Compact Spacelike Surfaces in the 3-Dimensional de Sitter Space
We establish several sufficient conditions for a compact spacelike surface in the 3-dimensional de Sitter space to be totally geodesic or spherical.
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irk-123456789-1065782016-10-01T03:02:02Z Compact Spacelike Surfaces in the 3-Dimensional de Sitter Space Borisenko, A.A. We establish several sufficient conditions for a compact spacelike surface in the 3-dimensional de Sitter space to be totally geodesic or spherical. 2006 Article Compact Spacelike Surfaces in the 3-Dimensional de Sitter Space / A.A. Borisenko // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 1. — С. 3-8. — Бібліогр.: 11 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106578 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We establish several sufficient conditions for a compact spacelike surface in the 3-dimensional de Sitter space to be totally geodesic or spherical. |
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Article |
| author |
Borisenko, A.A. |
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Borisenko, A.A. Compact Spacelike Surfaces in the 3-Dimensional de Sitter Space Журнал математической физики, анализа, геометрии |
| author_facet |
Borisenko, A.A. |
| author_sort |
Borisenko, A.A. |
| title |
Compact Spacelike Surfaces in the 3-Dimensional de Sitter Space |
| title_short |
Compact Spacelike Surfaces in the 3-Dimensional de Sitter Space |
| title_full |
Compact Spacelike Surfaces in the 3-Dimensional de Sitter Space |
| title_fullStr |
Compact Spacelike Surfaces in the 3-Dimensional de Sitter Space |
| title_full_unstemmed |
Compact Spacelike Surfaces in the 3-Dimensional de Sitter Space |
| title_sort |
compact spacelike surfaces in the 3-dimensional de sitter space |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2006 |
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http://dspace.nbuv.gov.ua/handle/123456789/106578 |
| citation_txt |
Compact Spacelike Surfaces in the 3-Dimensional de Sitter Space / A.A. Borisenko // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 1. — С. 3-8. — Бібліогр.: 11 назв. — англ. |
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Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT borisenkoaa compactspacelikesurfacesinthe3dimensionaldesitterspace |
| first_indexed |
2025-07-07T18:40:05Z |
| last_indexed |
2025-07-07T18:40:05Z |
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1837014561678426112 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2006, vol. 2, No. 1, pp. 3�8
Compact Spacelike Surfaces in the 3-Dimensional
de Sitter Space
A.A. Borisenko
Department of Mechanics and Mathematics, V.N. Karazin Kharkov National University
4 Svobody Sq., Kharkov, 61077, Ukraine
E-mail:borisenk@univer.kharkov.ua
Received September 14, 2005
We establish several su�cient conditions for a compact spacelike surface
in the 3-dimensional de Sitter space to be totally geodesic or spherical.
Key words: de Sitter space, compact spacelike surface, second fundamen-
tal form, Gaussian curvature; totally umbilical round sphere.
Mathematics Subject Classi�cation 2000: 53C42 (primary); 53B30, 53C45
(secondary).
Let E4
1
be a 4-dimensional Lorentz�Minkowski space, that is, the space E4
1
endowed with the Lorentzian metric tensor h ; i given by
h; i = (dx1)
2 + (dx2)
2 + (dx3)
2 � (dx0)
2;
where (x1; x2; x3; x0) are the canonical coordinates of E4
1
. The 3-dimensional
unitary de Sitter space is de�ned as the following hyperquadric of E4
1
:
S3
1 = fx 2 R4 : hx; xi = 1g:
As it is well known, S3
1
inherits from E4
1
a time-orientable Lorentzian metric which
makes it the standard model of a Lorentzian space of constant sectional curvature
one. A smooth immersion : F ! S3
1
� E4
1
of a 2-dimensional connected mani-
foldM is said to be a spacelike surface if the induced metric via is a Riemannian
metric on M , which, as usual, is also denoted by h ; i. The time-orientation of S3
1
allows us to de�ne a (global) unique timelike unit normal �eld n on F , tangent
to S3
1
, and hence we may assume that F oriented by n. We will refer to n as the
Gauss map of F .
