Generalization of the H.A. Schwarz Theorem on Stability of Minimal Surfaces
We proved two theorems on stability of minimal submanifolds in a Riemannian space, which can be included in a regular family of minimal submanifolds.
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Цитувати: | Generalization of the H. A. Schwarz theorem on stability of minimal surfaces / Yu. Aminov, J. Witkowska // Журн. мат. физики, анализа, геометрии. — 2007. — Т. 3, № 4. — С. 399-410. — Бібліогр.: 16 назв. — англ. |
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irk-123456789-76152010-04-07T12:01:04Z Generalization of the H.A. Schwarz Theorem on Stability of Minimal Surfaces Aminov, Yu. Witkowska, J. We proved two theorems on stability of minimal submanifolds in a Riemannian space, which can be included in a regular family of minimal submanifolds. Доведено дві теореми про стійкість мінімальних підбагатовидів в рімановому просторі, якщо мінімальний підбагатовид можливо включити в регулярну сім'ю мінімальних підбагатовидів. 2007 Article Generalization of the H. A. Schwarz theorem on stability of minimal surfaces / Yu. Aminov, J. Witkowska // Журн. мат. физики, анализа, геометрии. — 2007. — Т. 3, № 4. — С. 399-410. — Бібліогр.: 16 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/7615 en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We proved two theorems on stability of minimal submanifolds in a Riemannian space, which can be included in a regular family of minimal submanifolds. |
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Article |
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Aminov, Yu. Witkowska, J. |
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Aminov, Yu. Witkowska, J. Generalization of the H.A. Schwarz Theorem on Stability of Minimal Surfaces |
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Aminov, Yu. Witkowska, J. |
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Aminov, Yu. |
| title |
Generalization of the H.A. Schwarz Theorem on Stability of Minimal Surfaces |
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Generalization of the H.A. Schwarz Theorem on Stability of Minimal Surfaces |
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Generalization of the H.A. Schwarz Theorem on Stability of Minimal Surfaces |
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Generalization of the H.A. Schwarz Theorem on Stability of Minimal Surfaces |
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Generalization of the H.A. Schwarz Theorem on Stability of Minimal Surfaces |
| title_sort |
generalization of the h.a. schwarz theorem on stability of minimal surfaces |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/7615 |
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Generalization of the H. A. Schwarz theorem on stability of minimal surfaces / Yu. Aminov, J. Witkowska // Журн. мат. физики, анализа, геометрии. — 2007. — Т. 3, № 4. — С. 399-410. — Бібліогр.: 16 назв. — англ. |
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AT aminovyu generalizationofthehaschwarztheoremonstabilityofminimalsurfaces AT witkowskaj generalizationofthehaschwarztheoremonstabilityofminimalsurfaces |
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2025-07-02T10:25:47Z |
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2025-07-02T10:25:47Z |
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1836530477910982656 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2007, vol. 3, No. 4, pp. 399�410
Generalization of the H.A. Schwarz Theorem
on Stability of Minimal Surfaces
Yuriy Aminov
Institute of Mathematics of the Polish Academy of Sciences
8 Sniadeckich, PO Box 21, 00-956, Warsaw, Poland
E-mail:Y.Aminov@impan.gov.pl
Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkiv, 61103, Ukraine
E-mail:aminov@ilt.kharkov.u
Joanna Witkowska
Bialystok University of Finance and Management, Faculty of Engineering
1 Grunwaldska Str., 19-300 Elk, Poland
Received January 19, 2007
We proved two theorems on stability of minimal submanifolds in
a Riemannian space, which can be included in a regular family of minimal
submanifolds.
Key words: Riemannian space, minimal submanifold, stability.
Mathematics Subject Classi�cation 2000: 53A10, 53A07, 53C42.
1. Introduction
H.A. Schwarz proved stability of a minimal surface in 3-dimensional Euclidean
space E3, when this minimal surface could be included in a regular family of
minimal surfaces [1]. It follows from this theorem that every compact domain on
a minimal surface z = z(x1; x2) is stable.
Notice, that the question of minimal surface stability was considered in [3�9].
The existence and applications of stable minimal surfaces were given in [10�16].
Here we give the generalizations of this theorem for the cases of minimal hy-
persurfaces in a Riemannian space and for 2-dimensional surfaces in 4-dimensional
Riemannian space.
