1) determine the most general lass of 2m -dimensional
n
submanifolds in E2n
with struture of a double ber bundle. This lass is a diret
generalization of the orresponding lasses of submanifolds of odimension two, studied
in [10, 11?.
There are three possible ases: 1) 2(n ? m) > m or 3m < 2n , 2) 2(n ? m) = m or
3m = 2n , 3) 2(n ? m) < m or 3m > 2n .
The ase 3) was studied in [12?. Let us study the ase 3m < 2n . Therefore the
following inequalities hold
3n < 6m < 4n.
This ondition is equivalent to the inequality 2m ? n < n ? m . Let us introdue
new indies ? = 1, . . . , 2m ? n ; ? = 2m ? n + 1, . . . , n ? m ; a = n ? m + 1, . . . , m . The
n
metri form of the total Rashevsky spae E2n
dened by the invariant bilinear losed
I
nondegenerate form d? = ? ? ?I indues the bilinear form
?
a
d?? = ? ? ? ?? + ? ? ? ?? + ? a ? ?a + ?2m?n+?
?? ? ? ? + ?2m?n+?
?a ? ? ?
on M .
(2.2)
N
ON THE GEOMETRY OF SUBMANIFOLDS IN E2N
219
Substitution of relations (2.1) into (1.3) and appliation of the above introdued
indies gives the following general struture equations of a submanifold M
?
d? ? = ??? ? ? ? + ??? ? ? ? + ?a? ? ? a + ?m+k
? ?2m?n+k ,
?
d? ? = ??? ? ? ? + ??? ? ? ? + ?a? ? ? a + ?m+k
? ?2m?n+k ,
a
? ?2m?n+k ,
d? a = ?ba ? ? b + ??a ? ? ? + ??a ? ? ? + ?m+k
d?? = ???? ? ?? ? ??? ? ?? ? ??a ? ?a ? ??m+k ? ? k ,
d?? = ???? ? ?? ? ??? ? ?? ? ??a ? ?a ? ??m+k ? ? k ,
d?a = ??ab ? ?b + ?a? ? ?? ? ?a? ? ?? ? ?am+k ? ? k ,
?
d??? = ??? ? ??? + ??? ? ??? + ?a? ? ??a + ?m+k
? ??m+k ,
?
d??? = ?µ? ? ??µ + ??? ? ??? + ?a? ? ??a + ?m+k
? ??m+k ,
d?ba = ?ca ? ?bc + ??a ? ?b? + ??a ?
?b?
a
+ ?m+k
?
(2.3)
?bm+k ,
?
d??? = ??? ? ??? + ??? ? ??? + ?a? ? ??a + ?m+k
? ??m+k ,
?
d?a? = ??? ? ?a? + ??? ? ?a? + ?ca ? ?ac + ?m+k
? ?am+k ,
?
d??? = ??? ? ??? + ??? ? ??? + ??? ? ??a + ?m+k
? ??m+k ,
?
d?a? = ??? ? ?a? + ??? ? ?a? + ?b? ? ?ab + ?m+k
? ?am+k ,
a
d??a = ??a ? ??? + ??a ? ??? + ?ba ? ??b + ?m+k
? ??m+k ,
a
d??a = ??a ? ??? + ??a ? ??? + ?ba ? ??b + ?m+i
? ??m+i ,
?
?
where the seondary forms ??? , ??? , ?ba , ??? , ?a? , ??? , ?a? , ??a , ??a and ?m+k
, ?m+k
,
m+k
a
m+k
m+k
(2)
?m+k , ?? , ??
, ?a
are dened on the manifold T M of seond order tangent
frames assoiated to the manifold M and adapted to its struture.
Taking into aount that the bilinear form d?? is losed and using exterior dierentiation of (2.2) with appliation of general struture equations (2.3), we arrive at the iden?
?
tity whih shows that, in the general ase, the forms ??? , ?a? , ?2m?n+?
??? ??2m?n+?
??? ,
?
?
a
a
a
b
?2m?n+?
??? ??2m?n+?
??a , ?2m?n+?
??? ??2m?n+?
?a? , ?2m?n+?
??? ??2m?n+?
?ba are prinipal. We will use these general onditions for more detailed researh of the dierentialgeometri struture on M .
Taking into aount that the submanifold M has a struture of a double ber bundle,
i. e., that the following systems of linear dierential equations
? ? = 0,
? ? = 0,
? a = 0,
? = 1, . . . , 2m ? n,
?? = 0,
?? = 0,
? = 2m ? n + 1, . . . , n ? m,
a = n ? m + 1, . . . , m;
? = 2m ? n + 1, . . . , n ? m,
a = n ? m + 1, . . . , m
?a = 0,
? = 1, . . . , 2m ? n,
are totally integrable, we arrive at the following system of identities
?
?m+k
? ?2m?n+k = 0,
??m+k ? ? k = 0,
?
?m+k
? ?2m?n+k = 0,
??m+k ? ? k = 0,
a
?m+k
? ?2m?n+k = 0,
?am+k ? ? k = 0.
(2.4)
220
S. HAROUTUNIAN
?
