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УЧЕНЫЕ ЗАПИСКИ КАЗАНСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА Том 151, кн. 4 Физико-математические пауки 2009
UDK 514.76
ON THE GEOMETRY OF SUBMANIFOLDS IN E2nn
S. Haroutunian
Abstract
A special class of 2m-dimensional submanifolds in a 2n-dimensional pseudo-Euclidean space with metric of signature (n, n), known as a pseudo-Euclidean Rashevsky space, is studied. For such submanifolds, canonical integrals and parametric equations are found.
Key words: even-dimensional submanifolds, pseudo-Euclidean Rashevsky space, double fiber bundle, canonical integral, differential-geometric structure, fibration, foliation.
One of the most characteristic features of modern Differential Geometry is the active application of its methods in the adjoining fields of the mathematical science. Essentially increased effectiveness of these methods which accumulated fundamental achievements, first of all from the general algebra and theory of differential equations, in combination with the tendency to consider various mathematical objects as differential-geometric structures on corresponding manifolds, has led. on the one hand, to the appearance of new directions of the differential geometric study, and. on the other hand, to a more fundamental, geometric interpretation of these objects.
The next step is the differential geometric analysis of these structures and identification of their most general characteristic (geometric) properties. Finally, on the last stage of research, these properties or their part become the foundation for generalizations and new problems in the initial theory.
Moreover, in accordance with [1|. they assume in particular the exact description of the category of the structures under study and also the identification of the category of algebraic systems necessary for their study.
All above mentioned is true for the geometry of multiple integral depending on parameters. The study of differential geometric structures defined by such an integral on the manifold of integration variables and parameters to a certain extent is similar to the study of the integral geometry [2 4]. At the same time the presence of parameters totally changes the cycle of arising problems and corresponding results. By systematic study of multiple integrals depending on parameters (in a special case when the number of parameters is equal to the number of variables) and corresponding integral transforms one can see a good number of interesting geometrical problems connected with the description of invariant properties of such integrals.
The present article is devoted to the study of a special class of 2m-dimensional submanifolds with structure of double fiber bundle in the 2n-dimensional pseudo-Euclidean space E2n with metric of index n. We find multiple integral depending on parameters, determining the structure of such a submanifold on the corresponding manifold of integration variables and parameters, also parametric equations of this submanifold.
1. Pseudo-Riemannian Rashevsky space
In 1925. Russian geometer P. A. Shirokov from Kazan State University introduced [5] the special class of even-dimensional symmetric spaces known as A-spaces or elliptic A-spaces. In 1933, E. Kahler [6] studied the same spaces known now as Kahler spaces.
Let us consider a 2n-dimensional manifold M with local coordinates x1,...,xn, yi,..., yn such that in all admissible transformations of coordinates two sets of n coordinates are separated: the transformed coordinates x1,..., xn are functions of x1,..., xn and the same is true for the second set of coordinates. Consider a real kern function U(x1,..., xn, y1,..., yn) and introduce the following values
is invariant under all admissible transformations of local coordinates. This matrix is
M
M
families are complex conjugate.
n
manifolds was introduced by P.K. Rashevsky [7]. He studied an invariant scalar field U(x1,..., xn, y1;..., yn) with nondegenerate matrix of second order derivatives:
nM
the corresponding pseudo-Riemannian connection. This space is known as a Rashevsky pseudo-Riemannian space. It has the following characteristic properties.
1. The scalar field U (x1,..., xn, y1;..., yn) generating the structure of a pseudo-M
M
of fibers. Fibers from different families have intersection in no more than one point.
3. The fibers of both the families are isotropic.
4. The fibers of each family have the property of absolute parallelism (auto parallelism): vectors tangent to fibers from one of the families remain tangent to them after parallel transfer along an arbitrary smooth curve.
It follows from each of the two latter properties that both the families of fibers are M
This space was studied by P.K. Rashevsky and other researchers as an example of a pseudo-Riemannian space only, without any relations to other fields of Mathematics and Physics.
Later, in 60-th, professor V.V. Vishnevsky from Kazan State University introduced AA
these structures.
In terms of a co-basis of linear differential forms w1,..., wn, ..., wn adapted to
2n
dxj dyi
d2U
It is easy to check that the matrix
U(xi,yj) U(xi,yj) + U1(xi) + U2(yj).
can bo presented in the form [9] dwp = wK A wK,
dwp = —^K A wk , I, K, P, T = 1,..., n (1-1)
dwK = wp A wK + RkpwP A wy,
where RKP are the nonzero components of the curvature tensor. The metric of this space is generated by the nondegenerate bilinear closed form [9]
d<^> = wp A w/ . (1-2)
It is known [9] that an integral of the form Aw1 A ... A wn induces a structure of
2n M
parameters under natural condition of nondegeneracy for the matrix of second order derivatives of the function ln A.
