Notes on the lightlike hypersurfaces along spacelike submanifolds
UDC 514.7 In the light of the method of construction of lightlike hypersurfaces along spacelike submanifolds, we give a relation between the second fundamental form of a spacelike submanifold and the screen second fundamental form of the corresponding lightlike hypersurface. In addition, we investi...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507274771955712 |
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| author | Ekmekci, F. N. Tuğ, G. Екмекці, Ф. Н. Туг, Г. |
| author_facet | Ekmekci, F. N. Tuğ, G. Екмекці, Ф. Н. Туг, Г. |
| author_sort | Ekmekci, F. N. |
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| datestamp_date | 2019-12-05T08:57:08Z |
| description | UDC 514.7
In the light of the method of construction of lightlike hypersurfaces along spacelike submanifolds, we give a relation between
the second fundamental form of a spacelike submanifold and the screen second fundamental form of the corresponding
lightlike hypersurface. In addition, we investigate the conditions for a lightlike hypersurface of this kind to be screen
conformal. |
| first_indexed | 2026-03-24T02:06:43Z |
| format | Article |
| fulltext |
UDC 514.7
G. Tuğ (Karadeniz Techn. Univ., Trabzon, Turkey),
F. N. Ekmekci (Ankara Univ., Turkey)
NOTES ON THE LIGHTLIKE HYPERSURFACES
ALONG SPACELIKE SUBMANIFOLDS
ПРО СВIТЛОПОДIБНI ГIПЕРПОВЕРХНI
ВЗДОВЖ ПРОСТОРОВОПОДIБНИХ ПIДМНОГОВИДIВ
In the light of the method of construction of lightlike hypersurfaces along spacelike submanifolds, we give a relation between
the second fundamental form of a spacelike submanifold and the screen second fundamental form of the corresponding
lightlike hypersurface. In addition, we investigate the conditions for a lightlike hypersurface of this kind to be screen
conformal.
У свiтлi методу побудови свiтлоподiбних гiперповерхонь уздовж просторовоподiбних пiдмноговидiв отримано спiв-
вiдношення мiж другою фундаментальною формою просторовоподiбного пiдмноговиду та екранною другою фунда-
ментальною формою вiдповiдної свiтлоподiбної гiперповерхнi. Крiм того, вивчено умови, за яких така свiтлоподiбна
гiперповерхня є екранно конформною.
1. Introduction. The theory of lightlike hypersurfaces has a special place in differential geometry
and theoretical physics. In the general relativity, lightlike hypersurfaces play an important role since
they are considered as the models for different horizon types of black holes. A black hole is a region
of spacetime which contains a huge amount of mass compacted into an extremely small volume. The
gravity inside a black hole is so strong that, even light with a remarkable speed can not escape, see
[1] . Since Einstein’s theory of gravitation was first published in 1915, so many research papers on
the mathematical and physical theory of black holes, have been published. For further information
about black holes and applications of lightlike hypersurfaces, see [2, 4, 6].
The lightlike hypersurfaces along spacelike submanifolds was introduced by Izumiya and Sato in
[7]. They constructed the lightlike hypersurfaces as ruled hypersurfaces based on spacelike subman-
ifolds with lightlike rulings. There are several types of ruled surfaces in the Lorentz – Minkowski
space (see, for example, [1, 8]). In this paper, we investigate the geometric properties of the lightlike
hypersurfaces along spacelike submanifolds defined by
LHM (p, \xi , t) = X(u) + tLG(nT )(u, \xi ).
Since we have degenerate metric on the tangent space, considering a lightlike hypersurface together
with its screen distribution provides simplicity. Thus, we define the screen second fundamental form
and give the lightcone Weingarten equations for the screen distribution of the lightlike hypersurface
mentioned above. Then we find a relation between the second fundamental form of the spacelike
submanifold and the screen second fundamental form of corresponding lightlike hypersurface. Also,
we put forward the conditions to be screen conformal of the lightlike hypersurface. As an example,
we show that the event horizon in Schwarzschild spacetime is actually the lightlike hypersurface
along a spacelike submanifold. Then we support our theory with some other examples.
c\bigcirc G. TUĞ, F. N. EKMEKCİ, 2019
968 ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
NOTES ON THE LIGHTLIKE HYPERSURFACES ALONG SPACELIKE SUBMANIFOLDS 969
2. Preliminaries. Let x = (x0, x1, . . . , xn), y = (y0, y1, . . . , yn) \in \BbbR n+1, then the pseudoscalar
product of x and y is defined by
\langle x, y\rangle = - x0y0 +
n\sum
i=1
xiyi.
