On Pseudospherical Surfaces in E⁴ with Grassmann Image of Prescribed Type
It is demonstrated that there exist surfaces of constant negative Gauss curvature in E⁴ whose Grassmann image consists of either hyperbolic or parabolic or elliptic points. As a consequence, there exist surfaces of constant negative Gauss curvature in E⁴ which do not admit Backlund transformations w...
Збережено в:
| Дата: | 2006 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2006
|
| Назва видання: | Журнал математической физики, анализа, геометрии |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/106588 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On Pseudospherical Surfaces in E⁴ with Grassmann Image of Prescribed Type / V. Gorkavyy // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 2. — С. 138-148. — Бібліогр.: 12 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-106588 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1065882016-10-01T03:02:19Z On Pseudospherical Surfaces in E⁴ with Grassmann Image of Prescribed Type Gorkavyy, V. It is demonstrated that there exist surfaces of constant negative Gauss curvature in E⁴ whose Grassmann image consists of either hyperbolic or parabolic or elliptic points. As a consequence, there exist surfaces of constant negative Gauss curvature in E⁴ which do not admit Backlund transformations with help of pseudospherical congruencies. A geometric representation for pseudospherical surfaces in E⁴ with parabolic Grassmann image is proposed. 2006 Article On Pseudospherical Surfaces in E⁴ with Grassmann Image of Prescribed Type / V. Gorkavyy // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 2. — С. 138-148. — Бібліогр.: 12 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106588 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
It is demonstrated that there exist surfaces of constant negative Gauss curvature in E⁴ whose Grassmann image consists of either hyperbolic or parabolic or elliptic points. As a consequence, there exist surfaces of constant negative Gauss curvature in E⁴ which do not admit Backlund transformations with help of pseudospherical congruencies. A geometric representation for pseudospherical surfaces in E⁴ with parabolic Grassmann image is proposed. |
| format |
Article |
| author |
Gorkavyy, V. |
| spellingShingle |
Gorkavyy, V. On Pseudospherical Surfaces in E⁴ with Grassmann Image of Prescribed Type Журнал математической физики, анализа, геометрии |
| author_facet |
Gorkavyy, V. |
| author_sort |
Gorkavyy, V. |
| title |
On Pseudospherical Surfaces in E⁴ with Grassmann Image of Prescribed Type |
| title_short |
On Pseudospherical Surfaces in E⁴ with Grassmann Image of Prescribed Type |
| title_full |
On Pseudospherical Surfaces in E⁴ with Grassmann Image of Prescribed Type |
| title_fullStr |
On Pseudospherical Surfaces in E⁴ with Grassmann Image of Prescribed Type |
| title_full_unstemmed |
On Pseudospherical Surfaces in E⁴ with Grassmann Image of Prescribed Type |
| title_sort |
on pseudospherical surfaces in e⁴ with grassmann image of prescribed type |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2006 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/106588 |
| citation_txt |
On Pseudospherical Surfaces in E⁴ with Grassmann Image of Prescribed Type / V. Gorkavyy // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 2. — С. 138-148. — Бібліогр.: 12 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT gorkavyyv onpseudosphericalsurfacesine4withgrassmannimageofprescribedtype |
| first_indexed |
2025-07-07T18:44:13Z |
| last_indexed |
2025-07-07T18:44:13Z |
| _version_ |
1837014824630878208 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2006, vol. 2, No. 2, pp. 138�148
On Pseudospherical Surfaces in E4 with Grassmann
Image of Prescribed Type
Vasyl Gorkavyy
Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov, 61103, Ukraine
E-mail:gorkavyy@ilt.kharkov.ua
Received October 5, 2005
It is demonstrated that there exist surfaces of constant negative Gauss
curvature in E
4 whose Grassmann image consists of either hyperbolic or
parabolic or elliptic points. As a consequence, there exist surfaces of con-
stant negative Gauss curvature in E4 which do not admit Backlund transfor-
mations with help of pseudospherical congruencies. A geometric represen-
tation for pseudospherical surfaces in E4 with parabolic Grassmann image
is proposed.