Rashovsky space can also be considered as a double fiborod manifold with two fanii-n
of two manifolds: a (2n + s)-dimensional smooth manifold M is said to be a double fiber bundle if two smooth mappings
n : M —^ Mi, i = 1, 2,
from M onto n- and n + s-dimensional smooth manifolds M1 and M2 are given, the fibers, i.e., full preimages of points from M1 and M2 under the map pings n1 and n2 respectively are smooth n + s- and n-dimensional submanifolds, and the tangent
spaces to the fibers of the bundles i^d n2 at an arbitrary point have only trivial
intersection:
Tp (n-^x)) n Tp (n-^y)) = p, n1(p) = x, n2(p) = y, x e M1, y e M2.
Therefore, the tangent space of M at an arbitrary point is a direct sum of n + sand n-dimensional subspaces. The case of Rashevsky spaces corresponds to s = 0.
If the curvature tensor of such a space is trivial, we have a psoudo-Euclidoan Rashevsky space which, in terms of a co-frame of principal exterior linear differential forms w1,..., wn, w1,..., wn adapted to the structure of a double fiber bundle on Enn and
E2nn
the following structure equations [9]
dwp = wK A wK,
dwp = — wK A wk , I, K, P = 1,..., n (1-3)
dwK ^ wp A wk ,
where the secondary forms wK are defined on the manifold T2Enn of second order
E2nn
a natural connection between such spaces and the Fourier transform [9]: this integral
E2nn
M
An (n + s)-tuple integral depending on n parameters is said to be a canonical integral of a differential-geometric structure on a 2n + s-dimensional manifold M if this
M n n
2n
nn constructed on parameters of integration generates the same structure of a Rashovsky 2n M
M
constant coefficient). This is the geometrical meaning of the invortibility of the corro-
2n
(n, n)
invortiblo integral transforms. One of the most important problems here is findind of an integral transform generating the structure of a given Rashovsky (Einstein) space M
2. 2m-dimensional submanifolds with structure of double fiber bundle in a pseudo-Euclidean space E£n
E2nn
corresponding canonical integrals follows from the problem of finding canonical integrals of Rashovsky (Einstein) spaces because in the special case when the space under study is pseudo-Euclidean (the curvature is equal to zero) the corresponding canonical integral coincides with the classical Fourier transform. Taking into account that an integral generates the corresponding differential-geometric structure in an invariant way and that the geometry of a pseudo-Riemannian space, in general, is defined by its curvature tensor, it is natural to suppose that the canonical integral of a Rashovsky (Einstein) space is related to the curvature tensor of this space in a special way.
Let us consider 2m-dimensional submanifold M with structure of a double fiber 2n E2nn n
n
2m > n.
Suppose that, in terms of a co-basis of linear differential forms w1, w2,..., wn, w1; w2 , . . . , wn E2nn
2m
M
w +' = w2m-n+^ wm+i = w\ i =1,...n m. (2-1)
2m
E2nn
generalization of the corresponding classes of submanifolds of codimonsion two. studied in [10. 11].
There are three possible cases: 1) 2(n — m) > m or 3m < 2n, 2) 2(n — m) = m or
3m = 2n, 3) 2(n — m) < m or 3m > 2n.
3m < 2n
following inequalities hold
3n < 6m < 4n.
2m — n < n — m
new indices a = 1,..., 2m — n; £ = 2m — n + 1,..., n — m; a = n — m + 1,..., m. The
E2nn
nondegenerate form d^ = wp A wp induces the bilinear form
d<£>* = wa A wa + w^ A w£ + wa A wa + ^2m-n+awc A w“ + ^2m-n+^wa A w^ (2.2)
M
Substitution of relations (2.1) into (1.3) and application of the above introduced
M
dwa = w| A w^ + w^ A w? + wa A wa + w”+k A w2m-n+k,
dw? = w? A w^ + w| A w“ + w? A wa + wi +k A w2m-n+k,
dwa = wJJ A w2 + w^ A wa + w| A w? + w” +k A w2m-n+k, dwa = —w^ A w^ — w? A w? — w^ A wa — wm+k A wk, dw? = —w^ A w^ — wa A wa — w?^ A wa — w? + A wk, dwa = — w^ A wb + wa A wa — w? A w? — wm+k A wk,
■a = waA w^+waA w?+wa A wa+wm+k
dwa = wa a wY + wa a w? + wa a wa + wm, k a wm+k dw? = wm a <+wi a wa+w? a <+wi+k a ^ (2-3)
dwa =wa a w6+ws a wa+wi a w?+wm+k a wm+k,
dwa = wi a w?i+wa a wn+wa a wi+wm+k a wm+k
dwa = wa A wf + wlA w? + w? A wC + wm+k A
dwl = w? A wa + wri A wS + w! A wS + wi +k A wm+k:
dw?=wi a wa+w? a wn+w? a wa+w!+k a wm+k, dwa = wa a w^+wa a wi+wa a wa+wm+k a wm+k,
dw?a = w^ A wa + w^ A w| + w^ A w^ + wm+i A w?”^,
where the secondary forms wa, w?, w£, w?*, wa, w?, w?, wJ2, w| and w”+k, w”+k .