Let F n be a minimal submanifold with boundary � in a Riemannian manifold
V N . We consider some submanifold �n with the same boundary �, which is close
to F n in the class C1.
c
Yu. Aminov and J. Witkowska, 2007
Yu. Aminov and J. Witkowska
We say that a compact domain D with nonempty boundary � on a minimal
submanifold F n is stable, if for all submanifolds �n with the same boundary �,
close to D in the class C1 but di�erent from D, the volume V ol(�n) is grater
than the volume of D
V ol(�n) > V ol(D):
Theorem 1. If a simple connected compact domain D on an orientable
minimal hypersurface F n in the Riemannian manifold V n+1 can be included in
a regular family of minimal hypersurfaces, then this domain D is stable.
Theorem 2. Let F 2 be an orientable minimal surface in 4-dimensional
Riemannian manifold. Let a simple connected compact domain D on F 2 can
be included in a 2-parametric regular family of minimal surfaces with integrable
distribution of normal planes. Then this domain D is stable.
Later we construct a 2-parametric family of stable minimal surfaces
in Euclidean space E4 with nonintegrable distribution of normal planes. From
another side, there exists a nonstable minimal surface in the Euclidean space E4,
which can be included in the regular family of minimal surfaces. In this case the
distribution of normal planes is nonintegrable, too. This example shows that the
second condition in Th. 2 is essential.
2. Minimal Hypersurface
Later under F n we understand the simple connected compact domain D.
Let F n be included in the regular family of minimal hypersurfaces F n(t) such
that F n(0) = F n. We introduce on F n a coordinate system with the coordi-
nates y1; : : : ; yn. With the help of orthogonal trajectories to the family F n(t) we
construct a coordinate system with the coordinates y1; : : : ; yn+1 in some neighbor-
hood of F n. Every F n(t) corresponds to the equation yn+1 = const. The metric
of the space V n+1 takes the following form:
ds2 =
nX
i=1
aijdy
idyj + an+1;n+1(dy
n+1)2; (1)
where all coe�cients depend on all coordinates as regular functions of the class
C1. Later 1 � i; j � n. Denote an+1;n+1 = h; yn+1 = t.
Lemma 1. The coe�cients Lij of the second quadratic form of F n(t) have the
following form:
Lij = �
1
2h
@aij
@t
; i; j = 1; : : : ; n: (2)
400 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4
Generalization of the H.A. Schwarz Theorem on Stability of Minimal Surfaces
This lemma is well known (see, for example, [2]).
Denote a = jaij j.
Lemma 2. If every F n(t) is a minimal hypersurface, then
@a
@t
= 0: (3)
For a minimal hypersurface the mean curvature H is equal to zero, and we
have
H =
1
n
Lija
ij = 0;
where aij are the elements of the inverse matrix to jjaij jj. As a consequence of
(2) and (3), we obtain
aij
@aij
@t
= 0:
For simplicity we denote
@aij
@t
= a
0
ij
. Introduce also the following vectors:
li = (a1i; : : : ; ani); l
0
i = (a
0
1i; : : : ; a
0
ni):
Later we write these vectors in the form of columns and denote the determinant
by [ ]. We have evidently
@a
@t
= [l
0
1; l2; : : : ; ln] + � � �+ [l1; : : : ; l
0
n
] = a
nX
i;j
aij
@aij
@t
= 0:
Let �n be some hypersurface, which is close to F n in the class C1. In this case
�n has one-to-one projection on F n and in the correspondent points its tangent
spaces are close. We can write the representation of �n in the evident form:
yn+1 = f(y1; : : : ; yn)
with the condition f j� = 0. Denote later @f
@yi
= fi. The �rst quadratic form
dl2 = bijdy
idyj of �n can be calculated with the help of metric form of V n+1
dl2 =
nX
i;j
(aijdy
idyj + h2fifjdy
idyj):
Hence
bij = aij + h2fifj:
Introduce the vectors
ai = (a1i; : : : ; ani); m = (f1; : : : ; fn):
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 401
Yu. Aminov and J. Witkowska
Later these vectors are written in the form of columns. We have
jbij j = [a1+h2f1m;a2+h2f2m; : : : ; an+h2fnm] = a+h2
nX
i
[a1; : : : ;m; : : : ; an]fi;
where in sum the vector m stays on the i-th place. Taking the decomposition of
every determinant in sum, we obtain
jbijj = a(1 + h2
nX
i;j=1
fifja
ij):
But the matrix aij is positively determined, so
nX
i;j=1
fifja
ij � 0; (4)
and the equality can be only in the case when all fi = 0. Therefore, f = const.