Key words: pseudospherical surface, Grassman image.
Mathematics Subject Classi�cation 2000: 53A07, 53A25, 53B25.
1. Introduction
The theory of surfaces with constant negative Gauss curvature K = �1 in
three-dimensional Euclidean space E3 is one of the most attractive branches of
the classical di�erential geometry. Initially the pseudospherical surfaces in E3
were of a great interest for many geometers since each surface with constant nega-
tive Gauss curvature realizes locally the hyperbolic geometry. Another important
reason for the studying of pseudospherical surfaces was the discovery of elegant
geometric transformations constructed and studied by L. Bianchi, A. Backlund,
G. Darboux and others. The corresponding geometric construction based on the
notion of pseudospherical congruencies in E3 has various nontrivial properties,
we shall mention two of them. Firstly, if two surfaces in E3 are connected by a
pseudospherical congruence, then both surfaces are of the same constant negative
Gauss curvature. Secondly, an arbitrary pseudospherical surface in E3 admits a
large family of Backlund transformations. So an iteration of Backlund transfor-
mations generates a family of pseudospherical surfaces from a given one [1�3].
c
Vasyl Gorkavyy, 2006
On Pseudospherical Surfaces in E4
with Grassmann Image of Prescribed Type
The pseudospherical surfaces in E3 may be interpreted via solutions of the
well-known sine-Gordon equation @xy' = sin', and the geometric Backlund
transformations of pseudospherical surfaces correspond to analytic transforma-
tions of solutions of SGE. This interpretation resulted in the general fundamental
idea of Backlund transformations for solutions of nonlinear partial di�erential
equations (integrable systems) [1�3].
Large majority of results from the classical theory of pseudospherical surfaces
in E3 and their Backlund transformations were generalized for n-dimensional
pseudospherical submanifolds in E2n�1 [4�7].
An attempt to generalize the mentioned constructions and ideas for two-
dimensional pseudospherical surfaces in four-dimensional Euclidean space E4 nec-
essarily leads to the consideration of Cartan surfaces. By de�nition, a Cartan
surface in E4 is characterized by the existence of a well-de�ned net of conjugate
curves. Besides, a Cartan surfaces in E4 may be de�ned in terms of the Grass-
man image (generalized Gauss image): the Grassmann image of a Cartan surface
consists of hyperbolic points (here we apply a classi�cation of points on surfaces
in E4 proposed by Yu. Aminov [1, Ch. 8, x 6]). It was remarked in [8�10], that
pseudospherical congruencies in E4 and corresponding Backlund transformations
may be constructed either for pseudospherical Cartan surfaces in E4 or for pseu-
dospherical hypersurfaces in E3 � E4 only.
On the other hand, there are many surfaces in E4 which are neither Cartan
surfaces nor hypersurfaces in E3 � E4. We speak about the surfaces whose Grass-
mann image consists of elliptic and/or parabolic points. By de�nition, a surface
with elliptic Grassmann image, an E-surface, is characterized by the absence of
conjugate directions in tangent planes. A surface with parabolic Grassmann im-
age, a P -surface, has a well-de�ned asymptotic direction at each its point. So
a P -surfaces in E4 is foliated in a unique way by asymptotic curves; the ruled
surfaces in E4 which are not hypersurfaces in E3 � E4 present a particular class
of P -surfaces.
The following question was of the major interest for us: do there exist sur-
faces of constant negative Gauss curvature in E4
which don't admit Backlund
transformations with help of pseudo-spherical congruencies? We reformulate this
question as follows: do there exist E-surfaces and/or P -surfaces with constant
negative Gauss curvature in E4
? It turns out that the answer is positive.
Theorem 1.
