Space-like surfaces in Minkowski space $E^4_1$ with pointwise 1-type Gauss map
We first classify space-like surfaces in the Minkowski space $E^4_1$, de Sitter space $S^3_1$, and hyperbolic space $H^3$ with harmonic Gauss map. Then we give a characterization and classification of space-like surfaces with pointwise 1-type Gauss map of the first kind. We also present some explici...
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Institute of Mathematics, NAS of Ukraine
2019
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507164910551040 |
|---|---|
| author | Dursun, U. Turgay, N. C. Дурсун, У. Тургау, Н. Ц. |
| author_facet | Dursun, U. Turgay, N. C. Дурсун, У. Тургау, Н. Ц. |
| author_sort | Dursun, U. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T08:54:16Z |
| description | We first classify space-like surfaces in the Minkowski space $E^4_1$, de Sitter space $S^3_1$, and hyperbolic space $H^3$ with harmonic
Gauss map. Then we give a characterization and classification of space-like surfaces with pointwise 1-type Gauss map of
the first kind. We also present some explicit examples. |
| first_indexed | 2026-03-24T02:04:58Z |
| format | Article |
| fulltext |
UDC 515.14
U. Dursun (Isik Univ. Sile Campus, Istanbul, Turkey),
N. C. Turgay (Istanbul Techn. Univ., Turkey)
SPACE-LIKE SURFACES IN MINKOWSKI SPACE \BbbE \bffour
\bfone
WITH POINTWISE 1-TYPE GAUSS MAP*
ПРОСТОРОВО-ПОДIБНI ПОВЕРХНI У ПРОСТОРI МIНКОВСЬКОГО \BbbE \bffour
\bfone
З ПОТОЧКОВИМ ГАУССОВИМ ВIДОБРАЖЕННЯМ ПЕРШОГО ТИПУ
We first classify space-like surfaces in the Minkowski space \BbbE 4
1, de Sitter space \BbbS 3
1, and hyperbolic space \BbbH 3 with harmonic
Gauss map. Then we give a characterization and classification of space-like surfaces with pointwise 1-type Gauss map of
the first kind. We also present some explicit examples.
Насамперед наведено класифiкацiю просторово-подiбних поверхонь у просторi Мiнковського \BbbE 4
1, просторi де Сiт-
тера \BbbS 3
1 i гiперболiчному просторi \BbbH 3 з гармонiчним гауссовим вiдображенням. Пiсля цього охарактеризовано
i наведено класифiкацiю просторово-подiбних поверхонь першого типу з поточковим гауссовим вiдображенням
першого типу. Також наведено деякi конкретнi приклади.
1. Introduction. In late 1970’s B. Y. Chen introduced the notion of finite type submanifolds of
Euclidean space [6]. Since then many works have been done to characterize or classify submanifolds
of Euclidean space or pseudo-Euclidean space in terms of finite type. Also, B. Y. Chen and P. Piccinni
extended the notion of finite type to differentiable maps, in particular, to Gauss map of submanifolds
in [12]. A smooth map \phi on a submanifold M of a Euclidean space or a pseudo-Euclidean space is
said to be of finite type if \phi can be expressed as a finite sum of eigenfunctions of the Laplacian \Delta
of M, that is, \phi = \phi 0 +
\sum k
i=1
\phi i, where \phi 0 is a constant map, \phi 1, . . . , \phi k are non-constant maps
such that \Delta \phi i = \lambda i\phi i, \lambda i \in \BbbR , i = 1, . . . , k.
If a submanifold M of a Euclidean space or a pseudo-Euclidean space has 1-type Gauss map \nu ,
then \nu satisfies \Delta \nu = \lambda (\nu +C) for some \lambda \in \BbbR and some constant vector C. In [12], B. Y. Chen and
P. Piccinni studied compact submanifolds of Euclidean spaces with finite type Gauss map. Several
articles also appeared on submanifolds with finite type Gauss map (cf. [2 – 5, 24, 25]).
However, the Laplacian of the Gauss map of several surfaces and hypersurfaces such as helicoids
of the 1st, 2nd, and 3rd kind, conjugate Enneper’s surface of the second kind and B-scrolls in a
3-dimensional Minkowski space \BbbE 3
1, generalized catenoids, spherical n-cones, hyperbolical n-cones
and Enneper’s hypersurfaces in \BbbE n+1
1 take the form
\Delta \nu = f(\nu + C) (1.1)
for some smooth function f on M and some constant vector C [17, 21]. A submanifold of a pseudo-
Euclidean space is said to have pointwise 1-type Gauss map if its Gauss map satisfies (1.1) for some
smooth function f on M and some constant vector C. In particular, if C is zero, it is said to be of
the first kind. Otherwise, it is said to be of the second kind (cf. [1, 10, 15, 16, 18, 20, 22]).
* This work which is a part of the second author’s doctoral thesis is partially supported by Istanbul Technical University.
c\bigcirc U. DURSUN, N. C. TURGAY, 2019
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1 59
60 U. DURSUN, N. C. TURGAY
Remark 1.1. The Gauss map \nu of a totally geodesic submanifold M in \BbbE m
1 is a constant vector
and \Delta \nu = 0, i.e., it is harmonic. For f = 0 if we write \Delta \nu = 0 \cdot \nu , then M has pointwise 1-type
Gauss map of the first kind. If we choose C = - \nu , then (1.1) holds for any non-zero smooth function
f. In this case M has pointwise 1-type Gauss map of the second kind. Therefore, a totally geodesic
submanifold in \BbbE m
1 is a trivial submanifold with pointwise 1-type Gauss map of both the first kind
and the second kind.
The complete classification of ruled surfaces in \BbbE 3
1 with pointwise 1-type Gauss map of the first
kind was obtained in [21]. Also, a complete classification of rational surfaces of revolution in \BbbE 3
1
satisfying (1.1) was recently given in [20], and it was proved that a right circular cone and a hyper-
bolic cone in \BbbE 3
1 are the only rational surfaces of revolution in \BbbE 3
1 with pointwise 1-type Gauss map
of the second kind. The first author studied rotational hypersurfaces in Lorentz – Minkowski space
with pointwise 1-type Gauss map [17], Moreover, in [23] a complete classification of cylinderical
and non-cylinderical surfaces in \BbbE m
1 with pointwise 1-type Gauss map of the first kind was obtained.
In this article, we study space-like surfaces in \BbbE 4
1 with pointwise 1-type Gauss map of the first
kind. Surfaces with harmonic Gauss map in \BbbE 4
1 are of global 1-type Gauss map of the first kind.
We first give a characterization and classification of maximal surfaces and non-maximal space-like
surfaces in \BbbE 4
1 with harmonic Gauss map. We also prove that oriented maximal surfaces and surfaces
with light-like mean curvature vector in \BbbE 4
1 with harmonic Gauss map are the only surfaces in \BbbE 4
1
with (global) 1-type Gauss map of the first kind.