w”+k > wm+k, w””+^ wm+k are defined on the manifold T(2)M of second order tangent
M
Taking into account that the bilinear form dip* is closed and using exterior differentiation of (2.2) with application of general structure equations (2.3). we arrive at the identity which shows that, in the general case, the forms wa, w?, £?m_n+jgw^ — ^2m-n+ aw? .
^2m-n+?wa —^2m-n+aw? i ^2m-n+awt) —^2m-n+nwo ! ^2m-n+?wn — ^2m-n+nwb are principal. We will use these general conditions for more detailed research of the difforontial-M
M
i.o., that the following systems of linear differential equations
wa = 0, w? = 0, wa = 0,
a = 1,. .., 2m — n, £ = 2m — n + 1,. .., n — m, a = n — m + 1,. .., m;
wa = 0, w? = 0, wa = 0,
a = 1, .. ., 2m — n, £ = 2m — n + 1, .. ., n — m, a = n — m + 1, .. ., m
are totally intograblo, we arrive at the following system of identities
w
a A w m-n+k = 0, wm? +k A w m-n+k = 0, wma +k A w m-n+k = 0,
m+k A w 2m-n+k = 0, wm+k A w 2m-n+k = 0, wm
^m+k A wk =0, wm+k A wk =0, wm+k A wk =0.
(2.4)
wma +i
w”+i, w”+^d wm+% wm+% wm+i are linear combinations of the basic linear differential forms w?, wa, £ = 2m — n + 1,..., n — m, a = n — m + 1,..., m and wa, w?. a = 1, .. ., 2m — n, £ = 2m — n + 1,. .., n — m respectively.
On the other hand, the exterior differentiation of relations (2.1). which are identities M
(wm+i | wm+k \ A wk , wm+i A wa + wa A w +
(wk + w2m-n+i) A w + wa A w + w2m-n+i A wa +
(wm+”+? + w2m-n+i) A w? + (wm+m +a + ^m-n+i) A wa = 0, (wm+k + wk) A wk + w2 A w2 + w”+i A wa +
+ ^wn-m+? + wm+i^ A w? + (wn-m+a + wm+0 A wa 0.
Taking into account identities (2.4). it is easy to check that this system is equivalent to the system of the following four identities
(w”+i + w2mm+-kn+i) A wk + w”+i A wa =0,
w2m-n+i A wa + (wm+,”+? + w2m-n+i) A w? + (^T-”+a + ^m-n+i) A wa = 0,
(wm+k+wi) a wk+wa a wa = 0,
wm+i A wa + ^wn-m+? + wm+i^ A w? + (wn-m+a + wm+0 A wa 0.
It follows directly from the obtained system that all the secondary forms w2 are equal to zero identically. Indeed it’s follows from the third identity from (2.5) that the secondary forms w2 are linear combinations of the basic principal differential forms w1, w2,..., wn. But it is easy to see from the second identity of system (2.5) that the same
w1 , w , . . . , wn
only. This is possible if and only if the forms w2 are equal to zero.
wma +i
have nontrivial expansions in terms of the basic principal forms w 2m-n+1,..., wn only. Substituting the corresponding expansions into the fourth identity of system (2.5). we
wma +i
Let us note now that, as follows from the last identity of system (2.4). the secondary forms wm+i are linear combinations of the basic principal forms w1, w2,..., wn-m.
Substitution of the corresponding expansions into the first identity of system (2.5)
shows that all the forms wm+i are vanishing too.
Using relations (2.4). it is easy to check that system of identities (2.5) is equivalent to the following system
wn-m+a a wk =0, (wn-m+? + w””+k) A wk =0,
w2 A wa + (wn_m+? + w|) A wn = 0
(<-”+? + w?) A w? = 0 w” + a A w? + w”+a A wa = 0, (2.50
(w”+?k + w?) A wk + w? A wa = ^
^wn-m+n + wm+?) A wn + ^wn-m+a + wm+?) A wa 0.
Exterior differentiation of relation (2.2) shows that the secondary forms w|, w?2 are principal forms. The application of the second and third identities of system (2.5/) gives the following expansions
wa = cap wp + can wn,
? ? ?
w? = C? wa + C? wn + C\w2
a aa 1 an 1 ab
with the symmetry conditions on the coefficients corresponding to Cartan’s lemma:
x'-f? _ x'-f?