\bigl(
\BbbR n+1, \langle , \rangle
\bigr)
is called Lorentz – Minkowski (n+1)-space and represented by \BbbR n+1
1 . A non-zero vector
x \in \BbbR n+1
1 is spacelike, timelike or lightlike if \langle x, x\rangle > 0, \langle x, x\rangle < 0 or \langle x, x\rangle = 0, respectively. The
norm of a non-null vector is defined by \| x\| =
\sqrt{}
| \langle x, x\rangle | . The canonical projection \pi : \BbbR n+1
1 - \rightarrow \BbbR n,
where \pi (x0, x1, . . . , xn) = (x1, . . . , xn). The lightcone with vertex a is defined as follows:
LCa =
\bigl\{
x = (x0, x1, . . . , xn) \in \BbbR n+1
1
\bigm| \bigm| \langle x - a, x - a\rangle = 0
\bigr\}
and we denote LC\ast = LC0\setminus \{ 0\} .
Let X : U \rightarrow \BbbR n+1
1 be a spacelike embedding of codimension k, where U \subset \BbbR s is an open sub-
set. Take NpM as the pseudonormal space of M at p in \BbbR n+1
1 , which is a k-dimensional Lorentzian
subspace of Tp\BbbR n+1
1 . On the pseudonormal space NpM, there are following pseudospheres:
Np(M ; - 1) = \{ v \in NpM | \langle v, v\rangle = - 1\} ,
Np(M ; 1) = \{ v \in NpM | \langle v, v\rangle = 1\} ,
so that it can be written the following unit spherical normal bundles over M :
N(M ; - 1) =
\bigcup
p\in M
Np(M ; - 1) and N(M ; 1) =
\bigcup
p\in M
Np(M ; 1).
There is always a future directed unit timelike normal vector field nT (u) \in Np(M ; - 1). One
can also choose a pseudonormal section nS(u) \in
\bigl(
\mathrm{S}\mathrm{p}
\bigl\{
nT (u)
\bigr\} \bigr) \bot \cap N(M ; 1) at least locally. Then
\langle nS , nS\rangle = 1 and \langle nS , nT \rangle = 0. A (k - 1)-dimensional spacelike unit sphere is defined by
N1(M)p[n
T ] =
\bigl\{
\xi \in Np(M ; 1) | \langle \xi , nT \rangle = 0
\bigr\}
and a spacelike unit k - 2 spherical bundle over M is defined by
N1(M)[nT ] =
\bigcup
p\in M
N1(M)p[n
T ].
The vector field nT + nS is taken as a lightlike normal vector field along M, see [7].
Definition 1. The mapping LG(nT ) : N1(M)[nT ] \rightarrow LC\ast , defined by
LG(nT )(u, \xi ) = nT (u) + \xi ,
is called the lightcone Gauss image of N1(M)[nT ] [7].
Definition 2. A hypersurface
LHM (nT ) : N1(M)[nT ]\times \BbbR \rightarrow \BbbR n+1
1
given by LHM (p, \xi , t) = X(u) + t(nT + \xi )(u) = X(u) + tLG(nT )(u, \xi ), where p = X(u), is
called the lightlike hypersurface along M relative to nT [7].
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
970 G. TUĞ, F. N. EKMEKCİ
3. Lightlike hypersurfaces. In the light of the information given in [5], we consider the lightlike
hypersurface along a spacelike submanifold M = X(U) in \BbbR n+1
1 mentioned in the previous section.
Take LHM (u, \xi , t) = Y (u, \xi , t) to ease the calculations. If we take \~U = N1(M)[nT ]\times \BbbR , we can
represent the hypersurface by Y ( \~U) = \~M . It is easy to see that the dimension of TP
\~M = n and
M is a submanifold of \~M . We can find the basis of TP
\~M by taking the partial derivatives as
Yui(u, \xi , t) = Xui(u) + t
\bigl(
nT
ui
+ \xi ui
\bigr)
(u), i = 1, . . . , s,
Y\xi m(u, \xi , t) = t\xi \xi m(u), m = 1, . . . , k - 2, (1)
Yt(u, \xi , t) = (nT + \xi )(u),
where u \in U and \xi \in N1(M)[nT ]. Here we note that the sections Yui and Y\xi m are spacelike and Yt
is lightlike.