Applying Cartan's lemma [1?, one an easily see that the seondary forms ?m+i
,
?
m+i
a
m+i
m+i
?m+i , ?m+i and ?? , ?? , ?a
are linear ombinations of the basi linear dierential forms ?? , ?a , ? = 2m ? n + 1, . . . , n ? m , a = n ? m + 1, . . . , m and ? ? , ? ? ,
? = 1, . . . , 2m ? n , ? = 2m ? n + 1, . . . , n ? m respetively.
On the other hand, the exterior dierentiation of relations (2.1), whih are identities
on M , gives the following two identities
m+k
?
?km+i + ?2m?n+i
? ? k + ?am+i ? ? a + ?2m?n+i
? ?? +
?
m+i
m+i
a
?n?m+?
+ ?2m?n+i
? ?? + ?n?m+a
+ ?2m?n+i
? ?a = 0,
m+k
?
?m+i
+ ?ki ? ? k + ?ai ? ? a + ?m+i
? ?? +
?
i
i
a
+ ?n?m+?
+ ?m+i
? ?? + ?n?m+a
+ ?m+i
? ?a = 0.
Taking into aount identities (2.4), it is easy to hek that this system is equivalent
to the system of the following four identities
m+k
?km+i + ?2m?n+i
? ? k + ?am+i ? ? a = 0,
?
m+i
m+i
?
a
?2m?n+i
? ?? + ?n?m+?
+ ?2m?n+i
? ?? + ?n?m+a
+ ?2m?n+i
? ?a = 0,
(2.5)
m+k
?m+i
+ ?ki ? ? k + ?ai ? ? a = 0,
?
?
i
i
a
?m+i
? ?? + ?n?m+?
+ ?m+i
? ?? + ?n?m+a
+ ?m+i
? ?a = 0.
It follows diretly from the obtained system that all the seondary forms ?a? are
equal to zero identially. Indeed it's follows from the third identity from (2.5) that the
seondary forms ?a? are linear ombinations of the basi prinipal dierential forms ? 1 ,
? 2 , . . . , ? n . But it is easy to see from the seond identity of system (2.5) that the same
forms have expansions in terms of the basi prinipal dierential forms ?1 , ?2 , . . . , ?n
only. This is possible if and only if the forms ?a? are equal to zero.
?
Besides the rst identity of system (2.4) shows that the seondary forms ?m+i
have nontrivial expansions in terms of the basi prinipal forms ?2m?n+1 , . . . , ?n only.
Substituting the orresponding expansions into the fourth identity of system (2.5), we
?
see that the seondary forms ?m+i
are vanishing.
Let us note now that, as follows from the last identity of system (2.4), the seondary
forms ?am+i are linear ombinations of the basi prinipal forms ? 1 , ? 2 , . . . , ? n?m .
Substitution of the orresponding expansions into the rst identity of system (2.5)
shows that all the forms ?am+i are vanishing too.
Using relations (2.4), it is easy to hek that system of identities (2.5) is equivalent
to the following system
?kn?m+a ? ? k = 0,
?kn?m+? + ??m+k ? ? k = 0,
n?m+?
??? ? ?? + ?n?m+?
+ ??? ? ?? = 0,
?
n?m+a
a
?n?m+?
+ ?a? ? ?? = 0, ?m+?
? ?? + ?m+?
? ?a = 0,
(2.5? )
m+k
?m+?
+ ?k? ? ? k + ?a? ? ? a = 0,
?
?
?
a
?n?m+?
+ ?m+?
? ?? + ?n?m+a
+ ?m+?
? ?a = 0.
221
N
ON THE GEOMETRY OF SUBMANIFOLDS IN E2N
Exterior dierentiation of relation (2.2) shows that the seondary forms ?a? , ??? are
prinipal forms. The appliation of the seond and third identities of system (2.5? ) gives
the following expansions
??? = C??? ?? + C??? ?? ,
(2.6)
?
?
?
?a? = Ca?
? ? + C??
? ? + Cab
?b
with the symmetry onditions on the oeients orresponding to Cartan's lemma:
?
?
C??? = C??? , Cab
= Cba
.
Applying Cartan's lemma [1? to identities of the rst and the last identities of system
(2.5), we obtain the following expansions
m+i ?
m+i ?
??m+i = C??
? + C??
? ,
m+i
m+i
C??
= C??
,
m+i ?
m+i ?
??m+i = C??
? + C??
? ,
m+i
m+i
C??
= C??
,
?
??
?a
?m+i
= Cm+i
?? + Cm+i
?a ,
??
??
Cm+i
= Cm+i
,
?a
a
ab
?m+i
= Cm+i
?? + Cm+i
?b ,
ab
ba
Cm+i
= Cm+i
.
(2.7)
Exterior dierentiation of expansions (2.6), (2.7) with further appliation of the
general struture equations of a submanifold M gives the following dierential identities
dC??? ? C??? ??? ? C??? ??? + C??? ??? ? ?? +
+ dC??? ? C??? ??? ? C??? ??? + C??? ??? ? C??? ??? ??? ?C??? ??a ??a ?C??? ??a ??a = 0,
?
? c
? ?
?
?
?
dCab
+ Cac
?bc + Ccb
?a ? Cab
?? + Ca?