But f j� = 0. Hence, f = 0. If we put a condition that �n is di�erent from F n,
then there exists some subset, where in (4) we have strong inequality. Denote by
G the domain of the coordinates y1; : : : ; yn. Now we can calculate the volume of
�n and compare it with the volume of F n
V ol(�n) =
Z
G
q
jbij(y1; : : : ; f)jdy
1 : : : dyn >
Z
G
p
a(y1; : : : ; f)dy1 : : : dyn
=
Z
G
p
a(y1; : : : ; 0)dy1 : : : dyn = V ol(F n):
Hence, F n is the stable minimal hypersurface.
The reviewer remarked that in the paper by H. Rosenberg [17] there were
some statements close to the ones of Th. 1. But in the paper there was indicated
only a weak stability. Besides, the consideration was too short and therefore not
clear enough.
3. Minimal Surface in a 4-Dimensional Riemannian Space
Let F 2 be a minimal surface in the Riemannian 4-dimensional space V 4. We
suppose that F 2 is included in a 2-parametric regular family of minimal surfaces
F 2(t; �) in some neighborhood D such that F 2(0; 0) = F 2. We say that it is the
�rst family. Through every point in the neighborhood D of F 2 there goes one
and only one surface from this family. Therefore, at this point the normal plane
is determined, and we have a distribution of normal planes.
402 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4
Generalization of the H.A. Schwarz Theorem on Stability of Minimal Surfaces
By the conditions of Th. 2, the distribution of these normal planes is integrable.
So, there exists the second family of the surfaces which are orthogonal to the
surfaces from the �rst family. With the help of these two families, in the same
way we can construct the coordinates in the considered neighborhood. We take
the coordinate system y1; y2 on the surface F 2 and take the surface N0 from
the second family, which goes through some point p0 2 F 2. We introduce the
coordinates y3; y4 on the surface N0. So, if a point p 2 D, then through this point
p there goes one surface from the second family, which intersects with F 2 at the
point with coordinates y1; y2 as well as one surface from the �rst family, which
intersects with N0 at the point with coordinates y3; y4. Hence the point p has
coordinates y1; : : : ; y4.
Later the Latin indexes have the value 1 or 2, and the Greek ones 3 or 4,
respectively. Then the �rst quadratic form of V 4 will be
ds2 =
2X
i;j=1
aijdy
idyj +
4X
�;�=3
a��dy
�dy� ;
where all coe�cients depend on y1; : : : ; y4.
Now let some surface �2 be close to F 2 and have the same boundary. We can
represent �2 in the following form
y� = f�(y1; y2); � = 3; 4;
and f� = 0 on the boundary. Denoting the metric of �2 by dl2 = bijdy
idyj
we obtain
bij = aij + a��y
�
;iy
�
;j
;
where y�
;i
are the derivatives with respect to coordinate yi. Let F n(t; �) be a mi-
nimal surface, which goes through the point with coordinates y1; y2; y3; y4 on the
surface �2.
Lemma 3. Determinant a of the �rst quadratic form of F 2(t; �) does not
depend on y3 and y4
@a
@y3
=
@a
@y4
= 0:
Let �k = f��
k
g; k = 1; 2, be an orthogonal basis of normal plane of F 2(t; �)
and Lk
ij
be the coe�cients of the second quadratic forms of F 2(t; �) with respect
to this basis. Following the de�nition of the second quadratic forms (see [2]), we
have two equations for � = 3; 4
y�;ij +
�����y
�
;i
y�:j = Lk
ij�
�
k
;
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 403
Yu. Aminov and J. Witkowska
where ���
��
are the Christo�el symbols of the metric of V 4. Here y�
;ij
are the second
covariant derivatives of the function y� with respect to the metric of F 2(t; �).
We notice that every surface of this kind has the representation
y3 = const; y4 = const:
Hence
y�
;i
= 0; y�
;ij
=
@2y�
@yi@yj
� �k
ij
y�
;k
= 0; � = 3; 4;
where �k
ij
are the Christo�el symbols of the metric of F n(t; �).
Besides, yi
;j
= 0. So we have
���ij = Lk
ij�
�
k
:
From the expressions of the Christo�el symbols we obtain
1
2
a��(
@a�i
@yj
+
@a�j
@yi
�
@aij
@y�
) = Lk
ij�
�
k
:
But a�i = 0. Therefore
�
1
2
a��
@aij
@y�
= Lk
ij
��
k
:
For a minimal surface we have
Lk
ij
aij = 0; k = 1; 2:
Therefore, we have the system of equations
@a
@y3
a33 +
@a
@y4
a34 = 0;
@a
@y3
a34 +
@a
@y4
a44 = 0:
From here Lemma 3 follows.