Then we obtain the necessary and sufficient conditions on non-maximal space-like surfaces in
\BbbE 4
1 with pointwise 1-type Gauss map of the first kind, and we give a classification of such surfaces.
Further, we prove that an oriented non-maximal space-like surface in \BbbE 4
1 has (global) 1-type Gauss
map of the first kind if and only if the surface has constant Gaussian curvature and parallel mean
curvature.
2. Prelimineries. Let \BbbE m
t denote the pseudo-Euclidean m-space with the canonical pseudo-
Euclidean metric tensor of index t given by
g = -
t\sum
i=1
dx2i +
m\sum
j=t+1
dx2j ,
where (x1, x2, . . . , xm) is a rectangular coordinate system in \BbbE m
t . We put
\BbbS m - 1
t (r2) =
\bigl\{
x \in \BbbE m
t : \langle x, x\rangle = r - 2
\bigr\}
,
\BbbH m - 1
t - 1 ( - r2) =
\bigl\{
x \in \BbbE m
t : \langle x, x\rangle = - r - 2
\bigr\}
,
where \langle , \rangle is the indefinite inner product of \BbbE m
t . Then \BbbS m - 1
t (r2) and \BbbH m - 1
t - 1 ( - r2), m \geq 3, are com-
plete pseudo-Riemannian manifolds of constant curvature r2 and - r2, respectively. The Lorentzian
manifolds \BbbE m
1 and \BbbS m - 1
1 (r2) are known as the Minkowski and de Sitter spaces, respectively. For
t = 1
\BbbH m - 1( - r2) =
\Bigl\{
x = (x1, . . . , xm) \in \BbbE m
1 : \langle x, x\rangle = - r - 2 and x1 > 0
\Bigr\}
is the hyperbolic space in \BbbE m
1 .
The light cone \scrL \scrC n - 1 with vertex at the origin in \BbbE m
t is defined to be
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
SPACE-LIKE SURFACES IN MINKOWSKI SPACE \BbbE 4
1 WITH POINTWISE 1-TYPE GAUSS MAP 61
\scrL \scrC n - 1 =
\bigl\{
x \in \BbbE m
t : \langle x, x\rangle = 0
\bigr\}
.
A vector v in \BbbE m
t is called space-like (resp., time-like) if \langle v, v\rangle > 0 (resp., \langle v, v\rangle < 0). A vector
v is called light-like if it is nonzero and it satisfies \langle v, v\rangle = 0.
Let M be an n-dimensional pseudo-Riemannian submanifold of the pseudo-Euclidean space \BbbE m
t .
We denote Levi-Civita connections of \BbbE m
t and M by \widetilde \nabla and \nabla , respectively. In this section, we
shall use letters X, Y, Z, W (resp., \xi , \eta ) to denote vectors fields tangent (resp., normal) to M. The
Gauss and Weingarten formulas are given, respectively, by\widetilde \nabla XY = \nabla XY + h(X,Y ), (2.1)
\widetilde \nabla X\xi = - A\xi (X) +DX\xi , (2.2)
where h, D and A are the second fundamental form, the normal connection and the shape operator
of M, respectively.
For each \xi \in T\bot
p M, the shape operator A\xi is a symmetric endomorphism of the tangent space
TpM at p \in M. The shape operator and the second fundamental form are related by \langle h(X,Y ), \xi \rangle =
= \langle A\xi X,Y \rangle .
The Gauss, Codazzi and Ricci equations are given, respectively, by
\langle R(X,Y, )Z,W \rangle = \langle h(Y, Z), h(X,W )\rangle - \langle h(X,Z), h(Y,W )\rangle , (2.3)
( \=\nabla Xh)(Y,Z) = ( \=\nabla Y h)(X,Z), (2.4)
\langle RD(X,Y )\xi , \eta \rangle = \langle [A\xi , A\eta ]X,Y \rangle , (2.5)
where R, RD are the curvature tensors associated with connections \nabla and D, respectively, and \=\nabla h
is defined by
( \=\nabla Xh)(Y,Z) = DXh(Y, Z) - h(\nabla XY,Z) - h(Y,\nabla XZ).
A submanifold M is said to have flat normal bundle if RD = 0 identically, and the second fun-
damental form h of M in \BbbE m
t is called parallel if \=\nabla h = 0. A submanifold with parallel second
fundamental form is also known as a parallel submanifold.
Let \{ e1, e2, . . . , em\} be a local orthonormal frame on M with \varepsilon A = \langle eA, eA\rangle = \pm 1 such that
e1, e2, . . . , en are tangent to M and en+1, en+2, . . . , em are normal to M. We use the following
convention on the range of indices: 1 \leq A,B,C, . . . \leq m, 1 \leq i, j, k, . . . \leq n, n+ 1 \leq \beta , \gamma , . . . \leq
\leq m.
Let \{ \omega AB\} with \omega AB + \omega BA = 0 be the connection 1-forms associated to \{ e1, . . . , em\} . Then
we have \widetilde \nabla ekei =
n\sum
j=1
\varepsilon j\omega ij(ek)ej +
m\sum
\beta =n+1
\varepsilon \beta h
\beta
ike\beta
and \widetilde \nabla eke\beta = -
n\sum
j=1
\varepsilon jh
\beta
kjej +
m\sum
\nu =n+1
\varepsilon \nu \omega \beta \nu (ek)e\nu ,
where h\beta ij ’s are the coefficients of the second fundamental form h.
The mean curvature vector H, the scalar curvature S and the squared length \| h\| 2 of the second
fundamental form h are defined by
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
62 U. DURSUN, N. C. TURGAY
H =
1
n
m\sum
\beta =n+1
\varepsilon \beta \mathrm{t}\mathrm{r}A\beta e\beta , (2.6)
\| h\| 2 =
\sum
i,j,\beta
\varepsilon i\varepsilon j\varepsilon \beta h
\beta
ijh
\beta
ji, (2.7)
S = n2\langle H,H\rangle - \| h\| 2, (2.8)
where \mathrm{t}\mathrm{r}A\beta denotes the trace of shape operator A\beta , i.e., \mathrm{t}\mathrm{r}A\beta =
\sum n
i=1
\varepsilon ih
\beta
ii.
The mean curvature vector H of a submanifold of M in \BbbE m
t is called parallel if DH = 0
identically.
The gradient of a smooth function f defined on M into \BbbR is defined by \nabla f =
\sum n
i=1
\varepsilon iei(f)ei
and the Laplace operator acting on M is \Delta =
\sum n
i=1
\varepsilon i(\nabla eiei - eiei). If the position vector x of
M in Em
s satisfies \Delta x \not = 0 and \Delta 2x = 0, then M is called biharmonic.
A surface M in \BbbE 4
1 is called space-like if every non-zero tangent vector on M is space-like. Let
\{ e1, e2, e3, e4\} be a local orthonormal frame on a space-like surface M such that e1, e2 are tangent
to M and e3, e4 are normal to M with \varepsilon \beta = \langle e\beta , e\beta \rangle , \beta = 3, 4.