?b? + Ca?
?b? ? ? b +
?
?
?
?
?
+ dCa?
+ Ca?
??? + Cb?
?ab ? Ca?
??? + Cab
??b ? ? ? +
?
?
?
?
µ ?
+ dCa?
+ Caµ
??µ + Cb?
?ab ? Ca?
?µ + Cab
??b ? ? ? = 0,
m+i
m+i ?
m+k m+i
m+i ?
m+i ?
m+i ?
?m+k + C??
?? + C??
?? ? ? ? +
?? + C??
?? ? C??
dC??
+ C??
m+i ??
m+i
m+i ?
m+k m+i
m+i ?
m+i ?
?m+k + C??
?? + C??
C? ?? ? ? ? = 0,
+ dC??
+ C??
?? + C??
?? ? C??
(2.8)
m+i
m+i ?
m+k m+i
m+i ?
m+i ??
m+i ?
dC??
+ C??
?? + C??
?? ? C??
?m+k + C??
?? + C??
C? ?? ? ? ? +
m+i
m+i µ
m+k m+i
m+i µ
m+i ??
m+i ??
+ dC??
+ C?µ
?? + Cµ?
?? ? C??
?m+k + C??
C? ?? + C??
C? ?? ?? ? = 0,
?a
? b
?
?a
??
??
µ?
??
m+k
dCm+i
? Cm+i
??? ? Cm+i
?µ? + Cm+k
?m+i
? Cm+i
Cab
? ? Cab
Cm+i
? b ? ?? +
?a
?b
?a
?a
??
?
m+k
ca
+ dCm+i
? Cm+i
?ba ? Cm+i
??? + Cm+k
?m+i
? Cm+i
??a ? Ccb
Cm+i
? b ? ?a = 0,
?a
?b
?a
?a
??
?
m+k
ca
dCm+i
? Cm+i
?ba ? Cm+i
??? + Cm+k
?m+i
? Cm+i
??a ? Ccb
Cm+i
? b ? ?? +
?a
?b
m+k
ab
ac
cb
ab
+ dCm+i
? Cm+i
?cb ? Cm+i
?ca + Cm+k
?m+i
? Cm+i
??b ? Cm+i
??a ? ?b = 0.
222
S. HAROUTUNIAN
?
It follows from rst two identities of this system that quantities Cab
and C??? are invariants and therefore their vanishing has an invariant geometri meaning. For example,
if C??? = 0 , then the system of linear dierential equations ? ? = 0 , ? = 1, . . . , 2m ? n
?
is totally integrable; the ondition Cab
= 0 haraterizes the total integrability of the
system of Pfa equations ?a = 0 , a = n ? m + 1, . . . , m .
??
m+i
m+i
Next identities of the system (2.8) show that the quantities C??
, Cm+i
, C??
,
ab
Cm+i are invariants, and the other quantities ourring in this system are not invariants.
?a
m+i
Therefore, without any loss of generality, the quantities C??
, Cm+i
an be onsidered
equal to zero.
There are three possible ases:
a) 2m ? n < 2n ? 3m , i. e., 5m < 3n , therefore 15n < 30m < 18n ,
b) 2m ? n = 2n ? 3m , i. e., 5m = 3n , therefore 15n < 30m = 18n ,
) 2m ? n > 2n ? 3m , i. e., 5m > 3n , therefore 18n < 30m < 20n .
Let us onsider the ase 15n < 30m = 18n . We note that, by virtue of the
fat that the ranges of the indies ? = 1, . . . , 2m ? n ; ? = 2m ? n + 1, . . . , n ? m ;
a = n ? m + 1, . . . , m are of the same length, the dimension m is divisible by 3.
?
Exterior dierentiation of identities ?a? = 0 , ?m+i
= 0 , ?am+i = 0 and appliation
of the general struture equations of a submanifold M gives the system of relations
??? ? ?a? = 0,
?
??? ? ?m+i
= 0,
??m+i ? ?a? = 0,
and, therefore, by virtue of expansions (2.6) and (2.7), the following system of algebrai
relations holds:
?
C??? Ca?
= 0,
?
C??? Cab
= 0,
?
C?ab Ca?
= 0,
?
C??? Ca?
= 0,
?
C??? Cab
= 0,
?
C??? Caµ
= 0,
?
= 0,
C?ab Ca?
?
C??c Cab
= 0,
m+i ?
C??
Cab = 0,
m+i ?
C??
Cab = 0,
m+i ?
m+i ?
C??
Ca? = C??
Ca? ,
??
C??? Cm+i
= 0,
m+i ?
m+i ?
C??
Caµ = C?µ
Ca? ,
?a
C??? Cm+i
= 0,
??
?a
C??a Cm+i
= C??? Cm+i
,
m+i ?
m+i ?
C??
Ca? = C??
Ca? ,
?
C??? Ca?
= 0,
(2.9)
??
??
C??? Cm+i
= C??? Cm+i
,
?a
?b
C??b Cm+i
= C??a Cm+i
.
m+?