C? = C? ? Ca2 = C2a '
Applying Cartan’s lemma fl] to identities of the first and the last identities of system (2.5). we obtain the following expansions
w”+i = Cm+iw^ + Cm?+iw?, Cm+i = C”+i,
a ap 1 a? ’ ap p a
i
(2.7)
a ?a a2 a2 2a
wm+i Cm+iw? + Cm+iW2, Cm+i Cm+i.
. ,m+i _ ^m+i a _i_ /^m+i n ^m+i ____ ^m+i
w? — C w + G?n w', G?n — CnC
w? — C?n w I C?a w C?n — cn?
wm+i — om+iwn + om+iWa> om+i — cm+i,
Exterior differentiation of expansions (2.6). (2.7) with further application of the general structure equations of a submanifold M gives the following differential identities
(dO“e - C“Yw^ - of w^ + C“ew?n) Л We+
+ (dC“n - 0?a vwn - + Canw?v - C“ewn) Лшч - C“eweЛwa -05“nwn Лш„ — 0,
(dCab + 0acwb + °cbWa - ^b^ + 0 aWb" + C!nWn) Л W&+
+ (dCla + CfeW^ + CbaWbb - Ca awj5 + CfbW^ Л W +
+ (dC!n + CImW^ + 05nwbb - CanWM + ^ Л wn — 0
(dom+i+om7+iwY+om+iwY - om+k wm+i+om+ w?+o^w?) л we+
+ (dom?+i+om+i w5n+om+iwe - om?+k wm+k++0™+ of w7) л w?—0,
(2.8)
(dom+i+oa;+iwn+om+iwe - om+k wm+i+c^wa+cm^cf w7) л wa+
+( dom+i+om+ч+omn+iwf - om+k wm+i+oa^op w7+oa^op w7) лwn—0,
(dom+i- om+iwa- om+iwM+om+fcwm+k - oma+ionbwb - ofbom+iwb) л wn+
+ (dom+i- omb+iwb- om+iwn+oma+k wm+i - om+iwa - o?boma+iwb) л wa—°, (dom+i- om+iwb- om+iw?+om+k wm+k - om+iwa- ^L^m+i^) л w?+
i l z^ac b /'-rcb a . /'-tab m+k /^?a b /^?b a\ л _ гч
+ ^d0m+i Cm+iWc Cm+iWc + Cm+kWm+i Cm+iW? Cm+iW? у Л Wb °*
It follows from first two identities of this system that quantities c22 and C?*p are invariants and therefore their vanishing has an invariant geometric meaning. For example, if C?2 P = 0, then the system of linear differential equations w 2 = 0, a =1,..., 2m — n
is totally integrable; the condition c22 = 0 characterizes the total integrability of the system of Pfaff equations wa = 0, a = n — m + 1,..., m.
Next identities of the system (2.8) show that the quantities C?"+% C,?^, Cm+i, Cm2+i are invariants, and the other quantities occurring in this system are not invariants. Therefore, without any loss of generality, the quantities Cm?+% Cm+i can be considered equal to zero.
There are three possible cases:
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,
c) 2m — n > 2n — 3m, i. e., 5m > 3n, therefore 18n < 30m < 20n.
Let us consider the case 15n < 30m = 18n. We note that, by virtue of the
fact that the ranges of the indices a = 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 differentiation of identities w2 = 0, w^+i = 0, wm+i = 0 and application of the general structure equations of a submanifold M gives the system of relations
w2 A w? = 0, w5m+i A w? = 0, w2 A wi+i = 0,
and, therefore, by virtue of expansions (2.6) and (2.7), the following system of algebraic relations holds:
C?“P Ca?7 =0, C?“P Ca?2 = 0, C?a2Ca?7 = 0, C?“n Ca?7 =0
C?“n Ca?2 = 0, C?“n Ca?M =0, C?a2Ca?n = 0, C?ac Ca?2 = 0,
C C = 0, C C = 0, C C = C C ,
a? a2 ’ ?n a2 ’ a? ap p? aa-
m+i ? m+i ? m+i ? m+i ?
C?n Caa = Ca? Can, C?n Ca^ = C?M Can,
.'-fa^^-f?a ^ /"tavr^?V /'-»an^-»?v
C? Cm+i = 0, C? Cm+i = 0, C? Cm+i = C? Cm+i,
/~iaa s~i?'n /'-fa^^-f?a /'-»a2^''*?a
C? Cm+i = C? Cm+i, C? Cm+i = C? Cm+i.
(2.5/) wmm++a?
wa w? wa wa w?
(and from the same identities) that the form w? is also principal, but it has expansion
wa w? wa wa
shows that the form w? has an expansion in terms of the principal forms w2, w? only.
Ca?2 = 0
Further classification of admissible differential-geometric structures is based on the analysis of algebraic relations (2.9).