The work supported by research grant DFFD of Ukrainian Ministry of Education and Sci-
ence, No. 01. 07/ 00132.
c
A.A. Borisenko, 2006
A.A. Borisenko
We note that Lobachevsky space L3 is the set of points
L3 = fx 2 E4
1 : hx; xi = �1; x0 > 0g:
It is well known that a compact spacelike surface in the 3-dimensional de Sitter
space S3
1
is di�eomorphic to a sphere S2. Thus, it is interesting to look for
additional assumptions for such a surface to be totally geodesic or totally umbilical
round sphere.
There are two possible kinds of geometric assumptions: extrinsic, that is re-
lative to the second fundamental form, and intrinsic, namely, concerning to the
Gaussian curvature of the induced metric. As regards to the extrinsic approach,
J. Ramanathan [10] proved that every compact spacelike surface in S3
1
of constant
mean curvature is totally umbilical. This result was generalized to hypersurface of
any dimension by S. Montiel [9]. J. Aledo and A. Romero characterize the compact
spacelike surfaces in S3
1
whose second fundamental form de�nes a Riemannian
metric. They studied the case of constant Gaussian curvature KII of the second
fundamental form, proving that the totally umbilical round spheres are the only
compact spacelike surfaces in S3
1
withK < 1 and constantKII [2]. With respect to
the intrinsic approach H. Li [8] obtained that compact spacelike surface of constant
Gaussian curvature is totally umbilical. And he proved there is no complete
spacelike surface in S3
1
with constant Gaussian curvature K > 1. J. Aledo and
A. Romero proved the same result without condition that Gaussian curvature is
constant [2]. But it is true more general result.
Theorem 1. Let F be a C2-regular complete spacelike surface in de Sitter
space S3
1
. If Gaussian curvature K > 1 the surface F is totally geodesic great
sphere with Gaussian curvature K = 1.
S.N. Bernshtein proved that an explicitly given saddle surface over a whole
plane in the Euclidean space E3 with slower than linear growth at in�nity must
be a cylinder. He proved this theorem for surfaces of class C2 [4], and it was
generalized to the nonregular case in [1].
A surface F 2 of smoothness class C1 in S3 may be projected univalently into
a great sphere S2
0
if the great spheres tangent to F 2 do not pass through points
Q1, Q2 polar to S2
0
.
The surface F 2 in S3 is called a saddle surface if any closed recti�able con-
tour L, that is in the intersection of F 2 with an arbitrary great sphere S2 in S3,
lies in an open hemisphere, and is deformable to a point in the surface can be
spanned by a two-dimensional simply connected surface Q contained in F 2 \ S2.
In other words, from the surface it is impossible to cut o� a crust by a great
sphere S2, that is, on F 2 there do not exist domains with boundary that lie in an
open great hemisphere of S2 and are wholly in one of the great hemispheres of S3
4 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1
Compact spacelike surfaces in the 3-dimensional de Sitter space
into which it is divided by the great sphere S2. In this case when F is a regular
surface of class C2, the saddle condition is equivalent to the condition that the
Gaussian curvature of F 2 does not exceed one. We have the following result.
Theorem 2 ([5�7]). Let F be an explicitly given compact saddle surface of
smoothness class C1 in the spherical space S3. Then F is a totally geodesic great
sphere.
This theorem is a generalization of a theorem of Bernshtein to a spherical
space. For regular space we obtain the following corollary.
Theorem 3 ([5, 6]). Let F be an explicitly given compact surface that is
regular of class C2 in the spherical space S3. If the Gaussian curvature K of F
satis�es K 6 1 then F is a totally geodesic great sphere.