It follows from identities (2.5? ) that the form ?m+?
is prinipal and that it has an
expansion in terms of the prinipal forms ?? , ?? , ?a , ? ? , ? ? . It follows from here
(and from the same identities) that the form ?a? is also prinipal, but it has expansion
in terms of the prinipal forms ?? , ?? , ?a , ? ? only. Comparison of these relations
shows that the form ?a? has an expansion in terms of the prinipal forms ? ? , ? ? only.
?
Therefore, in partiular, Cab
= 0.
Further lassiation of admissible dierential-geometri strutures is based on the
analysis of algebrai relations (2.9).
A) At least for one value of the index i (= 1, . . . , n ? m) ,
??
m+i
det C??
6= 0 6= det Cm+i
.
B) At least for one value of the index i (= 1, . . . , n ? m) ,
??
m+i
det C??
6= 0 = det Cm+i
.
N
ON THE GEOMETRY OF SUBMANIFOLDS IN E2N
223
C) At least for one value of the index i(= 1, . . . , n ? m) ,
??
m+i
det C??
= 0 = det Cm+i
.
The ase(A) was
studied in [13?. Let us study the ase B. Taking into aount that
m+i
the matrix C??
is of maximal rank, it is easy to see from algebrai relations (2.9)
that
?
?
?
Cab
= 0, Ca?
= 0, Ca?
= 0, therefore ?a? = 0.
m+i
It follows from identities (2.8) that the values C??
an be onsidered equal to zero:
m+i
?
C?? = 0 , then the forms ?? beome prinipal. Using the same proedure, we arrive
??
at the relation Cm+i
= 0 but values C??? remain arbitrary. This gives a reason to
onsider system of algebrai relations (2.9) one more and suppose that, for a xed
value of the index x ( = 2m ? n + 1, . . . , n ? m ), the matrix C??? is nondegenerate:
??
det C??? 6= 0 . It follows from this ondition that Cm+i
= 0 . Let us note that this
m+i
ondition an be obtained from the relation det C??
6= 0 (for xed value of the
index i (= 1, . . . , n ? m) . It is easy to see now that the system of linear dierential
equations
??? = 0, ?a? = 0, ??? = 0, ??? = 0, ?ba = 0
is ompletely integrable. We an rewrite the system of struture equations of M in the
following form
d? ? = C??? ?? ? ? ? ,
d?? = ???a ? ?a ,
d? ? = 0,
d? a = ??a ? ? ? + ??a ? ? ? ,
d?? = ???a ? ?a ,
m+i
ab
C??
?b ? ? ? ,
d??a = Cm+i
d?a = 0,
(2.3? )
m+i
ab
d??a = C??? ??a ? ?? + Cm+i
C??
?b ? ? ? ,
where the oeients satisfy equations (2.8). Exterior dierentiation of the identity
??? = 0 gives the following algebrai ondition
m+i ??
C??
C? = 0,
m+i
therefore, C??
= 0 . It is obvious now that the system of linear dierential equations
a
?? = 0 is ompletely integrable. We obtain the nal form of the system of struture
equations
d? ? = C??? ?? ? ? ? , d? ? = 0, d? a = ??a ? ? ? ,
d?? = 0,
d?? = ???a ? ?a ,
d?a = 0,
(2.3?? )
m+i
ab
d??a = Cm+i
C??
?b ? ? ? .
dC??? = C???? ?? ,
m+?
m+? µ
dC??
= C??µ
? ,
m+µ
m+? ??
m+? µ
dC??
= ?C??
Cµ ?? + C??µ
? ,
(2.8? )
ab
ab
abc
dCm+?
= Cm+?
C??? ?? + Cm+?
?c ,
ab
abc
dCm+?
= Cm+?
?c .
It is easy to see that the system of prinipal and seondary dierential forms ? ? ,
m+i
ab
? , ? a , ?? , ?? , ?a , ??a and funtions C??? , C??
, Cm+i
satisfying equations (2.3?? ) ,
?
(2.8 ) is losed and, therefore, by virtue of the Cartan Laptev theorem [14? the following
statement is true.
?
224
S. HAROUTUNIAN
Theorem 2.1. The metri onnetion of a 2n -dimensional pseudo-Eulidean Ran
shevsky spae E2n
indues a dierential-geometri struture of speial type ane onnetion determined by the system of dierential forms ? ? , ? ? , ? a , ?? , ?? , ?a , ??a
m+i
ab
and funtions C??? , C??
, Cm+i
, ?, ? = 1, . . . , 2m ? n , ?, ? = 2m ? n + 1, . . . , n ? m ,
a, b = n ? m + 1, . . . , m , i = 1, . . . , n ? m satisfying equations (2.3?? ) , (2.8? ) on 2m dimensional (15n < 30m = 18n) submanifold M dened by equations (2.1) on onm+i
dition that, at least for one value of the index i (= 1, . . . , n ? m) , det C??
6= 0 =
??
= det Cm+i
, det C??? 6= 0 .
The struture of this ane onnetion an be studied using struture equations
(2.3? ) . Let us note that it has nontrivial urvature tensor
m+i
ab
ab
R??
= Cm+i
C??
.
To study the struture on M , let us note that the system of linear dierential
equations ? ? = 0 is ompletely integrable and, therefore, it determines submanifolds of
dimension n ? m . Therefore the following result is established.