The Gaussian curvature K is defined by K = R(e1, e2; e2, e1). Note that scalar curvature S and
Gaussian curvature of M satisfies S = 2K. Thus, (2.8) implies
K = 2\langle H,H\rangle - \| h\| 2/2. (2.9)
From Gauss equation (2.3) we have K = \varepsilon 3(\mathrm{d}\mathrm{e}\mathrm{t}A3 - \mathrm{d}\mathrm{e}\mathrm{t}A4). If K vanishes identically, M is said
to be flat. On the other hand, M is called maximal if H = 0. A surface M is called pseudo-umbilical
if its second fundamental form h and the mean curvature vector H satisfies \langle h(X,Y ), H\rangle = \rho \langle X,Y \rangle
for a smooth function \rho . Moreover, if the equation h(X,Y ) = \langle X,Y \rangle H is satisfied, then M is said
to be totally umbilical.
If we put hij,k = (\nabla ekh)(ei, ej), then for a space-like surface M in \BbbE 4
1 the Codazzi equation
given by (2.4) becomes
h\beta ij,k = h\beta jk,i, i, j, k = 1, 2, \beta = 3, 4,
h\beta jk,i = ei(h
\beta
jk) +
4\sum
\gamma =3
\varepsilon \gamma h
\gamma
jk\omega \gamma \beta (ei) -
2\sum
\ell =1
\Bigl(
\omega j\ell (ei)h
\beta
\ell k + \omega k\ell (ei)h
\beta
j\ell
\Bigr)
.
(2.10)
Let G(m - n,m) be the Grassmannian manifold consisting of all oriented (m - n)-planes through
the origin of \BbbE m
t and
\bigwedge m - n \BbbE m
t the vector space obtained by the exterior product of m - n vectors in
\BbbE m
t . Let fi1 \wedge . . .\wedge fim - n and gi1 \wedge . . .\wedge gim - n be two vectors in
\bigwedge m - n \BbbE m
t , where \{ f1, f2, . . . , fm\}
and \{ g1, g2, . . . , gm\} are two orthonormal bases of \BbbE m
t . Define an indefinite inner product \langle , \rangle on\bigwedge m - n \BbbE m
t by \bigl\langle
fi1 \wedge . . . \wedge fim - n , gi1 \wedge . . . \wedge gim - n
\bigr\rangle
= \mathrm{d}\mathrm{e}\mathrm{t}(\langle fi\ell , gjk\rangle ). (2.11)
Therefore, for some positive integer s, we may identify
\bigwedge m - n \BbbE m
t with some pseudo-Euclidean
space \BbbE N
s , where N =
\bigl(
m
m - n
\bigr)
. Let e1, . . . , en, en+1, . . . , em be an oriented local orthonormal frame
on an n-dimensional pseudo-Riemannian submanifold M in \BbbE m
t with \varepsilon B = \langle eB, eB\rangle = \pm 1 such that
e1, . . . , en are tangent to M and en+1, . . . , em are normal to M. The map \nu : M \rightarrow G(m - n,m) \subset
\subset \BbbE N
s from an oriented pseudo-Riemannian submanifold M into G(m - n,m) defined by
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
SPACE-LIKE SURFACES IN MINKOWSKI SPACE \BbbE 4
1 WITH POINTWISE 1-TYPE GAUSS MAP 63
\nu (p) = (en+1 \wedge en+2 \wedge . . . \wedge em)(p) (2.12)
is called the Gauss map of M that is a smooth map which assigns to a point p in M the oriented
(m - n)-plane through the origin of \BbbE m
t and parallel to the normal space of M at p [22]. We put
\varepsilon = \langle \nu , \nu \rangle = \varepsilon n+1\varepsilon n+2 . . . \varepsilon m = \pm 1 and
\widetilde MN - 1
s (\varepsilon ) =
\left\{ \BbbS N - 1
s (1) in \BbbE N
s , if \varepsilon = 1,
\BbbH N - 1
s - 1 ( - 1) in \BbbE N
s , if \varepsilon = - 1.
Then the Gauss image \nu (M) can be viewed as \nu (M) \subset \widetilde MN - 1
s (\varepsilon ).
3. Space-like surfaces in \BbbE \bffour
\bfone with harmonic Gauss map. The Laplacian of the Gauss map
of an n-dimensional oriented submanifold M of a Euclidean space \BbbE n+2 was obtained in [19]. By
a similar calculation, for the Laplacian of the Gauss map \nu given by (2.12) of an n-dimensional
oriented submanifold M of a pseudo-Euclidean space \BbbE n+2
t we have the following lemma.
Lemma 3.1. Let M be an n-dimensional oriented submanifold of a pseudo-Euclidean space
\BbbE n+2
t . Then the Laplacian of Gauss map \nu = en+1 \wedge en+2 is given by
\Delta \nu = \| h\| 2\nu + 2
\sum
1\leq j<k\leq n
\varepsilon j\varepsilon kR
D(ej , ek; en+1, en+2)ej \wedge ek+
+\nabla (\mathrm{t}\mathrm{r}An+1) \wedge en+2 + en+1 \wedge \nabla (\mathrm{t}\mathrm{r}An+2)+
+n
n\sum
j=1
\varepsilon j\omega (n+1)(n+2)(ej)H \wedge ej , (3.1)
where \| h\| 2 is the squared length of the second fundamental form, RD is the normal curvature
tensor and \nabla \mathrm{t}\mathrm{r}Ar is the gradient of \mathrm{t}\mathrm{r}Ar.
Remark 3.1. From (3.1) we see that if an n-dimensional submanifold M of \BbbE n+2
t has pointwise
1-type Gauss map of the first kind, then equation (1.1) is satisfied for f = \| h\| 2 and C = 0.
The Gauss map of a surface M in \BbbE 4
1 is said to be harmonic if \Delta \nu = 0. Clearly, a harmonic
Gauss map is of (global) 1-type of the first kind. In the Euclidean space \BbbE 4, a plane is the only surface
with harmonic Gauss map. However, in the Minkowski space \BbbE 4
1 there are non-planar surfaces with
harmonic Gauss map.
Lemma 3.2 [8]. Let M be a space-like surface with parallel mean curvature vector H in \BbbE 4
1.
Then we have:
(a) \langle H,H\rangle is constant,
(b) [AH , A\xi ] = 0 for any normal vector field \xi .
By combining the part (b) of Lemma 3.2 and the Ricci equation (2.5), we state the following
lemma for later use.
Lemma 3.3. Let M be a non-maximal space-like surface in the Minkowski space \BbbE 4
1. If the
mean curvature vector H of M is parallel, then the normal bundle of M is flat, i.e., RD \equiv 0.
Proposition 3.1. Let M be an oriented maximal surface in the Minkowski space \BbbE 4
1. Then the
Gauss map \nu of M is harmonic if and only if M is a flat surface in \BbbE 4
1 with flat normal bundle.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
64 U. DURSUN, N. C. TURGAY
Proof. Let M be a maximal surface in \BbbE 4
1, i.e., H \equiv 0. Then, from (2.9) we have \| h\| 2 = - 2K.