1. There exist P -surfaces with constant negative Gauss curvature in E4.
2. There exist E-surfaces with constant negative Gauss curvature in E4.
3. There exist Cartan surfaces with constant negative Gauss curvature in E4.
The E-surfaces as well as Cartan surfaces form two general classes of surfaces
in E4. Hence the existence of pseudospherical E-surfaces in E4 seems to be
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 139
Vasyl Gorkavyy
rather expected. We demonstrate that each pseudospherical E-surface in E4 may
be represented by one function of two variables and two functions of one variable.
The same situation is valid for the Cartan surfaces of constant negative curvature
in E4. As for the P -surfaces, they form a very particular class of surfaces in E4.
So the existence of pseudospherical P -surfaces is very surprising. We demonstrate,
that each pseudospherical P -surface in E4 may be represented by four functions
of one variable. From the geometric point of view, the corresponding initial data
for constructing of a pseudospherical P -surface F 2 in E4 are a generic curve
in
some E3
0 � E4 and a generic �eld of two-planes � along
which are tangent to
and transversal to E3
0 ; the surface F
2 passes through
and at each point x of
the tangent plane TxF
2 coincides with �(x), (see Th. 2 in Sect. 4).
The proven theorem leads us directly to the positive answer for the principal
question stated above.
Corollary. There exist pseudospherical surfaces in E4 which don't admit
Backlund transformations with help of pseudospherical congruencies.
It is an open question whether a similar statement holds for pseudo-spherical
Cartan surfaces in E4.
In order to deduce Corollary from Theorem 1, one can apply, for instance,
Theorem 4 from [8] which asserts that a pseudospherical E-surface in E4 doesn't
admit linear congruencies. The same is true for pseudospherical P -surfaces in E4.
Thus the direct generalization of Backlund transformations with help of pseu-
dospherical congruencies can not be applied to all pseudospherical surfaces in E4.
It would be very interesting to �nd an analogue of Backlund transformations for
pseudospherical E- and P -surfaces in E4, such an analogue has to be constructed
without any use of pseudospherical congruencies.
2. Classi�cation of Points on Surfaces in E4
Let F 2 be a regular two-dimensional surface in the four-dimensional Euclidean
space E4. Choose two normal �elds, ~n1, ~n2, on F 2 which form a frame in the
normal planes NxF
2. Let II� = L�
ij
duiduj , � = 1; 2, stand for the second funda-
mental forms of F 2 corresponding to the above choice of normals ~n1, ~n2.
Similarly to the classical case, various properties of the second fundamental
forms II� may be applied to classify points of F 2 � E4. We brie�y describe one
such classi�cation, which is based on the consideration of conjugate and asymp-
totic directions in tangent planes to F 2.
Let us recall some fundamental notions. The point codimension of F 2 at a
point x 2 F 2 (the dimension of the �rst normal space to F 2 at x) is de�ned by
140 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
On Pseudospherical Surfaces in E4
with Grassmann Image of Prescribed Type
the following formula:
codimx = Rank
�
L1
11 L1
12 L1
22
L2
11 L2
12 L2
22
�
:
Two directions X = (X1 : X2), Y = (Y 1 : Y 2) in the tangent plane TxF
2 are
called conjugate if
L1
11X
1Y 1 + L1
12(X
1Y 2 +X2Y 1) + L1
22X
2Y 2 = 0; (1)
L2
11X
1Y 1 + L2
12(X
1Y 2 +X2Y 1) + L2
22X
2Y 2 = 0: (2)
A self-conjugate direction X = (X1 : X2) in TxF
2 is called an asymptotic direc-
tion, its coordinates solve the following equations:
L1
11(X
1)2 + 2L1
12X
1X2 + L1
22(X
2)2 = 0;
L2
11(X
1)2 + 2L2
12X
1X2 + L2
22(X
2)2 = 0:
The existence of conjugate directions in TxF
2 depends on the solvability of
the system of algebraic equations (1)�(2). Write (1)�(2) as a system of two linear
equations with respect to Y 1, Y 2:�
L1
11X
1 + L1
12X
2 L1
12X
1 + L1
22X
2
L2
11X
1 + L2
12X
2 L2
12X
1 + L2
22X
2
��
Y 1
Y 2
�
=
�
0
0
�
: (3)
There exists a nonzero solution Y = (Y 1 : Y 2) of (3) if and only if
det
�
L1
11X
1 + L1
12X
2 L1
12X
1 + L1
22X
2
L2
11X
1 + L2
12X
2 L2
12X
1 + L2
22X
2
�
= 0;
i.e., if X = (X1 : X2) solves the following second-order homogeneous equation:
(X1)2(L1
11L
2
12�L
1
12L
2
11)+X
1X2(L1
11L
2
22�L
1
22L
2
11)+(X2)2(L1
12L
2
22�L
1
22L
2
12) = 0:
(4)
By symmetry, the same equation is satis�ed by Y = (Y 1 : Y 2).
