* Blanchet [2014] L. Blanchet, Gravitational radiation from post-newtonian sources and inspiralling compact binaries, Living Rev. Relativ. 17, 2 (2014), arXiv:1310.1528 .
  * [89] The ancillary files modes_3.5PNspin.dat.m and factorized_modes_3.5PNspin.dat.m contain the waveform modes with 3.5PN spin contributions, written in a PN expansion and in a factorized form.
  * Thorne [1980] K. Thorne, Multipole expansions of gravitational radiation, Rev. Mod. Phys. 52, 299 (1980).
  * Kidder [2008] L. Kidder, Using full information when computing modes of post-newtonian waveforms from inspiralling compact binaries in circular orbits, Phys. Rev. D 77, 044016 (2008), arXiv:0710.0614 .
  * Blanchet  _et al._ [2008] L. Blanchet, G. Faye, B. R. Iyer, and S. Sinha, The Third post-Newtonian gravitational wave polarisations and associated spherical harmonic modes for inspiralling compact binaries in quasi-circular orbits, Class. Quant. Grav. 25, 165003 (2008), [Erratum: Class.Quant.Grav. 29, 239501 (2012)], arXiv:0802.1249 [gr-qc] .
  * Henry  _et al._ [2021] Q. Henry, G. Faye, and L. Blanchet, The current-type quadrupole moment and gravitational-wave mode ($\ell$, m) = (2, 1) of compact binary systems at the third post-Newtonian order, Class. Quant. Grav. 38, 185004 (2021), arXiv:2105.10876 [gr-qc] .
  * Henry  _et al._ [2020] Q. Henry, G. Faye, and L. Blanchet, Tidal effects in the gravitational-wave phase evolution of compact binary systems to next-to-next-to-leading post-Newtonian order, Phys. Rev. D 102, 044033 (2020), arXiv:2005.13367 [gr-qc] .
  * Blanchet  _et al._ [2005] L. Blanchet, T. Damour, G. Esposito-Farèse, and B. R. Iyer, Dimensional regularization of the third post-newtonian gravitational wave generation of two point masses, Phys. Rev. D 71, 124004 (2005), gr-qc/0503044 .
  * Blanchet and Faye [2000] L. Blanchet and G. Faye, Hadamard regularization, J. Math. Phys. 41, 7675 (2000), gr-qc/0004008 .
  * Blanchet and Faye [2001] L. Blanchet and G. Faye, General relativistic dynamics of compact binaries at the third post-newtonian order, Phys. Rev. D 63, 062005 (2001), gr-qc/0007051 .
  * Marchand  _et al._ [2020] T. Marchand, Q. Henry, F. Larrouturou, S. Marsat, G. Faye, and L. Blanchet, The mass quadrupole moment of compact binary systems at the fourth post-Newtonian order, Class. Quant. Grav. 37, 215006 (2020), arXiv:2003.13672 [gr-qc] .
  * Blanchet and Iyer [2004] L. Blanchet and B. R. Iyer, Hadamard regularization of the third post-newtonian gravitational wave generation of two point masses, Phys. Rev. D 71, 024004 (2004), gr-qc/0409094 .
  * Blanchet  _et al._ [1998] L. Blanchet, G. Faye, and B. Ponsot, Gravitational field and equations of motion of compact binaries to 5/2 post-newtonian order, Phys. Rev. D 58, 124002 (1998), gr-qc/9804079 .
  * Faye  _et al._ [2015] G. Faye, L. Blanchet, and B. R. Iyer, Non-linear multipole interactions and gravitational-wave octupole modes for inspiralling compact binaries to third-and-a-half post-newtonian order, Class. Quant. Grav. 32, 045016 (2015), arXiv:1409.3546 .
  * Blanchet  _et al._ [2022] L. Blanchet, G. Faye, and F. Larrouturou, The Quadrupole Moment of Compact Binaries to the Fourth post-Newtonian Order: From Source to Canonical Moment, (2022), arXiv:2204.11293 [gr-qc] .
  * Blanchet and Schäfer [1993] L. Blanchet and G. Schäfer, Gravitational wave tails and binary star systems, Class. Quant. Grav. 10, 2699 (1993).
  * Arun  _et al._ [2008] K. Arun, L. Blanchet, B. Iyer, and M. Qusailah, Inspiralling compact binaries in quasi-elliptical orbits: The complete 3PN energy flux, Phys. Rev. D 77, 064035 (2008), arXiv:0711.0302 .
  * Gradshteyn and Ryzhik [1980] I. Gradshteyn and I. Ryzhik,  _Table of Integrals, Series and Products_ (Academic Press, 1980).
  * Favata [2009] M. Favata, Post-newtonian corrections to the gravitational-wave memory for quasicircular, inspiralling compact binaries, Phys. Rev. D 80, 024002 (2009), arXiv:0812.0069 .
  * Tanaka  _et al._ [1996] T. Tanaka, Y. Mino, M. Sasaki, and Shibata, Gravitational waves from a spinning particle in circular orbits around a rotating black hole, Phys. Rev. D 54, 3762 (1996).
  * Regge and Wheeler [1957] T. Regge and J. A. Wheeler, Stability of a Schwarzschild singularity, Phys. Rev. 108, 1063 (1957).
  * Zerilli [1970] F. J. Zerilli, Effective potential for even parity Regge-Wheeler gravitational perturbation equations, Phys. Rev. Lett. 24, 737 (1970).
  * Poisson [1993] E. Poisson, Gravitational radiation from a particle in circular orbit around a black hole. 1: Analytical results for the nonrotating case, Phys. Rev. D 47, 1497 (1993).
  * Blanchet [1998] L. Blanchet, Gravitational wave tails of tails, Class. Quant. Grav. 15, 113 (1998), [Erratum: Class.Quant.Grav. 22, 3381 (2005)], arXiv:gr-qc/9710038 .
  * Asada and Futamase [1997] H. Asada and T. Futamase, Propagation of gravitational waves from slow motion sources in Coulomb type potential, Phys. Rev. D56, 6062 (1997), arXiv:gr-qc/9711009 [gr-qc] .
  * Pan  _et al._ [2011b] Y. Pan, A. Buonanno, M. Boyle, L. T. Buchman, L. E. Kidder, H. P. Pfeiffer, and M. A. Scheel, Inspiral-merger-ringdown multipolar waveforms of nonspinning black-hole binaries using the effective-one-body formalism, Phys. Rev. D 84, 124052 (2011b), arXiv:1106.1021 [gr-qc] .
  * [114] Sergei Ossokine, Private communication (2022).