Thus, (3.1) implies
\Delta \nu = - 2K\nu + 2RD(e1, e2; e3, e4)e1 \wedge e2. (3.2)
Therefore, \nu is harmonic if and only if K = 0 and RD = 0.
Proposition 3.1 is proved.
Next, we obtain a non-planar maximal surface in \BbbE 4
1 with harmonic Gauss map.
Example 3.1 [11]. Let \Omega be an open, connected set in \BbbR 2 and \phi : \Omega \rightarrow \BbbR a smooth function.
We consider the surface M in the Minkowski space \BbbE 4
1 given by
x(u, v) = (\phi (u, v), u, v, \phi (u, v)). (3.3)
This surface lies in the degenerate hyperplane \scrH 0 = \{ (x1, x2, x3, x4) \in \BbbE 4
1 | x1 = x4\} . By a direct
calculation, we see that M is a flat surface with flat normal bundle and the mean curvature vector H
of M in \BbbE 4
1 is given by
H = (\Delta \phi , 0, 0,\Delta \phi ). (3.4)
Therefore, M is maximal if and only if \phi is harmonic.
Hence, Proposition 3.1 implies that if \phi is a harmonic function, then the surface given by (3.3)
has harmonic Gauss map.
Proposition 3.2. A non-planar flat maximal surface in the Minkowski space \BbbE 4
1 with flat normal
bundle is congruent to the surface given by (3.3) for a smooth harmonic function \phi : \Omega \subset \BbbR 2 \rightarrow \BbbR ,
where \Omega is an open set in \BbbR 2.
Proof. Let M be a non-planar flat maximal surface in \BbbE 4
1 with flat normal bundle.
Since M is flat, there exist local coordinates (u, v) on M such that e1 = \partial /\partial u, e2 = \partial /\partial v.
Then the induced metric tensor is given by g = du2 + dv2 and also \omega 12 \equiv 0. Let \{ e3, e4\} be a local
orthonormal normal frame on M with \varepsilon 3 = - \varepsilon 4 = 1, where \varepsilon \beta = \langle e\beta , e\beta \rangle . As M is maximal, we
have A\beta = (h\beta ij) with h\beta 11+h\beta 22 = 0, \beta = 3, 4, that is, \mathrm{t}\mathrm{r}A3 = \mathrm{t}\mathrm{r}A4 = 0. Moreover, since M is flat,
from the Gauss equation (2.3) we get \mathrm{d}\mathrm{e}\mathrm{t}A3 = \mathrm{d}\mathrm{e}\mathrm{t}A4. Therefore, the eigenvalues of A3 and A4 are
equal which imply that A3 = \mp A4 as RD = 0. Without loss of generality, we may take A3 = A4.
Let \Omega be an open set in \BbbR 2 and x : \Omega \rightarrow M \subset \BbbE 4
1 be an isometric immersion. From the Gauss
formula we obtain
xuu = h311(e3 - e4), xuv = h312(e3 - e4), xvv = - h311(e3 - e4) (3.5)
as \omega 12 \equiv 0. Also, the first and second equations in (3.5) imply that
xuu + xvv = 0. (3.6)
Moreover, xuu, xuv and xvv are pairwise linearly dependent light-like vector fields.
On the other hand, by a direct calculation, using the Weingarten formula and (3.5), we get
xuuu =
\bigl(
\partial u
\bigl(
h311
\bigr)
+ \omega 34(\partial u)h
3
11
\bigr)
(e3 - e4), (3.7)
xuuv =
\bigl(
\partial v
\bigl(
h311
\bigr)
+ \omega 34(\partial v)h
3
11
\bigr)
(e3 - e4). (3.8)
Now we define a vector valued function y = (y1, y2, y3, y4) : \Omega \rightarrow \BbbE 4
1 as y = xuu. From
equations (3.5), (3.7) and (3.8) we have yu = \gamma 1y and yv = \gamma 2y for some smooth functions \gamma 1 and
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SPACE-LIKE SURFACES IN MINKOWSKI SPACE \BbbE 4
1 WITH POINTWISE 1-TYPE GAUSS MAP 65
\gamma 2. Thus, the coordinate functions of y satisfy
yiu = \gamma 1y
i, yiv = \gamma 2y
i, i = 1, 2, 3, 4. (3.9)
By solving these equations, we get yj = cjy
1, i = 2, 3, 4, for some constants cj \in \BbbR . Thus, we
obtain
xuu = y1\eta 0, (3.10)
where \eta 0 = (1, c2, c3, c4) is a constant light-like vector. In a similar way, we get
xuv = \phi 2\eta 0 and xvv = \phi 3\eta 0, (3.11)
where \phi 2, \phi 3 : \Omega \subset \BbbR 2 \rightarrow \BbbR are some smooth functions. By integrating (3.10) and (3.11), we obtain
x(u, v) = \phi (u, v)\eta 0 + u\eta 1 + v\eta 2,
where \phi : \Omega \rightarrow \BbbR is a smooth function and \eta 1, \eta 2 are constant vectors such that \langle \eta 0, \eta i\rangle = 0,
\langle \eta i, \eta j\rangle = \delta ij , i, j = 1, 2. Equation (3.6) implies that \phi is harmonic. By choosing \eta 0 = (1, 0, 0, 1),
\eta 1 = (0, 1, 0, 0) and \eta 2 = (0, 0, 1, 0), the proof is completed.
By combining Propositions 3.1 and 3.2, we state the following classification theorem for maximal
surfaces in \BbbE 4
1 with harmonic Gauss map.
Theorem 3.1. An oriented maximal surface with harmonic Gauss map in the Minkowski space
\BbbE 4
1 is either an open part of a space-like plane or congruent to a surface given by (3.3) for a smooth
harmonic function \phi : \Omega \subset \BbbR 2 \rightarrow \BbbR , where \Omega is an open set in \BbbR 2.
Now we investigate non-maximal space-like surfaces in \BbbE 4
1 with harmonic Gauss map.
Theorem 3.2. Let M be an oriented non-maximal space-like surface in the Minkowski space
\BbbE 4
1. Then the Gauss map \nu of M is harmonic if and only if M is flat in \BbbE 4
1 with light-like and
parallel mean curvature vector.
Proof. Let M be an oriented non-maximal space-like surface in \BbbE 4
1 with harmonic Gauss map
\nu . Then we have \Delta \nu = 0. From (3.1) we obtain \| h\| 2 = 0 and RD = 0. That is, the normal bundle
is flat. So we can choose a local parallel orthonormal normal frame \{ e3, e4\} on M. Thus we have
\omega 34 = 0, and from (3.1)
\nabla (\mathrm{t}\mathrm{r}A3) \wedge e4 + e3 \wedge \nabla (\mathrm{t}\mathrm{r}A4) = 0 (3.12)
which implies that \mathrm{t}\mathrm{r}A3 and \mathrm{t}\mathrm{r}A4 are constants. Therefore, DH = 0, that is, H is parallel, and
\langle H,H\rangle is constant because of part (a) of Lemma 3.2.