Thus, the conjugate directions in TxF
2 are determined by solutions of (4).
The solvability of (4) depends on the coe�cients L�
ij
. Firstly, it is easy to see
that (4) is nondegenerate if and only if codimx is equal to 2. Secondly, in the
nondegenerate case the number of solutions of (4) depends on the sign of the
following discriminant:
D = (L1
11L
2
22 � L1
22L
2
11)
2 � 4(L1
11L
2
12 � L1
12L
2
11)(L
1
12L
2
22 � L1
22L
2
12):
If D > 0, then there exist two independent solutions X = (X1 : X2) and
Y = (Y 1 : Y 2), they determine a well-de�ned pair of independent conjugate
directions in TxF
2; in this case the point x 2 F 2 is said to be hyperbolic.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 141
Vasyl Gorkavyy
If D = 0, then there exist a unique solution X = (X1 : X2), it determines a
well-de�ned asymptotic direction in TxF
2; in this case the point x 2 F 2 is said
to be parabolic.
If D < 0, then there no exist nonzero solutions of (4), so the tangent plane
TxF
2 doesn't contain neither conjugate directions nor asymptotic ones; in this
case the point x 2 F 2 is said to be elliptic.
If codimx is equal to 1, the left hand side of (4) vanishes. It means that for
an arbitrary direction X = (X1;X2) in TxF
2 there exist a well-de�ned conjugate
direction Y = (Y 1; Y 2), which may be determined from (3).
If codimx is equal to 0, then the equations (1)�(2) degenerate, so arbitrary
directions in TxF
2 are conjugate.
Thus, we described �ve classes of points in F 2 � E4 with very di�erent
extrinsic-geometric properties depending on the point codimension and on the
sign of the discriminant D. The proposed classi�cation is well known, it cor-
responds to the so-called a�ne and Grassmannian classi�cations introduced by
A.A. Borisenko and by Yu.A. Aminov respectively (see [1, Ch. 8, x 6], [11, Ch. 3,
x 1]).
If a surface in E4 consists of hyperbolic points, it is called a Cartan surface;
such a surface carries a well-de�ned net of conjugate curves. A surface in E4 is
called a P -surface, if it consists of parabolic points; such a surface is foliated in
a unique way by asymptotic curves. A surface in E4 is called an E-surface, if it
consists of elliptic points; such a surface does not admit neither conjugate tangent
directions nor asymptotic ones.
If the point codimension is less than 2 at all points of a surface in E4, then
either such a surface belongs to some hyperplane E3 � E4 or it is ruled and has
a degenerate Grassmann image.
3. Pseudospherical Surfaces in E
4
The surface F 2 � E4 is said to be pseudospherical if its Gauss curvature K is
equal to �1. In view of the classi�cation discussed in the previous section, it is
naturally to analyze what kind of points may belong to F 2.