Theorem 2.2. The submanifold M is a double ber bundle. The bers of the rst
bundle are foliations with n ? m -dimensional at leaves. The bers of the seond bundle
n
are ross produts of 2m ? n - and n ? m -dimensional planes in E2n
.
It is easy to see that the system of Pfa equations ? ? = 0 , ?? = 0 , ? = 1, . . . , 2m?n
is ompletely integrable and determines in M submanifolds N of dimension 2(n ? m) .
3. Canonial integral
It is known [9? that a k -tuple integral depending on k parameters indues a struture
of a pseudo-Riemannian Rashevsky spae on the 2k -dimensional manifold N of integration variables and parameters. The inverse problem of nding a k -tuple integral
depending on k parameters induing a given admissible struture on N , known as
a anonial integral of this dierential-geometri struture, is muh more interesting.
If the urvature tensor is trivial, this integral leads to the Fourier transform. It is evident that, in all other ases, obtained integrals are natural generalizations of the Fourier
transform.
Let us nd a anonial integral of a dierential-geometri struture dened by equations (2.3?? ) , (2.8? ) on N , i. e., an n ? m -tuple integral of the form
? = ?? 2m?n+1 ? · · · ? ? m
(3.1)
depending on n ? m parameters induing the dierential-geometri struture
d? ? = 0,
d? a = ??a ? ? ? ,
d?? = ???a ? ?a ,
d?a = 0,
m+i
ab
d??a = Cm+i
C??
?b ? ? ? ,
m+? µ
m+?
dC??
= C??µ
? ,
m+µ
m+? µ
dC??
= C??µ
? ,
ab
abc
dCm+?
= Cm+?
?c ,
ab
abc
dCm+?
= Cm+?
?c .
(3.2)
N
ON THE GEOMETRY OF SUBMANIFOLDS IN E2N
225
on the 2(n?m) -dimensional manifold of integration variables and parameters. Following
the results obtained in [9?, this proedure inludes solution of the system of dierential
equations
d ln ? = ?? ? ? + ?a ? a + ?? ?? + ?a ?a ,
a
d ?? ?? + ?a ?a = ? ? ? ?? + ? a ? ?a + ?2m?n+?
?a ? ? ? .
(3.3)
Using struture equations (2.3?? ) , let us present the essential prinipal and seondary
forms as linear ombinations of dierentials of variables:
1 a
C?m+i dx? ,
? ? = dx? , ? a = dxa ? Cm+i
2
a
?? = dy? ? Cm+i
C?m+i dya , ?a = dya ,
(3.4)
1 a
m+i
ab
??a =
Cm+i C??
dx? ? Cm+i
C?m+i dyb
2
a
a
where the smooth funtions Cm+i
= Cm+i
(y2m?n+1 , . . . , ym ) , C?m+i
= C?m+i (x2m?n+1 , . . . , xm ) are solutions of the following dierential equations
=
a
ab
dCm+i
= Cm+i
dyb ,
m+i
dC?m+i = C??
dx? .
It is easy to hek that forms (3.4) satisfy struture equations (2.3?? ) .
Let us introdue the following formal expansions of the dierentials of the oeients
of equations (3.3):
d?? = ??? ? ? + ??a ? a + ??? ?? + ??a ?a ,
d?a = ?a? ? ? + ?ab ? b + ?a? ?? + ?ab ?b ,
d?? = µ?? ? ? + µ?a ? a + µ?? ?? + µa? ?a ,
(3.5)
d?a = µa? ? ? + µab ? b + µ?a ?? + µba ?b .
Let us substitute now the expressions of basi forms into these expansions. Then
we substitute these expansions into the seond relation and into the result of exterior
dierentiation of the rst relation of system (3.3). As a result we obtain the following
system of algebrai relations:
µ?? = µ?? ,
µ?a = µa? ,
??? = µ?? = ??? ,
µa? =
µab = µba ,
?ab = µab = ?ba ,
1 ab
a
C
C m+i ?b ? ?2m?n+?
,
2 m+i ?
??? = ??? ,
??a = ?a? ,
µ?a = ??a = 0,
?a? =
1 a
a
C
C m+i ?? ? ?2m?n+?
.
2 m+i ??
Substitution of the obtained relations into the system of expansions (3.5) gives the
following system of dierential equations
???
= ??? ,
?x?
???
= 0,
?xa
???
= ??? ,
?y?
???
a
= ???? Cm+i
C?m+i ;
?ya
??a
1 a
1 a
m+i ?
a
= ??2m?n+?
+ Cm+i
C??
? ? Cm+i
C?m+i ;
?x?
2
2
??a
= ?ba ,
?xb
??a
= ??a ,
?y?
??a
b
= ?ab ? ??a Cm+i
C?m+i ;
?yb
226
S. HAROUTUNIAN
???
1
a
= µ?? ? µ?a Cm+i
C?m+i ,
?x?
2
???
= µ?a ,
?xa
???
= ??? ,
?y?
???
1 ab
1 a
a
= Cm+i
C?m+i ?b ? Cm+i
C?m+i ? ?2m?n+?