# On contact screen conformal null submanifolds

Samuel Ssekajja  *  School of Mathematics
University of Witwatersrand
Private Bag 3, Wits 2050
South Africa<EMAIL_ADDRESS>
###### Abstract.

First, we prove that indefinite Sasakian manifolds do not admit any screen
conformal $r$-null submanifolds, tangent to the structure vector field. We,
therefore, define a special class of null submanifolds, called; contact screen
conformal $r$-null submanifold of indefinite Sasakian manifolds. Several
characterisation results, on the above class of null submanifolds, are proved.
In particular, we prove that such null submanifolds exists in indefinite
Sasakian space forms of constant holomorphic sectional curvatures of $-3$.

###### Key words and phrases:

Contact screen conformal submanifold, Null submanifolds, Integrable screens

###### 2010 Mathematics Subject Classification:

Primary 53C25; Secondary 53C40, 53C50

##  1\. Introduction

In the theory of non-degenerate submanifolds, the second fundamental forms and
their respective shape operators are related by means of the metric tensor.
Contrary to this, there are interrelations between the second fundamental
forms of null submanifold and its screen distribution and their respective
shape operators. These interrelations indicates that the null geometry depends
on a choice of screen distribution as explained in [5]. While we know that the
second fundamental forms of the null submanifolds are independent of a screen
(see Theorem 5.1.2 of [5, p. 199]), the same is not true for the fundamental
forms of the screens, which is the main cause of non-uniqueness anomaly in the
null geometry. Since, in general, it is impossible to remove this anomaly, the
authors in [5] considered null hypersurfaces and half null submanifolds for
which the null and screen second fundamental forms are conformally related.
Such classes of null submanifolds are called screen conformal (see Definition
2.2.1 of [5, p. 51] and Definition 4.4.1 of [5, p. 179]). However, this
condition can not be used for the case of general $r$-null submanifolds. For
this reason, Duggal-Sahin [5], extended the concept of screen conformal to
general null submanifolds of semi-Riemannian manifolds (see Definition 5.2.2
of [5]).

In case the ambient manifold is an indefinite Sasakian manifold, we note that
the above concept is not applicable due to some obvious contradictions
asproved in Theorem 3.3. Therefore, we introduce the notion of contact screen
conformal null submanifolds to cover this gap. Null submanifolds have numerous
applications in mathematical physics, particularly in general relativity and
electromagnetism, see [4, 5] for amore details. Many research papers have been
published on null geometry, for example [1, 2, 3, 6, 7, 8, 9, 10, 11], and
many more references cited therein. The main objective of this paper is define
the concept of contact screen conformal $r$-null submanifolds of indefinite
Sasakian manifolds. We prove that such null submanifolds exists in indefinite
Sasakian space forms of constant holomorphic sectional curvatures of $-3$. The
rest of the paper is arranged as follows; In section 2 we give basic notions
needed in the rest of the paper and in Section 3, we present our main results.