Let us suppose that F 2 � E4 is represented explicitly:
x1 = u; x2 = v; x3 = U(u; v); x4 = V (u; v); (5)
where U(u; v), V (u; v) are some Ck, k � 2, functions de�ned on a neighborhood of
the origin (0; 0) 2 R2. An easy calculation provides us with the following simple
formula for the Gauss curvature K [1, Ch. 6, x 7, p. 176]:
K =
�
(1 + (Uu)
2 + (Uv)
2)(VuuVvv � (Vuv)
2)
+(1 + (Vu)
2 + (Vv)
2)(UuuUvv � (Uuv)
2)
142 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
On Pseudospherical Surfaces in E4
with Grassmann Image of Prescribed Type
�(UuVu + UvVv)(UuuVvv � 2UuvVuv + UvvVuu))
=
�
1 + (Uu)
2 + (Uv)
2 + (Vu)
2 + (Vv)
2 + (UuVv � UvVu)
2
�2
:
Therefore the explicit surface F 2 � E4 is pseudospherical, K � �1, if and only
if the functions U(u; v) and V (u; v) satisfy the following second order partial
di�erential equation:
(1 + (Uu)
2 + (Uv)
2)(VuuVvv � (Vuv)
2) + (1 + (Vu)
2 + (Vv)
2)(UuuUvv � (Uuv)
2)
�(UuVu + UvVv)(UuuVvv � 2UuvVuv + UvvVuu)
+
�
1 + (Uu)
2 + (Uv)
2 + (Vu)
2 + (Vv)
2 + (UuVv � UvVu)
2
�2
= 0: (6)
It is easy to see, that (6) may be written in the following form:
Uuu =
A(Uu; Uv ; Uuv; Uvv ; Vu; Vv; Vuu; Vuv; Vvv)
(1 + (Vu)2 + (Vv)2)Uvv � (UuVu + UvVv)Vvv
; (7)
where A denotes some polynomial. By a corresponding existence and unique-
ness theorem from PDE theory [12, Ch. I, x 2, p. 24], for an arbitrary choice of
analytical functions V (u; v), U(0; v) = P (v), Uu(0; v) = Q(v) which obeys
(1 + (Vu)
2 + (Vv)
2)Uvv � (UuVu + UvVv)Vvv j(0;0)
= (1 + (Vu)
2 + (Vv)
2)Pvv � (QVu + PvVv)Vvv j(0;0) 6= 0;
there exists a unique analytical solution U(u; v) of (7) de�ned in a neighborhood
of (0; 0) 2 R2. Such a solution describes some pseudospherical surface in E4 via
the explicit representation (5).
In order to control the kind of points on F 2, we have to write the discriminant
D in terms of functions U(u; v) and V (u; v):
D = (UuuVvv � VuuUvv)
2 � 4(UuuVuv � VuuUuv)(UuvVvv � VuvUvv): (8)
As for the point codimension codim, it is determined by the following formula:
codim(u;v) = Rank
�
Uuu Uuv Uvv
Vuu Vuv Vvv
�
: (9)
Therefore, the surface F 2 � E4 consists of elliptic points if and only if the
functions U(u; v) and V (u; v) satisfy two conditions:
(UuuVvv � VuuUvv)
2 � 4(UuuVuv � VuuUuv)(UuvVvv � VuvUvv) < 0; (10:1)
Rank
�
Uuu Uuv Uvv
Vuu Vuv Vvv
�
= 2: (10:2)
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 143
Vasyl Gorkavyy
Moreover, in order to construct a pseudospherical E-surface in E4 the function
V (u; v) and the initial data U(0; v) = P (v), Uu(0; v) = Q(v) for (7) has to be cho-
sen in such a way that (10.1)�(10.2) hold at (0; 0). (For example, one can choose
V (u; v), P (v), Q(v) with Vuu(0; 0) = Vvv(0; 0) = 0, Vuv(0; 0) = 1=2, Pvv(0) 6= 0,
jQv(0)j <
p
3=2.) For such a choice of initial data there exists a solution U(u; v)
of (7), which satis�es (10.1) and (10.2) in a neighborhood of (0; 0) 2 R2. Then
(5) will describe a pseudospherical E-surface in E4.