;
?ya
2
2
??a
1
a
= µ?a ? µab Cm+i
C?m+i ,
?
?x
2
??a
= µab ,
?xb
??a
= 0,
?y?
??a
= ?ab .
?yb
Solving this system, we represent the solution in the following form
?? = x? + ? ? (y),
1 a
a
C?m+i x? + ? a (y),
?a = xa ? ?2m?n+?
? Cm+i
2
1 a
a
?? = y? ? Cm+i
C?m+i ya ? ?2m?n+?
ya + ?? (x),
2
?a = ya + ?a (x),
where ?? (x) , ?a (x) , ? ? (y) , ? a (y) , are smooth funtions of the orresponding variables.
Substitution of these expressions into the rst equation of system (3.3) gives the formula
a
a
ln ? = x? y? + xa ya ? ?2m?n+?
x? ya ? Cm+i
C?m+i x? ya + ?(x) + ?(y),
where ?(x) = ?(x1 , . . . , xm ) and ?(y) = ?(y1 , . . . , ym ) are smooth funtions on the
orresponding bers of the double bundle N . Therefore, the following result holds.
Theorem 3.1. An n ? m -tuple integral depending on n ? m parameters induing
a dierential-geometri struture (3.2) on the 2(n ? m) -dimensional submanifold N
of variables and parameters an be redued to an integral of the form
h
i
a
a
? = P (x)Q(y) exp x? y? + xa ya ? ?2m?n+?
x? ya ? Cm+i
C?m+i x? ya Ч
Ч dx2m?n+1 ? · · · ? dxm
(3.6)
where P (x) = P (x2m?n+1 , . . . , xm ) and Q(y) = Q(y2m?n+1 , . . . , ym ) are the exponents
of funtions ?(x) = ?(x2m?n+1 , . . . , xm ) and ?(y) = ?(y2m?n+1 , . . . , ym ) respetively.
m+i
ab
In the speial ase when the values C??
, Cm+i
are onstants, we arrive at the
formulas
m+i ?
a
ab
Cm+i
= Cm+i
yb , C?m+i = C??
x ,
and expression (3.6) an be rewritten in the following more symmetri form
h
i
m+i ? ?
a
ab
? = P (x)Q(y) exp x? y? + xa ya ? ?2m?n+?
x? ya ? Cm+i
C??
x x ya yb Ч
Ч dx2m?n+1 ? · · · ? dxm . (3.7)
a
Let us note that the partial derivatives of the funtions Cm+i
, C?m+i ourring
in the anonial integral of a dierential-geometri struture (3.6) ompose the urvature
tensor of the orresponding ane onnetion. Besides, the omponents of the urvature
tensor are oeients of monoms of the fourth degree.
N
ON THE GEOMETRY OF SUBMANIFOLDS IN E2N
227
4. Parametri equations
To nd parametri equations of the submanifold M under onsideration, let us
integrate struture equations (2.3?? ) . To do this, let us onsider the equations of innitesimal displaement of a moving frame (P, e? , e? , ea , em+i , e? , e? , ea , em+i ) in the
n
spae of ane onnetion E2n
?
de? = ??? e? + ??? e? + ?a? ea + ?m+i
em+i ,
?
de? = ??? e? + ??? e? + ?a? ea + ?m+i
em+i ,
a
dea = ??a e? + ??a e? + ?ba eb + ?m+i
em+i ,
m+i m+k
dem+i = ??m+i e? + ??m+i e? + ?am+i ea + ?m+k
e
,
de? = ???? e? ? ??? e? ? ??a ea ? ??m+k em+k ,
dea = ??a? e? ? ?a? e? ? ?ab eb ? ?am+k em+k ,
?
m+k
?
a
dem+i = ??m+i
e? ? ?m+i
e? ? ?m+i
ea ? ?m+i
em+k .
The submanifold M is haraterized by the identities
??? = 0,
?ba = 0,
?a? = 0,
?, ? = 1, 2m ? n;
??? = 0,
?
?m+i
= 0,
?a? = 0,
??a = 0,
?
?m+i
= 0,
??m+i = 0,
?, ? = 2m ? n + 1, . . . , n ? m;
??? = 0,
?am+i = 0,
a, b = n ? m + 1, . . . , m.
Substitution of these relations into the previous system gives the following equations of
innitesimal displaement of a moving frame (P, e? , e? , ea , em+i , e? , e? , ea , em+i ) on the
spae of ane onnetion M :
de? = C??? dy? e? ,
de? = 0,
dea =
1 a
m+i
ab
ab
Cm+i C??
dx? ? Cm+i
C?m+i dyb e? + Cm+i
dyb em+i ,
2
m+i
m+i
m+i m+k
dem+i = C??
dx? e? + C??
dx? e? + ?m+k
e
,
de? = 0,
de? = ?C??? dy? ea ?
1 a
m+i
m+i
ab
Cm+i C??
dx? ? Cm+i
C?m+i dyb ea ? C??
dx? em+i ,
2
dea = 0,
m+k
ab
dem+i = ?Cm+i
dyb ea ? ?m+i
em+k .
m+i
Replaing the seondary forms ?m+k
by ??ik and solving the obtained, we obtain
the following expressions for the basis vetors
e? = C?? (e? )0 + (e? )0 ,
e? = (e? )0 ,
228
S. HAROUTUNIAN
h
i
a
a
ea = Cm+i
C?m+i ? Cm+?