Now we will show that H is light-like. Suppose that H is not light-like, that is, \langle H,H\rangle \not = 0. As
\| h\| 2 = 0 we have K = 2\langle H,H\rangle \not = 0 from (2.9). Thus, M is not flat.
On the other hand, since the normal bundle is flat and \langle H,H\rangle \not = 0, we can choose a local
orthonormal frame field \{ e1, e2, e3, e4\} on M such that e3 = H/\alpha and e4 are parallel, the shape
operators are diagonalized, and e1, e2 are eigenvectors of A3, where \alpha =
\sqrt{}
| \langle H,H| \rangle . So we have
A3 = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}
\bigl(
h311, h
3
22
\bigr)
, A4 = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}
\bigl(
h411, - h411
\bigr)
, h311 + h322 = 2\alpha and \omega 34 = 0. Considering these, it
follows from Codazzi equation (2.10) that
e1(h
3
22) = - e1(h
3
11) = \omega 12(e2)
\bigl(
h311 - h322
\bigr)
, (3.13)
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66 U. DURSUN, N. C. TURGAY
e1(h
4
22) = - e1(h
4
11) = 2\omega 12(e2)h
4
11, (3.14)
e2(h
3
11) = - e2(h
3
22) = \omega 12(e1)
\bigl(
h311 - h322
\bigr)
, (3.15)
e2(h
4
11) = - e2(h
4
22) = 2\omega 12(e1)h
4
11. (3.16)
As \| h\| 2 = 0, we have \bigl(
h311
\bigr) 2
+
\bigl(
h322
\bigr) 2
= 2
\bigl(
h411
\bigr) 2
, (3.17)
from which we obtain
h311e1(h
3
11) + h322e1(h
3
22) = 2h411e1(h
4
11),
h311e2(h
3
11) + h322e2(h
3
22) = 2h411e2(h
4
11).
Using (3.13) – (3.16), the above equations become
- \omega 12(e2)
\bigl(
(h311 - h322)
2 - 4(h411)
2
\bigr)
= 0, (3.18)
\omega 12(e1)
\bigl(
(h311 - h322)
2 - 4(h411)
2
\bigr)
= 0. (3.19)
Since M is not flat, at least one of \omega 12(e1) and \omega 12(e2) is not zero. Therefore, (3.18) and (3.19)
imply that
(h311 - h322)
2 = 4(h411)
2.
Considering this and (3.17) we obtain h311h
3
22 + (h411)
2 = 0. Therefore, the Gauss curvature K =
= \varepsilon 3(h
3
11h
3
22 + (h411)
2) = 0 and hence 2\langle H,H\rangle = K = 0 which is a contradiction. As a result, H
is light-like. Since \langle H,H\rangle = 0 and \| h\| 2 = 0, (2.9) implies K = 0, i.e., M is flat.
Conversely, we assume that M is a flat surface in \BbbE 4
1 with parallel and light-like mean curvature
vector H, that is, K = \langle H,H\rangle = 0. So we have \| h\| 2 = 0 from (2.9). On the other hand,
Lemma 3.3 implies that M has flat normal bundle, i.e., RD = 0. Therefore, there exists a local
parallel orthonormal frame \{ e3, e4\} of normal bundle of M with \varepsilon 3 = - \varepsilon 4 = 1 and the shape
operators A3 and A4 can be diagonalized simultaneously by choosing a proper frame \{ e1, e2\} of
tangent bundle of M, namely, we have
A\beta = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}
\bigl(
h\beta 11, h
\beta
22
\bigr)
, \beta = 3, 4,
also, \omega 34 \equiv 0. Moreover, since H is light-like, we get
\mathrm{t}\mathrm{r}A3 = \mathrm{t}\mathrm{r}A4 = \mu \not = 0 and H =
\mu
2
(e3 - e4).
In addition, since H is parallel and \omega 34 = 0, \mu is a constant. Thus, we obtain \nabla (\mathrm{t}\mathrm{r}A3) = \nabla (\mathrm{t}\mathrm{r}A4) =
= 0. Therefore, equation (3.1) gives \Delta \nu = 0.
Theorem 3.2 is proved.
A space-like surface in the Minkowski space \BbbE 4
1 is called marginally trapped (or quasi-minimal)
if its mean curvature vector is light-like at each point on the surface. We will use the following
classification theorem of marginally trapped surfaces with parallel mean curvature vector in the
Minkowski space \BbbE 4
1 obtained in [14].
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SPACE-LIKE SURFACES IN MINKOWSKI SPACE \BbbE 4
1 WITH POINTWISE 1-TYPE GAUSS MAP 67
Theorem 3.3 [14]. Let M be a marginally trapped surface with parallel mean curvature vector
in the Minkowski space-time \BbbE 4
1. Then, with respect to suitable Minkowskian coordinates (t, x2, x3, x4)
on \BbbE 4
1, M is an open part of one of the following six types of surfaces:
(i) a flat parallel biharmonic surface given by
x(u, v) =
\biggl(
1 - b
2
u2 +
1 + b
2
v2, u, v,
1 - b
2
u2 +
1 + b
2
v2
\biggr)
, b \in \BbbR ;
(ii) a flat parallel surface given by
x(u, v) = a(\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}u, \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}u, \mathrm{c}\mathrm{o}\mathrm{s}u, \mathrm{s}\mathrm{i}\mathrm{n}u), a > 0; (3.20)
(iii) a non-parallel flat biharmonic surface with constant light-like mean curvature vector, lying
in the hyperplane \scrH 0 = \{ (t, x2, x3, t)\} , but not in the light cone \scrL \scrC ;
(iv) a non-parallel flat surface lying in the light cone \scrL \scrC ;
(v) a non-parallel surface lying in the de Sitter space-time S3
1(r
2) for some r > 0 such that the
mean curvature vector H \prime of M in S3
1(r
2) satisfies \langle H \prime , H \prime \rangle = - r2;
(vi) a non-parallel surface lying in the hyperbolic space H3( - r2) for some r > 0 such that the
mean curvature vector H \prime of M in H3( - r2) satisfies \langle H \prime , H \prime \rangle = r2.
Conversely, all surfaces of type (i) – (vi) above give rise to marginally trapped surfaces with parallel
mean curvature vector in \BbbE 4
1.
Remark 3.2 [9]. We can combine cases (i) and (iii) of Theorem 3.3 into a single case, namely,
flat surfaces defined by (3.3) such that \phi is a function satisfying \Delta \phi = c for some real number
c \not = 0.
The surfaces type (i) and (ii) in Theorem 3.3 are two explicit examples for Theorem 3.2. In the
next theorem we determine flat surfaces in \BbbS 31(r2) \subset \BbbE 4
1 with parallel and light-like mean curvature
vector in \BbbE 4
1.