##  2\. Preliminaries

Let $(\overline{M},\overline{g})$ be a real $(m+n)$-dimensional semi-
Riemannian manifold of constant index $\nu$ such that $m,n\geq 1$,
$1\leq\nu\leq m+n-1$, and let $(M,g)$ be an $m$-dimensional submanifold of
$\overline{M}$. In case $g$ is degenerate on the tangent bundle $TM$ of $M$,
we say that $M$ a null submanifold [4]. We denote the set of smooth sections
of a vector bundle $\Xi$ by $\Gamma(\Xi)$. For a degenerate metric tensor
$g=\overline{g}_{|{TM}}$, there exists locally a non-zero vector field
$\xi\in\Gamma(TM)$ such that $g(\xi,X)=0$, for any $X\in\Gamma(TM)$. Then, for
each tangent space $T_{x}M$, $x\in M$, we have $T_{x}M^{\perp}=\\{u\in
T_{x}\overline{M}:\overline{g}(u,v)=0,\;\forall\,v\in T_{x}M\\},$ which is a
degenerate $n$-dimensional subspace of $T_{x}\overline{M}$. The radical or
null subspace of $M$ is denoted by $\mathrm{Rad}\,T_{x}M$ and is given by
$\mathrm{Rad}\,T_{x}M=\\{\xi_{x}\in T_{x}M:g(\xi_{x},X)=0,\;\forall\,X\in
T_{x}M\\}$. Notice that $\mathrm{Rad}\,T_{x}M=T_{x}M\cap T_{x}M^{\perp}$ and
its dimension depends on $x\in M$. A submanifold $M$ of $\overline{M}$ is
called $r$-null if the mapping
$\mathrm{Rad}\,TM:x\longrightarrow\mathrm{Rad}\,T_{x}M,$ defines a smooth
distribution of rank $r>0$, where $\mathrm{Rad}\,TM$ is called the radical
(null) distribution on $M$. Let $S(TM)$ be a screen distribution which is a
semi-Riemannian complementary distribution of $\mathrm{Rad}\,TM$ in $TM$, and
is given by $TM=\mathrm{Rad}\,TM\perp S(TM)$. Note that the distribution
$S(TM)$ is not unique and canonically isomorphic to the factor vector bundle
$TM/\mathrm{Rad}\,TM$ [4]. Choose a screen transversal bundle $S(TM^{\perp})$,
which is semi-Riemannian complementary to $\mathrm{Rad}\,TM$ in $TM^{\perp}$.
Since, for any local basis $\\{\xi_{i}\\}$ of $\mathrm{Rad}\,TM$, there exists
a local null frame $\\{N_{i}\\}$ of sections with values in the orthogonal
complement of $S(TM^{\perp})$ in $S(TM)^{\perp}$ such that
$\overline{g}(\xi_{i},N_{j})=\delta_{ij}$, it follows that there exists a null
transversal vector bundle  $l\mathrm{tr}(TM)$ locally spanned by $\\{N_{i}\\}$
[4]. Let $\mathrm{tr}(TM)$ be complementary (but not orthogonal) vector bundle
to $TM$ in $T\overline{M}$. Then,

$\displaystyle\mathrm{tr}(TM)$ $\displaystyle=l\mathrm{tr}(TM)\perp
S(TM^{\perp}),$ $\displaystyle T\overline{M}$ $\displaystyle=S(TM)\perp
S(TM^{\perp})\perp\\{\mathrm{Rad}\,TM\oplus l\mathrm{tr}(TM)\\}$
$\displaystyle=TM\oplus\mathrm{tr}(TM).$

We say that a null submanifold $M$ of $\overline{M}$ is

  1. (1)

$r$-null if $1\leq r<\min\\{m,n\\}$,

  2. (2)

co-isotropic if $1\leq r=n<m$, $S(TM^{\perp})=\\{0\\}$,

  3. (3)

isotropic if $1\leq r=m<n$, $S(TM)=\\{0\\}$,

  4. (4)

totally null if $r=n=m$, $S(TM)=S(TM^{\perp})=\\{0\\}$.