C?? C?m+? (e? )0 +
1 a
a
C?m+i + Cm+i
C?m+i C?? + C?m+i (e? )0 +
+ Cm+i
2
a
a
a
+ Cm+?
? Cm+?
C?? (em+? )0 + Cm+?
(em+? )0 + (ea )0 ,
em+? = C?m+? (e? )0 + C?m+? C?? + C?m+? (e? )0 + (em+? )0 ,
em+? = C?m+? ? C?? C?m+? (e? )0 + C?m+? C?? + C?m+? (e? )0 ? C?? (em+? )0 + (em+? )0 ,
e? = (e? )0 ,
1 m+i a
a
e? = ?C?? (e? )0 +
C? Cm+i + C?m+? C?? Cm+?
(ea )0 ? C?m+? (em+? )0 +
2
+ C?m+? + C?m+? C?? (em+? )0 + (e? )0 ,
ea = (ea )0 ,
a
a
em+? = ? Cm+?
C?? Cm+?
(ea )0 + C?? (em+? )0 + (em+? )0 ,
a
em+? = ?Cm+?
(ea )0 + (em+? )0 .
where P, (e? )0 , (e? )0 , (ea )0 , (em+? )0 , (em+? )0 , (e? )0 , (e? )0 , (ea )0 , (em+? )0 , (em+? )0 is
n
a xed orthonormal frame in E2n
. Substitution of these relations into the equation
dP = ?? ? e? ? ? ? e? ? ? a ea ? ?2m?n+i em+i ? ?? e? ? ?? e? ? ?a ea ? ? i em+i
and further integration gives the equality
P = ?x? (e? )0 ? x? (e? )0 ?
h
?
a
a
? xa ? C?m+? Cm+? C?? ? ?2m?n+?
Cm+i
+ C?? Cm+?
y? +
?
a
a
+ ?2m?n+?
Cm+?
+ C?? Cm+?
Cm+i C?m+i ?
i
b
a
a
b
a
? ?2m?n+?
Cm+?
+ C?? Cm+?
yb ? ?2m?n+?
Cm+?
yb (ea )0 ?
h
i
?
?
a
? ?2m?n+?
y? + Cm+? + ?2m?n+?
ya ? ?2m?n+?
Cm+i C?m+i (em+? )0 ?
h
?
?
a
a
? ?2m?n+?
ya + ?2m?n+?
C?? ya ? ?2m?n+?
Cm+i C?m+i C?? + ?2m?n+?
C?? y? +
+ C m+? + C m+? C??
h
i
(em+? )0 ? y? + C m+? ? C?? C?m+? x? +
i
+ Cm+i C?m+i ? Cm+? C?? C?m+? + C? + C?? C?m+? x? (e? )0 ?
h
1
m+i
m+i ?
m+i
? y? + C? + ? Cm+i C?
+ Cm+i C? C? + C?
+ C? C?? + C? +
2
i
+ C?? C?m+? C?? x? + C?m+? + (· · · + C? ) (e? )0 ? ya (ea )0 ?
? x? + Cm+? ? Cm+? C?? (em+? )0 ? x? + Cm+? (em+? )0 .
229
N
ON THE GEOMETRY OF SUBMANIFOLDS IN E2N
Therefore, the following statement is true.
Theorem 4.1. The submanifold M has be given by parametri equations of the
form
X ? = x? ,
X ? = x? ,
?
a
a
X a = xa ? C?m+? Cm+? C?? ? ?2m?n+?
Cm+i
+ C?? Cm+?
y? +
?
a
a
+ ?2m?n+?
Cm+?
+ C?? Cm+?
Cm+i C?m+i ?
b
a
a
b
a
Cm+?
yb ,
? ?2m?n+?
Cm+?
+ C?? Cm+?
yb ? ?2m?n+?
?
?
a
X m+? = ?2m?n+?
y? + Cm+? + ?2m?n+?
ya ? ?2m?n+?
Cm+i C?m+i ,
?
a
a
X m+? = ?2m?n+?
ya + ?2m?n+?
C?? ya ? ?2m?n+?
Cm+i C?m+i C?? +
?
+ ?2m?n+?
C?? y? + C m+? + C m+? C?? ,
Ya = y? + C m+? ? C?? C?m+? x? + Cm+i C?m+i ? Cm+? C?? C?m+? +
+ C? + C?? C?m+? x? ,
1
Y? = y? + C? + Cm+i C?m+i + Cm+i C?m+i C?? + C?m+i + C? C?? +
2
m+? ? ?
?
+ C? C?
C? x + C?m+? + C?m+? C?? ,
Ya = ya ,
where
Ym+? = x? + Cm+? ? Cm+? C?? ,
C m+? = C m+? (x1 , . . . , x2m?n ),
C? = C? (x1 , . . . , x2m?n ),
Cm+? = Cm+? (yn?m+1 , . . . , ym ),
Ym+? =? +Cm+? ,
C m+? = C m+? (x1 , . . . , x2m?n ),
C? = C? (yn?m+1 , . . . , ym ),
Cm+? = Cm+? (yn?m+1 , . . . , ym )
are smooth funtions satisfying the following dierential equations
dC m+? = C?m+? dx? ,
a
dCm+? = Cm+?
dya ,
dC m+? = C?m+? dx? ,
dC? = C?m+? dx? ,
a
dCm+? = Cm+?
dya .