Theorem 3.4. Let M be a space-like surface in the de Sitter space \BbbS 31(r2) \subset \BbbE 4
1 for some r > 0.
If M is a flat surface with parallel and light-like mean curvature vector in \BbbE 4
1, then M is congruent
to the surface given by
x(u, v) =
\biggl(
r
2
(u2 + v2), u, v,
r
2
(u2 + v2) - 1
r
\biggr)
. (3.21)
Proof. Suppose that M is a flat space-like surface in \BbbS 31(r2) \subset \BbbE 4
1 with parallel and light-like
mean curvature vector H in \BbbE 4
1. Since M is flat, there exist local coordinates u and v on M such
that the induced metric tensor is g = du2 + dv2. Let x : \Omega \rightarrow M \subset \BbbS 31(r2) \subset \BbbE 4
1 be an isometric
immersion, where \Omega is an open set in \BbbR 2. Then, we have \langle x, x\rangle = r - 2. Thus, a local frame field
\{ e1, e2, e3, e4\} on M can be chosen as e1 = \partial u, e2 = \partial v, e3 = rx, and e4 is a unit normal vector
field orthogonal to e3 such that H = - r(e3 - e4) as H is light-like (see [8], Lemma 2.2).
From the Weingarten formula (2.2), we have \widetilde \nabla \partial ue3 = r\partial u and \widetilde \nabla \partial ve3 = r\partial v which imply
A3 = - rI, where I is identity operator acting on tangent bundle of M. Moreover, since M is flat
and H = - r(e3 - e4), we have \mathrm{d}\mathrm{e}\mathrm{t}A3 = \mathrm{d}\mathrm{e}\mathrm{t}A4, \mathrm{t}\mathrm{r}A3 = \mathrm{t}\mathrm{r}A4 from which and A3 = - rI we
obtain A3 = A4. Thus, M is pseudo-umbilical. Theorem 4 [7] implies that M is biharmonic.
As M is a biharmonic surface with light-like mean curvature vector, from the proof of [11]
(Theorem 6.1), one can see that x is of the form
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68 U. DURSUN, N. C. TURGAY
x(u, v) = (\phi (u, v), u, v, \phi (u, v) - \phi 0) (3.22)
for a smooth function \phi and constant \phi 0 \not = 0. As \langle x, x\rangle = r - 2, from (3.22) we obtain
- 2\phi 0\phi + \phi 2
0 + u2 + v2 = r - 2.
By considering this equation and a linear isometry of \BbbE 4
1, we may assume that
\phi (u, v) =
r
2
(u2 + v2) and \phi 0 =
1
r
(3.23)
from which and (3.22) we have (3.21).
Theorem 3.4 is proved.
Similarly, we state that the following theorem holds true.
Theorem 3.5. Let M be a space-like surface in the hyperbolic space \BbbH 3( - r2) \subset \BbbE 4
1 for some
r > 0. If M is a flat surface with parallel and light-like mean curvature vector in \BbbE 4
1, then M is
congruent to the surface given by
x(u, v) =
\biggl(
1
r
+
r
2
(u2 + v2), u, v,
r
2
(u2 + v2)
\biggr)
. (3.24)
The proof of this theorem is similar to the proof of Theorem 3.4.
Corollary 3.1. Up to linear isometries in \BbbE 4
1, the surface given by (3.21) (resp., (3.24)) is the
only surface in \BbbS 31(r2) \subset \BbbE 4
1 (resp., \BbbH 3(r2) \subset \BbbE 4
1) with harmonic Gauss map.
By combining the results given in this section, we state that the following theorem holds true.
Theorem 3.6. Let M be an oriented space-like surface in the Minkowski space \BbbE 4
1. Then the
Gauss map \nu of M is harmonic if and only if M is congruent to one of the following six types of
surfaces:
(i) an open part of a space-like plane;
(ii) the flat surface given by (3.3) for a smooth function \phi : \Omega \rightarrow \BbbR satisfying \Delta \phi = c, where \Omega
is an open set in \BbbR 2 and c \in \BbbR ;
(iii) the flat surface given by (3.20);
(iv) a non-parallel flat surface lying in the light cone \scrL \scrC ;
(v) the flat surface given by (3.21) lying in the de Sitter space-time \BbbS 31(r2);
(vi) the flat surface given by (3.24) lying in the hyperbolic space \BbbH 3( - r2).
4. Space-like surfaces in \BbbE \bffour
\bfone with pointwise 1-type Gauss map of the first kind. Let M be an
oriented space-like surface in the Minkowski space \BbbE 4
1 with harmonic Gauss map \nu . Then \nu satisfies
(1.1) for f = 0 and C = 0. Thus, a harmonic Gauss map \nu is of pointwise 1-type of the first kind.
In this section, we obtain a characterization of surfaces in \BbbE 4
1 with pointwise 1-type Gauss map of
the first kind.
Theorem 4.1. Let M be an oriented maximal surface in the Minkowski space \BbbE 4
1. Then M has
pointwise 1-type Gauss map of the first kind if and only if M has flat normal bundle. Moreover, the
Gauss map \nu satisfies (1.1) for f = \| h\| 2 and C = 0.
Proof. If M is maximal, then the Gauss map \nu satisfies (3.2). Hence, \nu is of pointwise 1-type
of the first kind if and only if RD = 0.
We now give the following lemma.
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1 WITH POINTWISE 1-TYPE GAUSS MAP 69
Lemma 4.1. Let M be an oriented maximal surface in the Minkowski space \BbbE 4
1. If M has
pointwise 1-type Gauss map of the first kind, then the function f = \| h\| 2 satisfies
e1(f) = - 4 \varepsilon \omega 12(e2)f, (4.1)
e2(f) = 4 \varepsilon \omega 12(e1)f, (4.2)
where \{ e1, e2\} is a local orthonormal frame for tangent bundle of M and \varepsilon \in \{ - 1, 1\} .
Proof. Let M be a maximal surface in \BbbE 4
1 with pointwise 1-type Gauss map of the first
kind. Then Theorem 4.1 implies that M has flat normal bundle. Thus, the shape operators can be
diagonalized simultaneously, i.e., there exists an orthornormal frame field \{ e1, e2, e3, e4\} on M such
that A\beta = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(h\beta 11, - h\beta 11), \beta = 3, 4, as H = 0. Therefore, we have from (2.7)
f = \| h\| 2 = 2
\bigl(
\varepsilon 3(h
3
11)
2 + \varepsilon 4(h
4
11)
2
\bigr)
. (4.3)
and Codazzi equation (2.10) yields
e1(h
3
11) - \varepsilon 4h
4
11\omega 34(e1) = - 2\omega 12(e2)h
3
11, (4.4)
e1(h
4
11) + \varepsilon 3h
3
11\omega 34(e1) = - 2\omega 12(e2)h
4
11, (4.5)
e2(h
3
11) - \varepsilon 4h
4
11\omega 34(e2) = 2\omega 12(e1)h
3
11, (4.6)
e2(h
4
11) + \varepsilon 3h
3
11\omega 34(e2) = 2\omega 12(e1)h
4
11. (4.7)
By multiplying (4.4) and (4.5), respectively, \varepsilon 3h311 and \varepsilon 4h
4
11 and adding them, we have
\varepsilon 3h
3
11e1(h
3
11) + \varepsilon 4h
4
11e1(h
4
11) = - 2\omega 12(e2)
\Bigl(
\varepsilon 3(h
3
11)
2 + \varepsilon 4(h
4
11)
2
\Bigr)
.