Similarly, the surface F 2 � E4 consists of hyperbolic points if and only if the
functions U(u; v) and V (u; v) satisfy two following conditions:
(UuuVvv � VuuUvv)
2 � 4(UuuVuv � VuuUuv)(UuvVvv � VuvUvv) > 0; (11:1)
Rank
�
Uuu Uuv Uvv
Vuu Vuv Vvv
�
= 2: (11:2)
Besides, in order to construct a pseudospherical Cartan surface in E4 the function
V (u; v) and the initial data U(0; v) = P (v), Uu(0; v) = Q(v) for (7) has to be cho-
sen in such a way that (11.1)�(11.2) hold at (0; 0). (For example, one can choose
V (u; v), P (v), Q(v) with Vuu(0; 0) = Vvv(0; 0) = 0, Vuv(0; 0) = 1=2, Pvv(0) 6= 0,
jQv(0)j >
p
3=2.) For such a choice of initial data there exists a solution U(u; v)
of (7), which satis�es (11.1) and (11.2) in a neighborhood of (0; 0) 2 R2. Then
(5) will describe a pseudospherical Cartan surface in E4.
Finally, the surface F 2 � E4 consists of parabolic points if and only if the
functions U(u; v) and V (u; v) satisfy two conditions
(UuuVvv � VuuUvv)
2 � 4(UuuVuv � VuuUuv)(UuvVvv � VuvUvv) = 0; (12:1)
Rank
�
Uuu Uuv Uvv
Vuu Vuv Vvv
�
= 2: (12:2)
Therefore, in order to construct a pseudospherical P -surface in E4, we have to
solve two second order nonlinear partial di�erential equations, (6) and (12.1),
completed by the partial di�erential relation (12.2).
Consider (6) and (12.1) as a system of algebraic equations with respect to the
partial derivatives Uuu and Vuu. The equation (6) is linear, whereas (12.1) is a
second-order equation of parabolic type. So it's easy to see, that if Uu, Uv, Uuv,
Uvv , Vu, Vv, Vuv, Vvv satisfy some polynomial inequality
�(Uu; Uv; Uuv; Uvv ; Vu; Vv; Vuv ; Vvv) > 0; (13)
then (6) and (12.1) may be solved with respect to Uuu and Vuu as follows:
Uuu = A1(Uu; Uv; Uuv; Uvv ; Vu; Vv; Vuv; Vvv); (14:1)
144 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
On Pseudospherical Surfaces in E4
with Grassmann Image of Prescribed Type
Vuu = A2(Uu; Uv; Uuv; Uvv ; Vu; Vv; Vuv ; Vvv); (14:2)
where A1 and A2 are some analytical functions de�ned on an open domain
D =
�
(y1; : : : ; y8) 2 R8 j�(y1; : : : ; y8) > 0
:
Choose analytical initial data
U(0; v) = P (v); Uu(0; v) = Q(v); V (0; v) = R(v); Vu(0; v) = S(v) (15)
in such a way that
(Q(0); Pv(0); Qv(0); Pvv(0); S(0); Rv(0); Sv(0); Rvv(0)) 2 D: (16)
Then by Cauchy�Kowalewskaya theorem [12, Ch. I, x 2, p. 24] there exists a unique
analytical solution U(u; v), V (u; v) of (14.1)�(14.2), which is de�ned in a neigh-
borhood of (0; 0) 2 R2.