It is easy to hek that the parametri equations of the submanifolds N ? M an
be written in the following form
X ? = x? ,
?
?
a
a
b
a
X a = xa ? ?2m?n+?
Cm+i
y? + ?2m?n+?
Cm+?
Cm+i C?m+i ? ?2m?n+?
Cm+?
yb ,
?
?
a
X m+? = ?2m?n+?
y? + Cm+? + ?2m?n+?
ya ? ?2m?n+?
Cm+i C?m+i ,
a
X m+? = ?2m?n+?
ya + C m+? ,
Ym+? = x? + Cm+? ,
Ym+?
1
Y? = y? + Cm+i C?m+i + Cm+i C?m+i ,
2
= x? + Cm+? ,
Ya = ya ,
where Cm+? = Cm+? (yn?m+1 , . . . , ym ) , Cm+? = Cm+? (yn?m+1 , . . . , ym ) are smooth
funtions satisfying the following dierential equations
a
dCm+? = Cm+?
dya ,
a
dCm+? = Cm+?
dya .
230
S. HAROUTUNIAN
езюме
С.Х. Арутюнян.
n
О геометрии подмногообразий в E2n
.
Изучается специальный класс 2m -мерных подмногообразий в 2n -мерном псевдоевклидовом пространстве с метрикой сигнатуры (n, n) , известном как псевдоевклидово
пространство ашевского. Для изучаемых подмногообразий найдены канонические интегралы и параметрические уравнения.
Ключевые слова: четномерное подмногообразие, псевдоевклидово пространство ашевского, двойное расслоение, канонический интеграл, диеренциально-геометрическая структура, расслоение, слоение.
References
1.
Vasilyev A.M. Theory of differential-geometric structures. ? M.: Moscow State Univ.
Press, 1987. ? 190 p. [in Russian].
2.
Cartan E. Les espaces m e? triques fond e? s sur la notion d?aire. ? Paris, 1933.
3.
Kawaguchi A. Theory of connections in the generalized Finsler manifold // Proc. Imp.
Acad. ? Tokyo, 1937. ? V. 7. ? P. 211?214.
R
Kawaguchi A. Geometry in an n -dimensional space with length s =
Ai (x, x? )x??i +
+B(x, x? )) dt // Trans. Amer. Math. Soc. ? 1938. ? V. 44. ? P. 153?167.
4.
5.
Shirokov P.A. Constant fields of vectors and tensors in Riemannian spaces // Proc.
Kazan Phys. Math. Soc. ? 1925. V. 2, No 25. ? P. 86?114 [in Russian].
6.
K a? hler E. U? ber eine bemerkenswerte Hermitesche Metrik // Abh. Math. Sem. Univ.
Hamburg. ? 1933. ? Bd. 9. ? S. 173?186.
7.
Rashevsky P.C. Scalar field in the fiber bundle space // Reports of the Sem. on Vector
and Tensor Analysis. ? M.: Moscow State Univ. Press, 1949. ? V. 6. ? P. 225?248 [in
Russian].
8.
Vishnevsky V.V. About parabolic analogue of A -spaces // Izv. Vuz. Matematika. ? 1968. ?
No 1. ? P. 29?38 [in Russian].
9.
Haroutunian S. Geometry of n multiple integrals depending on n parameters // Problems
of Geometry. ? M.: VINITI Acad. Sci. USSR, 1990. ? V. 22. ? P. 37?58 [in Russian].
10.
Haroutunian S. On some classes of differential-geometric structures on submanifolds of
n+1
pseudoeuclidean space E2(n+1)
// Izv. Vuz. Matematika. ? 1989. ? No 10. ? P. 3?11
[in Russian].
11.
Haroutunian S. On some classes of submanifolds of codimension two in pseudoeuclidean
n+1
space E2(n+1)
// Izv. Vuz. Matematika. ? 1990. ? No 3. ? P. 3?11 [in Russian].
12.
Haroutunian S. Geometry of submanifolds with structure of double fiber bundle in Rashevsky pseudoeuclidean space // Izv. Vuz. Matematika. ? 2005. ? No 5. ? P. 33?40
[in Russian].
13.
n
Haroutunian S. On geometry of one class of submanifolds in E2n
// Tensor, N.S. ? 2005. ?
V. 66, No 3. ? P. 229?244.
14.
Laptev G.F. Differential geometry of immersed manifolds // Mosc. Math. Soc. Works. ?
1953. ? V. 2. ? P. 275?382 [in Russian].
Поступила в редакцию
08.09.09
Арутюнян Самвел Христоорович доктор изико-математических наук, проессор, заведующий каедрой высшей алгебры и геометрии Армянского государственного педагогического университета, г. Ереван, еспублика Армения.
E-mail: S_Haroutuniannetsys.am
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