From now, we assume a1 = u1, . . . , as = us, as+1 = \xi 1, . . . , an - 1 = \xi k - 2, an = t for the
easement of the calculations. We have a pseudo-Riemannian metric on \~M = Y ( \~U) which is the
lightcone first fundamental form defined by ds2 =
\sum n
i=1
\~gijdaidaj , where \~gij = \langle Yai , Yaj \rangle . Now
let V \in \mathrm{R}\mathrm{a}\mathrm{d}(T \~M) and we choose a screen distribution of \~M as S(T \~M) = \mathrm{S}\mathrm{p} \{ Yui , Y\xi m\} . Then
the lightcone second fundamental form with respect to the pair
\bigl(
nT , \xi
\bigr)
of S(T \~M) is
\~hij(n
T ) = \langle - Vai , Yaj \rangle , where i = 1, . . . , n, j = 1, . . . , n - 1. (2)
We know that the radical distribution of the lightlike hypersurface \~M is \mathrm{R}\mathrm{a}\mathrm{d}
\bigl(
T \~M
\bigr)
= \mathrm{S}\mathrm{p} \{ Yt\} . If
we use the global null splitting theorem [1], to figure out the lightcone Weingarten operator of the
screen distribution S(T \~M) of \~M, we can give the following theorem.
Theorem 1. The lightcone Weingarten equations of the screen distribution S(T \~M) of \~M are
\Pi (Vai) = -
n - 1\sum
j=1
\~hji (n
T )Yaj , i = 1, . . . , n,
where \~hji (n
T ) =
\Bigl(
\~hik(n
T )
\Bigr) \bigl(
\~gkj
\bigr)
, \~gkj = (\~gkj)
- 1 and \Pi is the canonical projection of \chi
\bigl(
T \~M
\bigr)
on
\chi
\bigl(
S(T \~M)
\bigr)
.
Corollary 1. Let \~M be the lightlike hypersurface along a spacelike submanifold M . Then the
following equation gives the relation between the lightcone second fundamental forms of S(T \~M)
and M :
\~hi\alpha =
s\sum
\beta =1
\~h\beta i
\left( g\alpha \beta - 2th\alpha \beta +
t2
\lambda 2
n - 1\sum
l,q=1
\~hl\beta
\~hq\alpha \~glq
\right) ,
where i = 1, . . . , n, \alpha = 1, . . . , s and \lambda \in \BbbR .
4. Screen conformality. Since the screen distribution is non degenerate, it is very important
when we investigate a lightlike hypersurface. Screen conformality provides getting information about
the structure of the lightlike hypersurface with the help of its screen distribution. In the following
theorem, we use the components of the lightlike vector field in the radical distribution to show the
screen conformality of a lightlike hypersurface. Hence, this method makes remarkable simplicity.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
NOTES ON THE LIGHTLIKE HYPERSURFACES ALONG SPACELIKE SUBMANIFOLDS 971
Theorem 2. A lightlike hypersurface \~M along a spacelike submanifold M is screen conformal,
if (nT - \xi )i \not = 0 and
\partial
\bigl(
\xi \pm nT
\bigr) i
\partial aj
= 0, where i \not = j, i, j = 1, . . . , n.
Proof. From now, we will denote the metric and connection of \BbbR n+1
1 by \langle , \rangle and \=\nabla , respectively.
There is a section N \in tr (TM) which is given by N =
1
2
(\xi - nT ) and it ensures \langle N,N\rangle = 0,
\langle N,Yui\rangle = \langle N,Y\xi m\rangle = 0, \langle N,Yt\rangle = 1. Since B(X,Yt) = 0, see [1], where X \in \chi ( \~M), we have
\=\nabla XYt = \nabla XYt. We can write Yt =
\sum n
A=1
(nT + \xi )A
\partial
\partial aA
and X =
\sum n
j=1
xj
\partial
\partial aj
, then we get
\=\nabla XYt =
n\sum
A=1
n\sum
j=1
xj
\partial (nT + \xi )A
aj
\partial
\partial aA
.