The constructed solution U(u; v), V (u; v) obeys the additional constraint
(12.2) in a neighborhood of (0; 0) 2 R2 if P (v), Q(v), R(v) and S(v) satisfy
the following inequality:
Qv(0)Rvv(0) � Pvv(0)Sv(0) 6= 0: (17)
Consider the open domain
~D =
�
(y1; : : : ; y8) 2 R8jy3y8 � y4y7 6= 0
� R8:
It's easy to verify that the intersection D \ ~D � R8 is nonempty, for instance
it contains the point (0; 0; 1; 1; 0; 0; 0; 1). Hence D? = D \ ~D is a nonempty
open domain in R8. Therefore, it is really possible to choose the initial data
P (v), Q(v), R(v), S(v) in such a way, that (16) and (17) hold. As consequence,
the corresponding solution U(u; v), V (u; v) of (14.1)�(14.2) will satisfy (12.2) in
a neighborhood of (0; 0) 2 R2. So, the explicit representation (5) will describe
a pseudospherical P -surface in E4, q.e.d.
R e m a r k 1. If a point x 2 F 2 is �xed, one can always specify the Cartesian
coordinates in E4 in such a way that P is the origin and TxF
2 is the (x1; x2)-plane.
For the explicit representation (5) such a speci�cation means that
U(0; 0) = 0; V (0; 0) = 0; Uu(0; 0) = 0; Vu(0; 0) = 0; Uv(0; 0) = 0; Vv(0; 0) = 0:
So, without loss of generality one can consider the initial data P (v), Q(v), R(v),
S(v) which satisfy
P (0) = 0; R(0) = 0; Q(0) = 0; S(0) = 0; Pv(0) = 0; Rv(0) = 0: (18)
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 145
Vasyl Gorkavyy
If (18) hold then the initial data P (v), Q(v), R(v), S(v) are said to be reduced.
It is easy to verify that
�(0; 0; y3; y4; 0; 0; y7; y8) = (y4)2 + (y8)2 � (y3y8 � y4y7)2:
Therefore if P (v), Q(v), R(v), S(v) are reduced then (16) reads
P 2
vv
+R2
vv
� (PvvSv �RvvQv)
2 > 0: (19)
The initial data P (v), Q(v), R(v), S(v) are referred to as appropriate if they obey
(17)�(19). For any choice of appropriate initial data corresponds a well-de�ned
pseudospherical P -surface in E4.
4. Geometric Representation of Pseudospherical P -Surfaces in E
4
The initial data P (v), Q(v), R(v), S(v) may be interpreted geometrically. Due
to (5), the functions P (v), R(v) represent a curve
2 F 2 explicitly given by
x1 = 0; x2 = v; x3 = P (v); x4 = R(v): (20)
The curve
is the intersection of F 2 with the hyperplane E3
0 � E4 given by
x1 = 0. At each point x of
the tangent plane TxF
2 is spanned by two vectors:
xu(0; v) = (1; 0; Uu(0; v); Vu(0; v)) = (1; 0; Q(v); S(v)) ; (20.1)
xv(0; v) = (0; 1; Uv(0; v); Vv(0; v)) = (0; 1; Pu(v); Ru(v)) : (20.2)
Obviously, xv(0; v) is the tangent vector to
. Besides, TxF
2 6� E3
0 . The curve
being given, the tangent planes to F 2 along
are one-to-one determined by Q(v)
and S(v).
The initial data P (v), Q(v), R(v), S(v) are reduced, i.e., (18) holds, if and
only if the origin O belongs to
and the tangent plane TOF
2 is the (x1; x2)-plane.
The analytic constraints (17) and (19) impose some restrictions on the dynamical
properties of
and TxF
2 at O.
Conversely, consider an arbitrary regular analytical curve
in some E3
0 � E4
and an analytical �eld of two-planes �2 along
. Suppose that
and �2 satisfy
the following conditions:
A1) the planes �2 are tangent to
, i.e., at each point x of
the tangent vector
to
belongs to �2(x);
A2) there is a point O 2
such that �2(O) doesn't belong to E3
0 .