Now we have
jbij j =
�����
a11 + a��y
�
;1y
�
;1
; a12 + a
�y
;1
y�
;2
a21 + a��y
�
;2y
�
;1
; a22 + a
�y
;2
y�
;2
����� :
Denote by y�ji = y
�
;k
aki and
p�� =
�����
y�
;1; y
�
;1
y�
;2; y
�
;2
����� :
404 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4
Generalization of the H.A. Schwarz Theorem on Stability of Minimal Surfaces
Then the expression of jbij j can be transformed to the following one:
jbij j = a(1 + a��(y
�j2y�;2 + y�;1y
�j1) +
1
2a
a��a
�p
�
p��):
Denote by grady� the gradient of the function y� with respect to coordinates
y1; y2 and the metric of F 2(t; �). So we obtain
jbij j = a[1 + a��(grady
�; grady�) +
1
2a
(a33a44 � (a34)
2)(p34)2]:
Here the brackets () at the second member in the right side denote the scalar
product in the metric aijdy
idyj at a point of F 2(t; �). It is clear that the third
term is nonnegative. Let us denote
A = (grady3)2; B = (grady3; grady4); C = (grady4)2:
The second term in the expression of jbij j has the form
a(Aa33 + 2Ba34 + Ca44):
We have evidently
AC �B2 � 0; a33a44 � (a34)
2 � 0:
Under these conditions the expression T = Aa33 + 2Ba34 + Ca44 � 0.
Hence jbij j � a. If there is an equality here, then p34 = 0. In this case there
exist some functions �(y1; y2) and ��(�) such that
y� = ��(�); � = 3; 4:
Under this condition the expression T has the form
T = jgrad�j2����a��:
So, from T = 0 we conclude that y� = const; � = 3; 4: But �2 is di�erent from
F 2. Therefore, we have some subset, where jbij j > a.
But a depends neither on y3, nor on y4. Therefore V ol(�2) > V ol(F 2).
Theorem 2 is proved.
4. One Example
Now we construct a 2-parametric family of minimal surfaces in E4 with the
nonintegrable distribution of normal planes.
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 405
Yu. Aminov and J. Witkowska
Denote by xk the coordinates in E4 and z1 = x1 + ix2, z2 = x3 + ix4:
Consider the family of minimal surfaces in E4 , which are given as level surfaces
of an analytical function of two complex variables
f(z1; z2) = c1 + ic2:
We have two real equations
�1 = Ref(z1; z2) = c1;
�2 = Imf(z1; z2) = c2:
It is a well-known fact that this surface is minimal and it is a holomorphic curve
in E4. Every compact domain is an absolutely minimized area. Normal plane is
determined by the following vectors:
Xi = grad�i; i = 1; 2:
Then the condition of integrability of distribution of normal planes has
the following form
rX2
X1 �rX1
X2 = �1X1 + �2X2 (5)
with some coe�cients �k. We take the particular example
f = z1z2 + z21 + z22 :
Then evidently we obtain
�1 = x1x3 � x2x4 + x21 � x22 + x23 � x24;
�2 = x2x3 + x1x4 + 2x1x2 + 2x3x4:
Consequently,
grad�1 = (x3 + 2x1;�x4 � 2x2; x1 + 2x3;�x2 � 2x4);
grad�2 = (x4 + 2x2; x3 + 2x1; x2 + 2x4; x1 + 2x3): (6)
For the simplicity of notation denote �1 = �; �2 = . By calculation we obtain
the matrices of the second derivatives for the functions � and
jj
@2�
@xi@xj
jj =
0
BB@
2; 0; 1; 0
0; �2; 0; �1
1; 0; 2; 0
0; �1; 0; �2
1
CCA ; jj
@2
@xi@xj
jj =
0
BB@
0; 2; 0; 1
2; 0; 1; 0
0; 1; 0; 2
1; 0; 2; 0
1
CCA : (7)
406 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4
Generalization of the H.A. Schwarz Theorem on Stability of Minimal Surfaces
Let us denote the second derivatives of functions, for example, �, by �ij. Intro-
duce the notation
ri =
X
j
(�ij j � ij�j): (8)
With the help of (5) - (8) we obtain the system of equations for �i
r1 = 10x2 + 8x4 = �1(x3 + 2x1) + �2(x4 + 2x2);
r2 = �10x1 � 8x3 = �1(�x4 � 2x2) + �2(x3 + 2x1);
r3 = 8x2 + 10x4 = �1(x1 + 2x3) + �2(x2 + 2x4);
r4 = �8x1 � 10x3 = �1(�x2 � 2x4) + �2(x1 + 2x3):
From the �rst two equations we have
�1 = �
80(x2x3 � x1x4)
(x1 + 2x3)2 + (x2 + 2x4)2
:
From the last two equations we �nd
�1 = �
80(x2x3 � x1x4)
(2x1 + x3)2 + (2x2 + x4)2
:
These expressions are di�erent, so the system does not have any solution.