By using (4.3) again in this equation, we obtain (4.1). In a similar way, we see that (4.6) and (4.7)
give (4.2).
Lemma 4.1 is proved.
Proposition 4.1. Let M be an oriented maximal surface in the Minkowski space \BbbE 4
1. Then M
has (global) 1-type Gauss map of the first kind if and only if the Gauss map \nu of M is harmonic.
Proof. We assume that M has (global) 1-type Gauss map \nu of the first kind. Then Theorem 4.1
implies that M has flat normal bundle. On the other hand, since \nu is (global) 1-type of the first kind,
(1.1) is satisfied for f = f0, where f0 is a constant. Moreover, Lemma 4.1 implies that f satisfies
(4.1) and (4.2) from which we obtain \omega 12(e1)f0 = \omega 12(e2)f0 = 0 that imply f0 = 0 or \omega 12 = 0. In
the case f0 = 0, we have \Delta \nu = f0\nu = 0, i.e., \nu is harmonic. Otherwise M is flat, and it follows
from Proposition 3.1 that \nu is harmonic.
The converse is obvious.
Proposition 4.1 is proved.
Now we study non-maximal space-like surfaces in \BbbE 4
1 with pointwise 1-type Gauss map of the
first kind.
Theorem 4.2. Let M be an oriented non-maximal space-like surface in \BbbE 4
1. Then M has point-
wise 1-type Gauss map of the first kind if and only if M has parallel mean curvature vector.
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70 U. DURSUN, N. C. TURGAY
Proof. Let M be an oriented non-maximal space-like surface in \BbbE 4
1. Suppose that M has
pointwise 1-type Gauss map of the first kind. Then (1.1) is satisfied for f = \| h\| 2 and C = 0. From
(1.1) and (3.1) we obtain that RD = 0 and
\nabla (\mathrm{t}\mathrm{r}A3) \wedge e4 + e3 \wedge \nabla (\mathrm{t}\mathrm{r}A4) + 2
2\sum
j=1
\omega 34(ej)H \wedge ej = 0. (4.8)
Since RD = 0, there exists a local orthonormal frame \{ e3, e4\} of normal bundle of M such that
\omega 34 = 0. So, it follows from (4.8) that \nabla \mathrm{t}\mathrm{r}A3 = \nabla \mathrm{t}\mathrm{r}A4 = 0, that is, \mathrm{t}\mathrm{r}A\beta = \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}, \beta = 3, 4,
from which and \omega 34 = 0 we have DH = 0.
Conversely, let H be parallel. From Lemma 3.3 we have RD = 0. Thus, there exists a local,
orthonormal frame \{ e3, e4\} of normal bundle of M such that \omega 34 \equiv 0. So, it follows from DH = 0
that \mathrm{t}\mathrm{r}A3 and \mathrm{t}\mathrm{r}A4 are constants. Therefore, equation (3.1) implies that \Delta \nu = \| h\| 2\nu , that is, M
has pointwise 1-type Gauss map of the first kind.
Theorem 4.2 is proved.
Example 4.1. Let M be a surface in \BbbE 4
1 given by
x(u, v) =
1\surd
2
(u \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}
\surd
2v, u \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\surd
2v,
\surd
2 \mathrm{s}\mathrm{i}\mathrm{n}
\surd
2u - u \mathrm{c}\mathrm{o}\mathrm{s}
\surd
2u,
\surd
2 \mathrm{c}\mathrm{o}\mathrm{s}
\surd
2u+ u \mathrm{s}\mathrm{i}\mathrm{n}
\surd
2u).
Then the mean curvature vector H of M is parallel and light-like [14]. Moreover, the Gaussian
curvature of M is K = u - 4 which implies \| h\| 2 = - 2u - 4 from (2.9). Therefore, M has proper
1-type Gauss map of the first kind because of Theorem 4.2, that is, (1.1) is satisfied for C = 0 and
f = \| h\| 2 = - 2u - 4.
In [13], a complete classification of space-like surfaces with parallel mean curvature vector was
given. By combining Theorem 3.1 [13] and Theorem 4.2, we have the following theorem.
Theorem 4.3. Let M be an oriented non-maximal space-like surface in \BbbE 4
1 with space-like or
time-like mean curvature vector. Then M has pointwise 1-type Gauss map of the first kind if and
only if M is a CMC surface lying in the light cone \scrL \scrC \subset \BbbE 4
1, a Euclidean hyperplane \BbbE 3 \subset \BbbE 4
1,
a Lorentzian hyperplane \BbbE 3
1 \subset \BbbE 4
1, the de Sitter space-time \BbbS 31(c2) \subset \BbbE 4
1, or the hyperbolic space
\BbbH 3( - c2) \subset \BbbE 4
1.
In the next proposition we obtain characterization of non-maximal surfaces with (global) 1-type
Gauss map of the first kind.
Proposition 4.2. Let M be an oriented space-like surface in the Minkowski space \BbbE 4
1 with light-
like mean curvature vector. Then M has (global) 1-type Gauss map of the first kind if and only if the
Gauss map \nu of M is harmonic.
Proof. Suppose that M has non-harmonic, (global) 1-type Gauss map of the first kind with
light-like mean curvature vector. Then (1.1) is satisfied for f = f0 and C = 0, where f0 \not = 0 is a
constant. Moreover, Theorem 4.2 implies that the mean curvature vector H of M is parallel. Since
\nu is non-harmonic, it follows from Theorems 3.2 and 3.3 that M is congruent to a non-flat surface
lying in either \BbbS 31(r2) or \BbbH 3( - r2), and if M is lying in \BbbS 31(r2) (resp., in \BbbH 3( - r2)), then its mean
curvature vector H \prime in \BbbS 31(c2) (resp., in \BbbH 3( - r2)) satisfies \langle H \prime , H \prime \rangle = - r2 (resp., \langle H \prime , H \prime \rangle = r2).
Also, from Remark 3.1 we have \| h2\| = f0.