A) At least for one value of the index i (= 1,..., n — m),
det (Ci+^ = 0 = det (Ci+i) .
B) At least for one value of the index i (= 1,..., n — m),
C?“p Ca?^ = 0,
(2.9)
det (C?i+^ = 0 = det (Ci+i)
C) At least for one value of the index i(= 1,..., n — m).
det (cm+i) = 0 = det (c^®) .
The case (A) was studied in [13]. Let us study the case B. Taking into account that the matrix j is of maximal rank, it is easy to see from algebraic relations (2.9)
that
Cfb = 0 = 0 = 0 therefore w| = 0.
It follows from identities (2.8) that the values C^® can be considered equal to zero: = 0, then the forms w1 become principal. Using the same procedure, we arrive
at the relation C^® = 0 but values C^ remain arbitrary. This gives a reason to consider system of algebraic relations (2.9) once more and suppose that, for a fixed value of the index x( = 2m — n + 1,...,n — m), the matrix C|ag is nondegenerate: det C|a g = 0. It follows from this condition that C^® = 0- Let us note that this
condition can be obtained from the relation det = 0 (for fixed value of the
index i (= 1,..., n — m). It is easy to see now that the system of linear differential equations
4 = 0, w| = 0, w- = 0, w« = 0, w6a = 0
is completely integrable. We can rewrite the system of structure equations of M in the following form
dw a = Cf wg A w1, dw1 = 0, dwa = wa A wa + w| A w1,
dwa = —wa A wa, dw| = —w| A wa, dwa = 0, (2.3')
dwa = Cm6+iCm+iw6 A wg, dwa = Cfwa A wg + Cm+iCm+iw5 A wn,
where the coefficients satisfy equations (2.8). Exterior differentiation of the identity w | = 0 gives the following algebraic condition
Cm+®C a g 0
therefore, Cm+® = 0. It is obvious now that the system of linear differential equations wa = 0 is completely integrable. We obtain the final form of the system of structure equations
dwa = C|agwg A w1, dw1 = 0, dwa = wa A w1,
dwa = 0, dw| = —wa A wa, dwa = 0, (2.3'')
dwa = Cm+®Cm+iw6 a wn. dC?“g = C5“g7 w7, dCm+a = C|"+a wM,
dC^+M = —Cm+-Cag wg+Cm+a w* (2.8')
rlf~'ab __ r^ab ag, . 1 siabc , .
dCm+a Cm+|C| wg + Cm+a wc,
ACab _____ z^abc
dCm+| = Cm+| wc-
w
w1, wa, wa, w^, wa, w|r and functions C|ag, C|"+% Cmb+® satisfying equations (2.3"), (2.8')
statement is true.
Theorem 2.1. The metric connection of a 2n-dimensional pseudo-Euclidean Ra-shevsky space Ей induces a differential-geometric structure of special type affine connection determined, by the system of differential forms wa, w2, wa, wa, wa, w|
and functions , cmb+i, а, в =!,•••, 2m — n, £, 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 defined by equations (2.1) on condition that, at least for one value of the index i (= 1,..., n — m), det j = 0 =
= det (cm+i), det (cf) =0.
The structure of this affine connection can be studied using structure equations (2.3'). Let us note that it has nontrivial curvature tensor
pab __ /'-tab /-rm+i
R2n cm+ic2n .
M
equations w2 = 0 is completely integrable and, therefore, it determines submanifolds of n— m
M
n— m
are cross products of 2m — n- and n — m-dimensional planes in .
It is easy to see that the system of Pfaff equations wa = 0, wa = 0, а = 1,..., 2m—n is completely integrable and determines in M submanifolds N of dimension 2(n — m).
3. Canonical integral
It is known [9] that a k-tuple integral depending on k parameters induces a structure of a pseudo-Riemannian Rashevsky space on the 2k-dimensional manifold N of inte-
k
kN a canonical integral of this differential-geometric structure, is much more interesting. If the curvature tensor is trivial, this integral leads to the Fourier transform. It is evident that, in all other cases, obtained integrals are natural generalizations of the Fourier transform.
Let us find a canonical integral of a differential-geometric structure defined by equations (2.3"), (2.8') on N, i.e., an n — m-tuple integral of the form
Л•• • Л wm
(3.1)
n— m
dw2 = 0, dwa = wa Л w2, dw2 = —wa Л wa, dwa = 0,
dwa = cm+icm+iwb л wn,
dc2m+a
dC ab
dcm+a
cm+aw^,
(3.2)
dc2m/M =
/'-»abc .
cm+2 wc
ibc
m+2w
c
2(n—m)
the results obtained in [9]. this procedure includes solution of the system of differential equations
d ln A = A^ w2 + Aawa + A2 w2 + Aawa,
(3.3)
d (A2W2 + Aawa) = w2 A W2 + wa A wa + ^2m-n+2Wa A w2.