Since
\partial (\xi + nT )i
\partial aj
= 0, where i \not = j, i, j = 1, . . . , n, we denote
\=\nabla XYt = Z =
\biggl(
x1
\partial (nT + \xi )1
a1
, . . . , xn
\partial (nT + \xi )n
an
\biggr)
,
where Z \in \chi
\bigl(
T \~M
\bigr)
. Therefore, we obtain
A\ast
Yt
X + \tau (X)Yt +\nabla XYt = 0.
It can be seen that \tau (X) = 0 by using the equations \langle \=\nabla XN,Yt\rangle = \tau (X), see [3] and (1). Then
A\ast
Yt
X = - PZ, where P is the projection on S(T \~M). For every X \in \chi
\bigl(
S(T \~M)
\bigr)
we can write
X =
\sum n
A=1
XA \partial
\partial aA
. Then we get
n\sum
A=1
XA(nT + \xi )A = 0,
\nabla YtX = \=\nabla YtX =
n\sum
A=1
n\sum
i=1
(nT + \xi )i
\partial XA
\partial ai
\partial
\partial aA
.
(3)
If we take the partial derivatives of (1), since
\partial (\xi + nT )i
\partial aj
= 0, where i \not = j, i, j = 1, . . . , n, we
obtain \langle \nabla YtX,Yt\rangle = 0. Therefore, \nabla YtX \in \chi
\bigl(
S(T \~M)
\bigr)
and ANYt = 0.
For X,W \in \chi
\bigl(
S(T \~M)
\bigr)
,
g (ANX,W ) = C(X,W ) = g(\nabla XW,N) =
\bigl\langle
\=\nabla XW,N
\bigr\rangle
.
If we write
\=\nabla XN =
1
2
n\sum
A=1
n\sum
i=1
XA\partial (nT - \xi )i
\partial aA
\partial
\partial ai
,
since
\partial (\xi - nT )i
\partial aj
= 0, i \not = j, i, j = 1, . . . , n, we have
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
972 G. TUĞ, F. N. EKMEKCİ
g (ANX,W ) = - \langle \=\nabla XN,W \rangle .
Hence, we write
ANX = - 1
2
\left[
X1\partial (n
T - \xi )1
\partial a1
...
Xn\partial (n
T - \xi )n
\partial an
\right]
n\times 1
A\ast
Yt
X = - PZ = -
\left[
X1\partial (n
T + \xi )1
\partial a1
...
Xn\partial (n
T + \xi )n
\partial an
\right]
n\times 1
.
Finding the map
\Phi =
\left[ \alpha 11 . . . \alpha 1n
...
. . .
...
\alpha n1 . . . \alpha nn
\right]
n\times n
which satisfies the equation ANX = \Phi A\ast
Yt
X, completes the proof. For this, from the equation
\alpha 11X
1\partial (n
T + \xi )1
\partial a1
+ . . .+ \alpha 1nX
n\partial (n
T + \xi )n
\partial an
= X1\partial (n
T - \xi )1
\partial a1
,
we see that all the coefficients are zero except the ones on diagonal line. We calculate \alpha ii as
\alpha 11 = - (nT + \xi )1
(nT - \xi )1
, \alpha 22 =
(nT + \xi )2
(nT - \xi )2
, . . . , \alpha nn =
(nT + \xi )n
(nT - \xi )n
.
Theorem 2 is proved.
Theorem 3. Let \~M be a lightlike hypersurface along a spacelike submanifold M. The Gauss
image LG(nT ) of M is a geodesic line in \~M .
5. Examples.
Example 1. The Schwarzschild spacetime in Eddington – Finkelstein coordinates (u, r, \vargamma , \varphi ) is
given by
ds2 = -
\biggl(
1 - 2M
r
\biggr)
du2 + 2dudr + r2d\Omega 2,
where M > 0 denotes the mass and d\Omega 2 = d\vargamma 2 + \mathrm{s}\mathrm{i}\mathrm{n}2 \theta d\varphi 2 denotes the volume element of the
standard sphere. The event horizon is the surface given by
r = r0 = 2M.
This is a lightlike hypersurface foliated by metric spheres of constant radius r = r0 and generated
by the lightlike vector field L = 2
\partial
\partial u
, see [9].
Now, let we consider the spacelike submanifold defined by
X(\vargamma , \varphi ) = (0, 2M,\vargamma , \varphi ).