This theorem was stated in [6]. Really Theorems 2 and 3 had been proved in
[7] but were formulated there for a centrally symmetric surfaces. The �nal version
was in [5].
It seems to us that the following conjecture must hold under a restriction on
the Gaussian curvature of the surface. Suppose that F is an embedded compact
surface, regular of class C2, in the spherical space S3. If the Gaussian curvature
K of F satis�es 0 < K 6 1, then F is a totally geodesic great sphere.
A.D. Aleksandrov [3] had proved that an analytical surface in Euclidean
space E3 homeomorphic to a sphere is a standard sphere if principal curvatures
satisfy the inequality
(k1 + c)(k2 + c) 6 0: (1)
This result had been generalized for analytic surfaces in spherical space S3 and
Lobachevsky space L3 [7]:
a) in S3 with additional hypothesis of positive Gaussian curvature;
b) in L3 under additional assumptions that principal curvatures k1; k2 satisfy
jk1j; jk2j > c0 > 1.
But in Lobachevsky space the result is true under weaker analytic restriction.
Theorem 4. Let F be a C3 regular surface homeomorphic to the sphere in the
Lobachevsky space L3. If jk1j; jk2j > c0 > 1 and principle curvatures k1 and k2
satisfy (1), then the surface is an umbilical round sphere in L3.
Analogical result it is true for surfaces in the de Sitter space S3
1
.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 5
A.A. Borisenko
Theorem 5. Let F be a C3 regular compact spacelike surface in the de Sitter
space S3
1
. If jk1j; jk2j < 1 and principal curvatures satisfy (1), then the surface is
an umbilical round sphere in S3
1
.
Let S3
1
be a simply-connected pseudo-Riemannian space of curvature 1 and
signature (+;+;�). It can be isometrically embedded in the pseudo-Euclidean
space E4
1
of signature (+;+;+;�) as the hypersurface given by the equation
x2
1
+x2
2
+x2
3
�x2
0
= 1. Together with E4
1
we consider the superimposed Euclidean
space E4 with unit sphere S3 given by the equation x2
1
+ x2
2
+ x2
3
� x2
0
= 1. We
specify a mapping of S3
1
into S3. To the point P of S3
1
with position vector r we
assign the point ~P with position vector ~r = r=
p
1 + 2x2
0
. Under the mapping, to
a surface F � S3
1
corresponds a surface ~F � S3. Let bij and ~bij be the coe�cients
of the second quadratic forms of F and ~F , and n = (n1; n2; n3; n0) be a normal
vector �eld on F .
Lemma 1 ([7]). ~bij = bij=
p
1 + 2x2
0
p
1 + 2n2
0
:
P r o o f o f T h e o r e m 1. From the condition K > 1 it follows that F is
a compact spacelike surface in the de Sitter space S3
1
. Locally a spacelike surface
is explicitly given over totally geodesic great sphere S2
0
� S3
1
and the orthogonal
projection p : F ! S2
0
in S3
1
is covering. Indeed, p is a local di�eomorphism. The
compactness of F and the simply connectedness of S2
0
imply that p is a global
di�eomorphism F on S2
0
and the surface F is globally explicitly given over S2
0
.
We map from a surface F in S3
1
to a surface ~F in S3. If F has a de�nite
metric and Gaussian curvature K > 1, then ~F has Gaussian curvature not greater
than 1. This follows immediately from Lemma 1, Gauss's formula and the fact
that hn; ni = �1 for normals to F . In a pseudo-Euclidean space, the analogous
correspondence between surfaces and their curvatures was used by Sokolov [11].
The surface ~F satis�es the conditions of Theorem 3. It follows that ~F is a
totally geodesic great sphere. By Lemma 1 the ranks of the second quadratic
forms of ~F and F coincide and we obtain that the surface F is a totally geodesic
surface in S3
1
.