Let x be the position vector of M in \BbbE 4
1 and \langle x, x\rangle = \varepsilon 3r
- 2, where \varepsilon 3 = \pm 1. We choose
a local orthonormal frame \{ e3, e4\} of the normal bundle of M such that e3 = rx and H =
= \varepsilon 3r
2(e3 - e4). Since e3 = rx is parallel, we have \omega 34 = 0. Moreover, RD = 0, i.e., M has flat
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SPACE-LIKE SURFACES IN MINKOWSKI SPACE \BbbE 4
1 WITH POINTWISE 1-TYPE GAUSS MAP 71
normal bundle. Thus, the shape operators of M are simultaneously diagonalizable. So, there exists a
local orthonormal frame \{ e1, e2\} of tangent bundle of M such that A\beta = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}
\bigl(
h\beta 11, h
\beta
22
\bigr)
, \beta = 3, 4,
and
h311 + h322 = h411 + h422 = 2r2. (4.9)
On the other hand, since \| h\| 2 and \langle H,H\rangle are constants, (2.9) implies the Gaussian curvature K of
M is constant, i.e., we have K0 = \varepsilon 3
\bigl(
h311h
3
22 - h411h
4
22
\bigr)
, where K0 \not = 0 is a constant, from which
and (4.9) we obtain
ei(h
3
11)h
3
22 + h311ei(h
3
22) = ei(h
4
11)h
4
22 + h411ei(h
4
22)
and
ei(h
\beta
11) = - ei(h
\beta
22), i = 1, 2, \beta = 3, 4. (4.10)
Using these equations we get
ei(h
3
11)(h
3
11 - h322) = ei(h
4
11)(h
4
11 - h422). (4.11)
In addition, considering (4.10), the Codazzi equation (2.4) yields
e1(h
\beta
11) = - \omega 12(e2)
\bigl(
h\beta 11 - h\beta 22
\bigr)
, (4.12)
e2(h
\beta
11) = \omega 12(e1)
\bigl(
h\beta 11 - h\beta 22
\bigr)
, \beta = 3, 4. (4.13)
So, it follows from these equations and (4.11) that
\omega 12(ei)
\Bigl( \bigl(
h311 - h322
\bigr) 2 - \bigl(
h411 - h422
\bigr) 2\Bigr)
= 0, i = 1, 2. (4.14)
As M is not flat, we have \omega 12 \not = 0. Thus, (4.14) implies
\bigl(
h311 - h322
\bigr) 2
=
\bigl(
h411 - h422
\bigr) 2
from which
and (2.9) we get f0 = \| h\| 2 = 0 which is a contradiction. Therefore, the Gauss map \nu is harmonic.
The converse is obvious.
Proposition 4.2 is proved.
Next we give a characterization for non-maximal space-like surfaces in the Minkowski space \BbbE 4
1
with (global) 1-type Gauss map of the first kind.
Theorem 4.4. Let M be an oriented non-maximal surface in the Minkowski space \BbbE 4
1. Then M
has (global) 1-type Gauss map of the first kind if and only if M has parallel mean curvature vector
and constant Gaussian curvature.
Proof. Let M be an oriented non-maximal surface in Minkowski space \BbbE 4
1. First we assume
that M has (global) 1-type Gauss map of the first kind. Then it follows from (1.1) and (3.1) that
\| h\| 2 = f0 for some constant f0. Also, Theorem 4.2 implies that M has parallel mean curvature
vector which implies \langle H,H\rangle is constant. Therefore, (2.9) implies that the Gaussian curvature K of
M is constant.
Conversely, let M has parallel mean curvature vector and constant Gaussian curvature. By
Theorem 4.2 we have \Delta \nu = \| h\| 2\nu . Also, equation (2.9) implies that \| h\| 2 is constant. Therefore,
the Gauss map of M is of 1-type of the first kind.
Theorem 4.4 is proved.
Next we give an example of a surface with non-harmonic (global) 1-type Gauss map of the first
kind.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
72 U. DURSUN, N. C. TURGAY
Example 4.2. Let M be a surface in \BbbE 4
1 given by
x(u, v) = (a \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}u, a \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}u, b \mathrm{c}\mathrm{o}\mathrm{s} v, b \mathrm{s}\mathrm{i}\mathrm{n} v), b2 - a2 \not = 0, ab \not = 0.
Let c =
\sqrt{}
| b2 - a2| . Then we have M = H1( - a - 1) \times S1(b - 1) \subset S3
1(c
- 2) \subset \BbbE 4
1 if b2 - a2 > 0,
and M = H1( - a - 1) \times S1(b - 1) \subset H3( - c - 2) \subset \BbbE 4
1 if b2 - a2 < 0. By a direct calculation, it
can be seen that M has parallel mean curvature vector and constant Gaussian curvature. Hence,
Theorem 4.4 implies M has (global) 1-type Gauss map of the first kind.
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Received 04.11.15
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
|
| id | umjimathkievua-article-1419 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:04:58Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/51/5c856342db7f88510f1d5e70752a4f51.pdf |
| spelling | umjimathkievua-article-14192019-12-05T08:54:16Z Space-like surfaces in Minkowski space $E^4_1$ with pointwise 1-type Gauss map Просторово-подiбнi поверхнi у просторi мiнковського $E^4_1$ з поточковим гауссовим вiдображенням першого типу Dursun, U. Turgay, N. C. Дурсун, У. Тургау, Н. Ц. We first classify space-like surfaces in the Minkowski space $E^4_1$, de Sitter space $S^3_1$, and hyperbolic space $H^3$ with harmonic Gauss map. Then we give a characterization and classification of space-like surfaces with pointwise 1-type Gauss map of the first kind. We also present some explicit examples. Насамперед наведено класифiкацiю просторово-подiбних поверхонь у просторi Мiнковського $E^4_1$, просторi де Сiттера $S^3_1$ i гiперболiчному просторi $H^3$ з гармонiчним гауссовим вiдображенням. Пiсля цього охарактеризовано i наведено класифiкацiю просторово-подiбних поверхонь першого типу з поточковим гауссовим вiдображенням першого типу. Також наведено деякi конкретнi приклади. Institute of Mathematics, NAS of Ukraine 2019-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1419 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 1 (2019); 59-72 Український математичний журнал; Том 71 № 1 (2019); 59-72 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1419/403 Copyright (c) 2019 Dursun U.; Turgay N. C. |
| spellingShingle | Dursun, U. Turgay, N. C. Дурсун, У. Тургау, Н. Ц. Space-like surfaces in Minkowski space $E^4_1$ with pointwise 1-type Gauss map |
| title | Space-like surfaces in Minkowski space $E^4_1$ with pointwise 1-type Gauss map |
| title_alt | Просторово-подiбнi поверхнi у просторi мiнковського $E^4_1$
з поточковим гауссовим вiдображенням першого типу |
| title_full | Space-like surfaces in Minkowski space $E^4_1$ with pointwise 1-type Gauss map |
| title_fullStr | Space-like surfaces in Minkowski space $E^4_1$ with pointwise 1-type Gauss map |
| title_full_unstemmed | Space-like surfaces in Minkowski space $E^4_1$ with pointwise 1-type Gauss map |
| title_short | Space-like surfaces in Minkowski space $E^4_1$ with pointwise 1-type Gauss map |
| title_sort | space-like surfaces in minkowski space $e^4_1$ with pointwise 1-type gauss map |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1419 |
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