Then we can find a basis of the tangent space as\biggl\{
X\vargamma =
\partial
\partial \vargamma
,X\varphi =
\partial
\partial \varphi
\biggr\}
.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
NOTES ON THE LIGHTLIKE HYPERSURFACES ALONG SPACELIKE SUBMANIFOLDS 973
Fig. 1
One can see that the vector field L is in the normal space of X. Then we find an orthogonal basis
of the normal space as
\biggl\{
\eta =
1
2
\partial
\partial u
+
\partial
\partial r
, \zeta =
1
2
\partial
\partial u
- \partial
\partial r
\biggr\}
. Here we choose \zeta = nT and \eta = \xi
since \zeta is timelike and \eta is spacelike. Hence, we have the lightcone Gauss image of the spacelike
submanifold X defined by (nT + \xi ) =
\partial
\partial u
. Finally, we can rewrite the event horizon as
Y (u, \vargamma , \varphi ) = (0, 2M,\vargamma , \varphi ) + u(nT + \xi ).
Example 2. In \BbbR 3
1 with the metric \langle x, y\rangle = - x0y0 + x1y1 + x2y2 we take the spacelike curve
\alpha (u) =
\bigl(
0, u, u2
\bigr)
. Then we have the Frenet frame
T =
1\surd
1 + 4u2
(0, 1, 2u),
N =
1\surd
1 + 4u2
(0, - 2u, 1),
B = ( - 1, 0, 0).
We can choose B = nT and N = \xi , then nT + \xi =
\biggl(
- 1,
- 2u\surd
1 + 4u2
,
1\surd
1 + 4u2
\biggr)
. Hence, the
lightlike hypersurface along \alpha is
Y (u, t) =
\biggl(
- t, u - 2ut\surd
1 + 4u2
, u2 +
t\surd
1 + 4u2
\biggr)
,
where t \in \BbbR . It can be seen in the Fig. 1.
If we use the notations in [3], we have
x0 = - t, x1 = u - 2ut\surd
1 + 4u2
, x2 = u2 +
t\surd
1 + 4u2
,
\partial
\partial u
=
\Biggl(
1 - 2t
(1 + 4u2)3/2
\Biggr)
\partial
\partial x1
+
\Biggl(
2u - 4ut
(1 + 4u2)3/2
\Biggr)
\partial
\partial x2
,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
974 G. TUĞ, F. N. EKMEKCİ
Fig. 2
Fig. 3
\partial
\partial t
= - \partial
\partial x0
- 2u\surd
1 + 4u2
\partial
\partial x1
+
1\surd
1 + 4u2
\partial
\partial x2
.
Then we calculate
nT + \xi = - \partial
\partial x0
- 2u\surd
1 + 4u2
\partial
\partial x1
+
1\surd
1 + 4u2
\partial
\partial x2
=
\partial
\partial t
.
It can be seen that the lightcone Gauss image LG(nT ) (u, \xi ) = nT (u) + \xi is in the direction of the
lightlike vector Yt \in \mathrm{R}\mathrm{a}\mathrm{d}T \~M = T \~M\bot . Figures 2 and 3 show the lightcone Gauss image of the
spacelike curve \alpha (u).
We can rewrite nT + \xi =
\bigl(
(nT + \xi )1, (nT + \xi )2
\bigr)
= (0, 1) according to the base
\biggl\{
\partial
\partial u
,
\partial
\partial t
\biggr\}
.
Since a1 = u, a2 = t we have
\partial (nT + \xi )1
\partial a2
=
\partial (nT + \xi )2
\partial a1
= 0.
According to Theorem 2, the lightlike hypersurface Y (u, t) is screen conformal.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
NOTES ON THE LIGHTLIKE HYPERSURFACES ALONG SPACELIKE SUBMANIFOLDS 975
Example 3. Let we take the spacelike curve \beta (u) = (0, \mathrm{c}\mathrm{o}\mathrm{s}u, \mathrm{s}\mathrm{i}\mathrm{n}u). Then the Frenet frame is
T = (0, - \mathrm{s}\mathrm{i}\mathrm{n}u, \mathrm{c}\mathrm{o}\mathrm{s}u),
N = (0, \mathrm{c}\mathrm{o}\mathrm{s}u, \mathrm{s}\mathrm{i}\mathrm{n}u),
B = ( - 1, 0, 0).