P r o o f o f T h e o r e m s 4 a n d 5. The normal n(u1; u2) to
F is chosen so that the principal curvature satisfy (1). In a neighborhood of an
arbitrary nonumbilical point P we choose coordinate curves consisting of the lines
of curvature, and an arbitrary orthogonal net in the case of umbilical point. At
P the coe�cients of the �rst quadratic form are e = g = 1; f = 0. Let F1 be the
surface with radius vector � = (r � cn)=
p
jc2 � 1j.
In both cases the surface F1 lies in S3
1
. Moreover,
�u1 =
(1 + ck1)p
jc2 � 1jru1 ; �u2 =
(1 + ck2)p
jc2 � 1jru2 :
6 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1
Compact spacelike surfaces in the 3-dimensional de Sitter space
The unit normal n1 =
cr � n
p
jc2 � 1j . From the conditions on the principal curvatures
of F in Theorems 4, 5 it follows that
h�u1 ; �u1i > 0; h�u2 ; �u2i > 0
and F1 is a spacelike surface in S3
1
. The coe�cients of the second quadratic form
of the surface F1 are
L1 =
(1 + ck1)(k1 + c)p
c2 � 1
; N1 =
(1 + ck2)(k2 + c)p
c2 � 1
:
The Gaussian curvature of F1 at the point P1 is equal to
K = 1� (k1 + c)(k2 + c)jc2 � 1j
(1 + k1c)2(1 + k2c)2
> 1:
The same is true in umbilical points too. The surface F1 satis�es the conditions
of Theorem 1. It follows that the surface F1 is a totally geodesic great sphere in
S3
1
and F is an umbilical surface in L3 or S3
1
.
References
[1] G.M. Adelson-Vel'sky, The generalization of one geometrical theorem of S.N. Bern-
shtein. � Dokl. Akad. Nauk USSR 49 (1945), 6. (Russian)
[2] J.A. Aledo and A. Romero, Compact spacelike surfaces in the 3-dimensional de Sitter
space with nondegenerate second fundamental form. � Di�. Geom. and its Appl.
19 (2003), 97�111.
[3] A.D. Aleksanrdov, On the curvature of surfaces. � Vestnik Leningr. Univ. Ser. Mat.
Mech. Astrs. 19 (1966) No. 4, 5�11. (Russian)
[4] S.N. Bernshtein , Ampli�cation of the theorem of surfaces with negative curvature.
(Sobr. Soch. V. 3.) Publ. House Akad. Nauk USSR, Moscow, 1960. (Russian)
[5] A.A. Borisenko, On explicitly given saddle surfaces in a spherical space. � Usp.
Mat. Nauk 54 (1999), No. 5, 151�152; Engl. transl.: Russian Math. Surveys 54
(1999), No. 5, 1021�1022.
[6] A.A. Borisenko, Complete l-dimensional surfaces of nonpositive extrinsic curvature
in a Riemannian space. � Mat. Sb. 104 (1977), 559�576; Engl. transl.: Mat. Sb. 33
(1977), 485�499.
[7] A.A. Borisenko, Surfaces of nonpositive extrinsic curvature in spaces of constant
curvature. � Mat. Sb. 114 (1981), 336�354; Engl. transl.: Mat. Sb. 42 (1982),
297�310.
[8] H. Li, Global rigidity theorems of hypersurfaces. � Ark. Mat. 35 (1997), 327�351.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 7
A.A. Borisenko
[9] S. Montiel, An integral inequality for compact spacelike hypersurfaces in de Sitter
space and applications to the case of constant mean curvature. � Indiana Univ.
Math. J. 37 (1988), 909�917.
[10] J. Ramanathan, Complete spacelike hypersurfaces of constant mean curvature in de
Sitter space. � Indiana Univ. Math. J. 36 (1987), 349�359.
[11] D.D. Sokolov, The structure of the limit cone of a convex surface in pseudo-Euclidean
space. � Usp. Mat. Nauk 30 (1975), No. 1(181), 261�262. (Russian)
8 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1
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