Hence, the distribution of normal planes is nonintegrable.
5. Minimal Surfaces in E4 with Nonparametric Representation
Let the minimal surface F 2 � E4 be given in the form
x3 = u(x1; x2);
x4 = v(x1; x2):
Later we denote derivatives in the form of ui; uij . The functions u and v of
a minimal surface satisfy two di�erential equations (see, for example, [10])
u11(1 + u22 + v22)� 2u12(u1u2 + v1v2) + u22(1 + u21 + v21) = 0;
v11(1 + u22 + v22)� 2v12(u1u2 + v1v2) + v22(1 + u21 + v21) = 0: (9)
It is easy to construct the family of minimal surfaces F 2(c1; c2)
x3 = u+ c1;
x4 = v + c2; ci = const:
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 407
Yu. Aminov and J. Witkowska
A normal plane is determined by vectors X1; X2
X1 = (u1; u2;�1; 0);
X2 = (v1; v2; 0;�1):
The condition of integrability of the distribution of normal planes is represented
by the following system of equations:
u11v1 + u12v2 � v11u1 � v12u2 = �1u1 + �2v1;
u12v1 + u22v2 � v12u1 � v22u2 = �1u2 + �2v2;
0 = ��1 + 0�2; (10)
0 = 0�1 � �2:
From here we have �1 = �2 = 0. Hence the condition (10) has the form
u11v1 + u12v2 = v11u1 + v12u2;
u12v1 + u22v2 = v12u1 + v22u2: (11)
Therefore, by Theorem 2 the minimal surface in E4 at nonparametric repre-
sentation is strongly stable if it satis�es the system of equations (11).
The reviewer proposed to construct an example of minimal surface which
would satisfy the system (9),(11).
To construct this example we put
u = �(x1) + �(x2); v = �(x1) + �(x2):
Then the system (9),(11) has the following form:
�00(1 + �02 + �02) + �00(1 + �02 + �02) = 0;
�00(1 + �02 + �02) + �00(1 + �02 + �02 = 0;
�00�0 � �00�0 = 0;
�00�0 � �00�0 = 0;
where 0 (prime) denotes the derivatives of function � or �; : : : with respect to their
arguments. From the third and forth equations we obtain
� = C1� + C2; � = C3� + C4;
408 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4
Generalization of the H.A. Schwarz Theorem on Stability of Minimal Surfaces
where Ci are constants. After substitution � and � into the �rst and the second
equations we conclude that C1 = C3 and the second equation can be rewritten in
the form of equation with separate arguments
�00
1 + �02a2
= �
�00
1 + �02a2
= k;
where k = const; a =
p
1 +C2
1
. By integration we obtain the equation of minimal
surface
u =
p
a2 � 1v; v =
1
ka2
ln
cos(kax2 + d2)
cos(kax1 + d1)
;
where di are constants. It is evidently that the surface is not determined on the
whole plane x1, x2.
In [7] M.J. Micallef proved the following Corollary 5.1 A complete stable
minimal surface in E4, which is an entire graph, is holomorphic. He indicated that
in [10] R. Osserman constructed the examples of entire two-dimensional minimal
graphs in E4, which were not holomorphic with respect to any orthogonal complex
structure on E4. These graphs are unstable by Cor. 5.1. So, on this surface there
exist the unstable domains. One of the Osserman surfaces has the following
representation:
x3 = u =
1
2
cos
x2
2
(ex1 � 3e�x1);
x4 = v = �
1
2
sin
x2
2
(ex1 � 3e�x1):
It is possible to include this surface in the family of minimal surfaces. The dis-
tribution of normal planes is not integrable, because the equations (11) for this
surface are not satis�ed. Therefore, the condition of integrability of the distribu-
tion of normal planes in Th. 2 is essential.
The Authors are thankful to the reviewer for helpful remarks.
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