We can choose B = nT and N = \xi , then nT + \xi = ( - 1, \mathrm{c}\mathrm{o}\mathrm{s}u, \mathrm{s}\mathrm{i}\mathrm{n}u). Hence, the lightlike
hypersurface along \beta is
Y (u, t) = ( - t, (t+ 1) \mathrm{c}\mathrm{o}\mathrm{s}u, (t+ 1) \mathrm{s}\mathrm{i}\mathrm{n}u) .
It can be calculated that nT + \xi = (t+1)2
\partial
\partial u
=
\bigl(
(t+ 1)2, 0
\bigr)
and this vector field is in the direction
of the lightlike vector Yt too. Since
\partial (nT + \xi )1
\partial a2
\not = 0, assertions of Theorem 2 do not hold.
References
1. Abdel-Baky A. R., Aldossary M. T. On the null scrolls in Minkowski 3-space E3
1 // IOSR J. Math. – 2013. – 7. –
P. 11 – 16.
2. Chandrasekhar S. The Mathematical theory of black holes. – Oxford Univ. Press, 1983.
3. Duggal K. L., Bejancu A. Lightlike submanifolds of semi-Riemannian manifolds and applications. – Dordrecht:
Springer Sci.+Business Media, 1996.
4. Duggal K. L. Foliations of lightlike hypersurfaces and their physical interpretation // Centr. Eur. J. Math. – 2012. –
10. – P. 1789 – 1800.
5. Duggal K. L., Şahin B. Differential geometry of lightlike submanifolds. – Basel etc.: Birkhäuser, 2010.
6. Hawking S. W. The event horizons in black holes. – Amsterdam: North Holland, 1972.
7. Izumiya S., Sato T. Lightlike hypersurfaces along spacelike submanifolds in Minowski space-time // J. Geom. and
Phys. – 2013.
8. Liu H., Yuan Y. Pitch functions of ruled surfaces and B -scrolls in Minkowski 3-space // J. Geom. and Phys. – 2012. –
62. – P. 47 – 52.
9. Sauter J. Foliations of null hypersurfaces and the Penrose inequality: Doct. thesis. – Zürich: ETH, 2008. – № 17842.
Received 23.05.16,
after revision — 30.05.17
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
|
| id | umjimathkievua-article-1490 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:06:43Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/a7/f4badfa853301fe6bc3a6d6b55af5ea7.pdf |
| spelling | umjimathkievua-article-14902019-12-05T08:57:08Z Notes on the lightlike hypersurfaces along spacelike submanifolds Про свiтлоподiбнi гiперповерхнi вздовж просторовоподiбних пiдмноговидiв Ekmekci, F. N. Tuğ, G. Екмекці, Ф. Н. Туг, Г. UDC 514.7 In the light of the method of construction of lightlike hypersurfaces along spacelike submanifolds, we give a relation between the second fundamental form of a spacelike submanifold and the screen second fundamental form of the corresponding lightlike hypersurface. In addition, we investigate the conditions for a lightlike hypersurface of this kind to be screen conformal. УДК 514.7 У свiтлi методу побудови свiтлоподiбних гiперповерхонь уздовж просторовоподiбних пiдмноговидiв отримано спiв- вiдношення мiж другою фундаментальною формою просторовоподiбного пiдмноговиду та екранною другою фунда- ментальною формою вiдповiдної свiтлоподiбної гiперповерхнi. Крiм того, вивчено умови, за яких така свiтлоподiбна гiперповерхня є екранно конформною. Institute of Mathematics, NAS of Ukraine 2019-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1490 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 7 (2019); 968-975 Український математичний журнал; Том 71 № 7 (2019); 968-975 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1490/474 Copyright (c) 2019 Ekmekci F. N.; Tuğ G. |
| spellingShingle | Ekmekci, F. N. Tuğ, G. Екмекці, Ф. Н. Туг, Г. Notes on the lightlike hypersurfaces along spacelike submanifolds |
| title | Notes on the lightlike hypersurfaces along spacelike submanifolds |
| title_alt | Про свiтлоподiбнi гiперповерхнi
вздовж просторовоподiбних пiдмноговидiв |
| title_full | Notes on the lightlike hypersurfaces along spacelike submanifolds |
| title_fullStr | Notes on the lightlike hypersurfaces along spacelike submanifolds |
| title_full_unstemmed | Notes on the lightlike hypersurfaces along spacelike submanifolds |
| title_short | Notes on the lightlike hypersurfaces along spacelike submanifolds |
| title_sort | notes on the lightlike hypersurfaces along spacelike submanifolds |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1490 |
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