(2.3'')
forms as linear combinations of differentials of variables:
w2 = dx2, wa = dxa — 1 cm+iC2”+idx2,
w2 = d^2 — Cm+icm+*dya’ w2 = dya’ (3-4)
„ 1
2 (cm+iCm;+idxn — cm6+iC2m+idy6)
where the smooth functions Cm+i = Cm+i(y2m-n+1,..., ym^ C2m+i =
= cm+i(x2m-”+1,..., xm) are solutions of the following differential equations
dcm+i = cm6+idyb:
dc^+i = C2”+idxn.
It is easy to check that forms (3.4) satisfy structure equations (2.3'').
Let us introduce the following formal expansions of the differentials of the coefficients of equations (3.3):
dA2 = A2 wn + A|w“ + A2n w^ + A2“wa.
dAa = A?w2 + Agw6 + Aa2 w2 + Aabwb,
n (3-5)
dA2 = M2n wn + M2a wa + M2 w2 + ^|wa,
dAa = Ma2 w2 + Mabw6 + + m2w^
Let us substitute now the expressions of basic forms into these expansions. Then we substitute these expansions into the second relation and into the result of exterior differentiation of the first relation of system (3.3). As a result we obtain the following system of algebraic relations:
M2n Mn2. M2a Ma2. Mab Mba. A2n A^2. A2 A 2.
An = . Aa = m2 = ^ m2 = a2 = 0.
a _ 1 ^tab rim+n ca \a _ 1 a /-im+Mn ra
M2 2 Cm+iC2 Ab °2m-n+2, A2 2 Cm+iC2n A °2m-n+2.
Substitution of the obtained relations into the system of expansions (3.5) gives the
following system of differential equations
dA2 = £2 dA2 =o dA2 = A2^ dA2 = —A2^ Ca Cm+®-
dxn = . dxa = 0 dyn = A . dya = A 5
dAa 1 1
d A ___ ca | 1 a /'rm+i \n 1 z''*a /'rm+i.
dx2 = °2m-n+2 + 2 °m+i°2n A 2 °m+i°2 5
a a a
dAb = ^a, dA. = A2a, = Aab — A2acm+iC2m+i;
dxb b dy2 dyb m+i 2 '
dA? _ , _ 1 , Ca Cm+i ^A? _ dA? _ rn
dxn _ ,n 2M«aCm+iCn > dx“ _ M?a’ dyn _ *?’
dA? _ 1 ab /nm+n 1 ^fa /^m+i ca .
dy _ 2 Cm+iC? Ab 2 Cm+iC? °2m_n+?;
dAa _ , _ 1 , ca Cm+i dAa _ ,, dAa _ o dAa _ rb
dx? _ ,a 2,abCm+iC? > dxb _ ,ab’ dy? _0, dyb _ da'
Solving this system, we represent the solution in the following form A? _ x? + p?(y).
Aa _ Xa _ ^2m—n+a _ 2Cm+iC?m+iX« + Pa(y),
A? _ y? _ 1 Cm+iC?m+iya _ ^2m-n+?ya + ^?(x),
Aa ya + ^a(x):
where ^? (x), ¥>a(x), p? (y), pa(y), are smooth functions of the corresponding variables. Substitution of these expressions into the first equation of system (3.3) gives the formula
In A _ x?y? + xaya _ ^m-n+?x?ya _ Cm+iCm+ix?ya + ^(x) +
where <^>(x) _ ^(x1,..., xm^d p(y) _ p(yi,..., ym) are smooth functions on the corresponding fibers of the double bundle N. Therefore, the following result holds.
Theorem 3.1. An n _ rn-tep/e integral depending on n _ m parameters inducing a differential-geometric structure (3.2) on the 2(n _ m) -dimensional submanifold N of variables and parameters can be reduced to an integral of the form
Q_ P(x)Q(y) exp x?y? + xaya _ ^m-n+?x?ya _ Cm+iCm+ix?ya X
X dx 2m_n+1 A • • • A dxm (3.6)
w/iere P(x) _ P(x2m n+1,..., xm) and Q(y) _ Q(y2m_n+1;... ,ym) are the exponents of functions ^(x) _ ^(x2m_n+1,..., xm) and p(y) _ p(y2m_n+1;... ,ym) respectively.
In the special case when the values Cm+i, Cmb+i are constants, we arrive at the formulas
Ca Cab y Cm+i cm+ixn
Cm+® _ Cm+iyb, C? _ C?n x ,
and expression (3.6) can be rewritten in the following more symmetric form
Q_ P(x)Q(y) exp x?y? + xaya _ ^2m_n+?x?ya _ Cm^C^x?xnyayb X
X dx2m_n+1 A •••A dxm (3.7)
Let us note that the partial derivatives of the functions Cm+i, 6^+® occurring in the canonical integral of a differential-geometric structure (3.6) compose the curvature tensor of the corresponding affine connection. Besides, the components of the curvature tensor are coefficients of nionoms of the fourth degree.