Introduce Cartesian coordinates x1; : : : ; x4 in E4 in such a way that O is the
origin, the tangent line to
at O is the x2-axe, and E3
0 � E4 is the hyperplane
x1 = 0. Since the tangent line to
at O is the x2-axe, the curve
may be
represented explicitly, x1 = 0, x2 = v, x3 = P (v), x4 = R(v), where P (v) and
146 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
On Pseudospherical Surfaces in E4
with Grassmann Image of Prescribed Type
Q(v) are some functions. Moreover, since �2(O) contains the vector tangent to
at O, i.e., (0; 1; Pv(0); Rv(0)) 2 �2(O), and it doesn't belong to the hyperplane
x1 = 0, one may conclude that the orthogonal projection from �2(O) to the
(x1; x2)-plane in E4 is bijective. The same is valid for all �2(x) at points x 2
su�ciently close to O. Therefore, at each such point x(v) 2
the plane �2(x) is
spanned by the vectors (0; 1; Pv(v); Rv(v)) and (1; 0; Q(v); S(v)). The functions
P (v), Q(v), R(v) and S(v) constructed from
and �2 satisfy P (0) = 0, Pv(0) = 0,
R(0) = 0, Rv(0) = 0. It is easy to see that these functions form reduced initial
data if and only if
and �2 satisfy the additional condition:
A3) the plane �2(O) is orthogonal to E3
0 , i.e., it contains the normal straight
line to E3
0 � E4 at O.
De�nition.
and �2 are referred to as appropriate geometric initial data if
they satisfy A1)�A3) and if P (v), Q(v), R(v) and S(v) corresponding to
and
�2 obey (17) and (19).
Thus, the following representation statement holds.
Theorem 2. Let
and �2 be appropriate geometric initial data. Then there
exists a unique pseudospherical P -surface F 2 � E4 which contains
and whose
tangent planes at points of
coincide with two-planes �2.
References
[1] Yu.A. Aminov, Geometry of Submanifolds. Gordon and Breach Sci. Publ., Amster-
dam, 2002.
[2] K. Tenenblat, Transformations of Manifolds and Applications to Di�erential Equa-
tions. Pitman Monographs and Surveys in Pure and Appl. Math. V. 93. Longman
Sci. Techn., Harlow, Essex; Wiley, New York, 2000.
[3] C. Rogers and W.K. Schief, Backlund and Darboux Transformations. Geometry
and Modern Applications in Soliton Theory. Ser. Cambridge Texts Appl. Math.,
No. 30. New South Wales University, Sydney, 2002.
[4] Yu.Aminov, Bianchi Transformations for Domains of Many-Dimensional Loba-
chevsky Space. � Ukr. Geom. Sb. 21 (1978), 3�5. (Russian)
[5] K. Tenenblat and C.-L. Terng, Backlund Theorem for n-Dimensional Submanifolds
of R2n�1. � Ann. Math. 111 (1980), 477�490.
[6] C.-L. Terng, A Higher Dimensional Generalization of the Sine-Gordon Equation
and its Soliton Theory. � Ann. Math. 111 (1980), 491�510.
[7] L.A.Masaltsev, Bianchi Pseudospherical Congruencies in E2n�1. � Mat. �z., analiz,
geom. 1 (1994), 505�512. (Russian)
[8] Yu. Aminov and A. Sym, On Bianchi and Backlund Transformations of Two-
Dimensional Surfaces in E4. � Math. Phys., Anal., Geom. 3 (2000), 75�89.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 147
Vasyl Gorkavyy
[9] V.A. Gorkavyy, On Pseudospherical Congruencies in E4. � Mat. �z., analiz, geom.
10 (2003), 498�504.
[10] V.A. Gorkavyy, Bianchi Congruencies for Two-Dimensional Surfaces in E4. � Mat.
Sb. 196 (2005), � 10, 98�119. (Russian)
[11] A.A. Borisenko, Intrinsic and Extrinsic Geometries of Many-Dimensional Subma-
nifolds. Examen, Moscow, 2003. (Russian)
[12] I.G. Petrovskiy, Lections on Partial Di�erential Equations. � Gos. Izd-vo Fiziko-
Mat. Lit., Moscow, 1961. (Russian)
148 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
|