4. Parametric equations
To find parametric equations of the submanifold M under consideration, let us integrate structure equations (2.3"). To do this, let us consider the equations of infinitesimal displacement of a moving frame (P, ea, e?, ea, em+®, ea, e?, ea, em+i) in the space of affine connection En„
dea _ w^fe^ + w^e? + w0“ea + ^m+iem+\
de? _ w|ep + w? e? + w? ea + .m+®em+i-
dea _ w2ea + w|e? + w6aeb + wm+iem+\
dem+® _ wm+®ea + w^e? + wm+®ea + wm+kem+k,
de? _ _waea _ w?r en _ w|ea _ Wm+kem+J,
dea _ _waae« _ w?e? _ wbeb _ wm+kem+k,
dem+i _ _wm+iea _ wm+ie? _ wm+iea _ wm+i em+k.
M
wa _o, w6a _ o, w? _ o, wa _ o, wm+® _o, w? _o,
w? _o, wm+® _o, wa _o, wm+® _o, wm+® _o,
a, _ 1, 2m _ n; £, n _ 2m _ n + 1, .. ., n _ m; a, b _ n _ m + 1,. .., m.
Substitution of these relations into the previous system gives the following equations of infinitesimal displacement of a moving frame (P, ea, e?, ea, em+®, ea, e?, ea, em+i) on the
M
dea _ C?*p dyp e?, de? _ o,
dea _ 1 (Cm+iC?m;+idxn _ Cmb+iC?m+idyb) e? + Cmb+idybem+\ dem+® _ Cm+®dxpea + Cm+idxne? + wm+kem+k,
ap 1 ?n m+k
dea _ o,
de? _ _C?aPdypea _ 2 (cm+ic?m+idxn _ Cmb+iC?m+idy^ ea _ C?^^em+i, dea _ o,
dem+i _ _Cm+idybea _ wm+j em+k.
Replacing the secondary forms wm+k by _w*k and solving the obtained, we obtain the following expressions for the basis vectors
ea _ C?a(e?)o + (ea)o, e? _ (e?)o,
s~ia /'-»m+i s~ia ^rtm+a
°m+i°p _ Cm+?C? Cp
_i_ 1 r^a /^m+i . r-ia
+ 2 °m+i°? + Cm+^vp v-'? 1“ v-'?
+ (cm+a _ cm+? c?a) (em+a)o+cm+? (em+? )o + (ea)o, em+a _ Cm+a(ep)o + (cm+ac| + Cm+a) (e?)o + (em+a)o,
em+? _
??
'P _ C?aCm+aJ (ep)o + (Cm+?Cna + Cm+?) (en)o _ C?a(em+a)o + (em+?)c e _ (e )o,
e? _ _Ca (ea)o + ^2C?m+®Cm+i + C?"^CaCm(ea)o _ (em+a)o +
+
n)o,
ea _ (ea)o,
em+a _ _ (Cm+aC? Cm+?) (ea)o + C? (em+?)o + (em+a)o, em+? _ _Cm+?(ea )o + (em+?)o.
where [P, (ea)o, (e?)o, (ea)o, (em+a)o, (em+?)o, (ea)o, (e?)o, (ea)o, (em+a)o, (em+?)o] IS a fixed orthonormal frame in £2n . Substitution of these relations into the equation
dP _ _w“ea _ w?e? _ waea _ w2m_n+iem+i _ wae“ _ w?e? _ waea _ w®em+®
and further integration gives the equality
P _ _xa(ea)o _ x?(e?)o_
xa _ C?m+aCm+nCna _ 4m_n+a (Cm+i + C^+J yn +
_l_ A? (na _i_ r^ar^a \ ry ^m+i_
+ y2m_n+a VCm+a + Cn Cm+n/ Cm+iC?
_ ^2m_n+a (Cm+a + CaCm+?) yb _ ^2m_n+?Cm+?yb (ea)o_
^?m_n+ay? + Cm+a + ^2m_n+aya _ ^fm_n+aCm+iC? + (em+a)o_
^2m_n+?ya + ^2m_n+aC? ya ^2m_n+aCm+iCn C? + ^2m_
n+aC?ayn +
+ (Cm+? + Cm+aC?“) (em+?)o _
ya +
-C
x?+
+ fCm+iCm+i _ Cm+?CPCm+p) + Ca + CpPCm+Px?l (ea)o_
y? + C? + ( _ ~ Cm+iCm'+i + Cm+i
? + c”+M + ( Cp C? + C?J +
+ C“ fcm+acpxn + C„m+a) + (• • • + C?)! (e?)o _ ya(ea)o_
_ (xa + Cm+a _ Cm+?C^) (em+a)o _ (x? + Cm+?) (em+?)o
em
Therefore, the following statement is true.
Theorem 4.1. The submanifold, M has be given by parametric equations of the form
Xa = xa, X2 = x2,
x a - cp+“cm+nca - 4m_„+a (Cm+i+cacm+n) yn+
2 Cm+nCn u2m-n+a VCm+i T Cn
_l_ x 2 fca I r-^ar^a \ r<m+i _
+ °2m-n+a VCm+a + Cn Cm+n / Cm+iC2
- A2m-n+a (Cm + a + C£ Cm+2) y2 - A2m-n+2Cm+2y2,
Xm+a = £2m_n+ay2 + Cm+a + A2m-n+aya - 4m-n+aCm+iCm+‘,
Xm+2 = ^am-n+2ya + ^2am-n+aCaya - <5^^^+^+^ +
+ ^"m-n+aCfyn + Cm+2 + Cm+aC?a,
Ya = ya + (Cm+2 - CfC^+f) X2 + (Cm+iCm+i - Cm+2Cf C^+f ) + + Ca + Cf Cm+f X2,
Y2 = y2 + C2 +
1 C /tm+i | ry f s~im+i s~if | s~im+i\
2 Cm+iC2 + Cm+i f C2 + C2 y
+ C“ (Cm+aCfxn + ^m+^ + Cm+nCa,
Ya ya? "^m+a x + Cm+a Cm+2C£ , "^m+2 2 H_Cm+2,
+ Cf Cf+
where
C m+a = Cm+aj^l X m-n) Cm+2 = C m+2 (X m-n)
Ca = Cajx1, . . . , X2m-n), C2 = C2(yn-m+1, . . . , ym),
Cm+a Cm+a(yn-m+1? • • • , ym)? Cm+2 Cm+2 (yn-m+1? • • • , ym)
are smooth functions satisfying the following differential equations
dCm+a = C^+adxf, dCm+2 = C^+2 dxa, dCa = Cm+f dxf,
dCm+a Cm +adya? dCm+2 Cm+2 dya-
It is easy to check that the parametric equations of the submanifolds N c M can
be written in the following form
X2 = x2,
Xa = xa - A2 Ca y + A2 Ca C , .Cm+i - A2 Ca y,
X x A 2 m-n+aCm+iyn + A 2 m-n+aCm+aCm+iC2 A 2 m-n+aCm+2 y2’
m+a a m+i
X = A 2 m-n+ay2 + Cm+a + A 2 m-n+aya A 2 m-n+aCm+iC2 ,
X m+2 = Aa y + Cm+2 Y^ = n, + - C ,Cm+® + C ,Cm+® Y = y
X — A2m-n+2 ya + C , Y2 — y2 + 2 Cm+iC2 + Cm+iC2 , Ya — ya7
^m+a = x + Cm+a? ^m+2 = x2 + Cm+2,
Where Cm+a = Cm+a(yn-m+1? • • • , ym) ; Cm+2 = Cm+2(yn-m+1? • • • , ym) <M*e SinOOth
functions satisfying the following differential equations
dCm+a Cm +adya? dCm+2 Cm+2dya •
Резюме
G.X. Арутюнян. О геометрии подмногообразий в EJn .
Изучается специальный класс 2т-мерных иодмногообразий в 2п-мерном псевдоев-клидовом пространстве с метрикой сигнатуры (п,п), известном как исевдоевклидово пространство Рашевского. Для изучаемых подмногообразий найдены канонические интегралы и параметрические уравнения.
Ключевые слова: четпомерпое подмногообразие, исевдоевклидово пространство Рашевского. двойное расслоение, канонический интеграл, дифферепциалыю-геометриче-ская структура, расслоение, слоение.
References
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+B(x,x')) dt // Trans. Amer. Math. Soc. - 1938. - V. 44. - P. 153-167.
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6. Kahler E. t/ber eine bemerkenswerte Hermitesche Metrik // Abh. Math. Sem. Univ. Hamburg. - 1933. - Bd. 9. - S. 173-186.
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10. Haroutunian S. On some classes of differential-geometric structures on submanifolds of pseudoeuclidean space Е‘П(+П+1) // Izv. Vuz. Matematika. - 1989. - No 10. - P. 3-11 [in Russian].
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Поступила в редакцию 08.09.09
Арутюнян Самвел Христофорович доктор физико-математических паук, профессор, заведующий кафедрой высшей алгебры и геометрии Армянского государственного педагогического университета, г. Ереван, Республика Армения.
Е-шаП: НагоиЫтапвпе^уя.ат