Details on the above classes of null submanifolds with examples are found in
[4, 5]. Let $M$ be a coisotropic null submanifold and consider a local quasi-
orthonormal fields of frames of $\overline{M}$ along $M$, on $\mathcal{U}$ as
$\\{\xi_{1},\cdots,\xi_{r},N_{1},\cdots,N_{r},Z_{r+1},\cdots,Z_{m}\\},$ where
$\\{Z_{r+1},\cdots,Z_{m}\\}$ is an orthogonal basis of
$\Gamma(S(TM)|_{\mathcal{U}})$ and that $\epsilon_{a}=g(Z_{a},Z_{a})$ is the
signature of $\\{Z_{a}\\}$. The following range of indices will be used.
$i,j,k\in\\{1,\cdots,r\\}$. Let $P$ be the projection morphism of $TM$ onto
$S(TM)$. Then, the Gauss-Weingartein equations [5] of a coisotropic
submanifold $M$ and $S(TM)$ are

$\displaystyle\overline{\nabla}_{X}Y$
$\displaystyle=\nabla_{X}Y+\sum_{i=1}^{r<m}h_{i}^{l}(X,Y)N_{i},$ (2.1)
$\displaystyle\overline{\nabla}_{X}N_{i}$
$\displaystyle=-A_{N_{i}}X+\sum_{j=1}^{r<m}\tau_{ij}(X)N_{j},$ (2.2)
$\displaystyle\nabla_{X}PY$
$\displaystyle=\nabla_{X}^{*}PY+\sum_{i=1}^{r<m}h_{i}^{*}(X,PY)\xi_{i},$ (2.3)
$\displaystyle\nabla_{X}\xi_{i}$
$\displaystyle=-A_{\xi_{i}}^{*}X-\sum_{j=1}^{r<m}\tau_{ji}(X)\xi_{j},$ (2.4)

for all $X,Y\in\Gamma(TM)$ where $\nabla$ and $\nabla^{*}$ are the induced
connections on $TM$ and $S(TM)$ respectively, $h_{i}^{l}$’s are symmetric
bilinear forms known as local null fundamental forms of $TM$. Also,
$h_{i}^{*}$’s are the local second fundamental forms of $S(TM)$. On the other
hand, $A_{N_{i}}$’s and $A_{\xi_{i}}^{*}$’s are linear operators on $TM$ while
$\tau_{ij}$’s are 1-forms on $TM$. It is easy to see from (2.1) that
$h_{i}^{l}(X,Y)=\overline{g}(\overline{\nabla}_{X}Y,\xi_{i})$, for all
$X,Y\in\Gamma(TM)$, from which we deduce the independence of $h_{i}^{l}$s on
the choice of $S(TM)$. It is easy to see that $\nabla^{*}$ is a metric
connection on $S(TM)$ while $\nabla$ is generally not a metric connection and
satisfies the relation

$(\nabla_{X}g)(Y,Z)=\sum_{i=1}^{r}\\{h_{i}^{l}(X,Y)\theta_{i}(Z)+h_{i}^{l}(X,Z)\theta_{i}(Y)\\},$
(2.5)

for any $X,Y\in\Gamma(TM)$ and 1-forms $\theta_{i}$ given by
$\theta_{i}(X)=\overline{g}(X,N_{i})$, for all $X\in\Gamma(TM)$. The above two
types of local second fundamental forms are related to their shape operators
by the following set of equations

$\displaystyle g(A_{\xi_{i}}^{*}X,Y)$
$\displaystyle=h_{i}^{l}(X,Y)+\sum_{j=1}^{r}h_{j}^{l}(X,\xi_{i})\theta_{j}(Y),\;\;\;\;\bar{g}(A_{\xi_{i}}^{*}X,N_{j})=0,$
(2.6) $\displaystyle g(A_{N_{i}}X,Y)$
$\displaystyle=h_{i}^{*}(X,PY),\;\;\;\theta_{j}(A_{N_{i}}X)+\theta_{i}(A_{N_{j}}X),\;\;X,Y\in\Gamma(TM).$
(2.7)

Let $(M,g,S(TM))$ be an $m$-dimensional $r$-null coisotropic submanifold of
$(\overline{M},\overline{g})$. Let $\overline{R}$ and $R$ denote the curvature
tensors of $\overline{\nabla}$ and $\nabla$ respectively. The following
curvature identities are needed in this paper (see [4] or [5] for details)

$\displaystyle\overline{R}(X,Y)Z=R(X,$ $\displaystyle
Y)Z+\sum_{i=1}^{r}\\{h_{i}^{l}(X,Z)A_{N_{i}}Y-h_{i}^{l}(Y,Z)A_{N_{i}}X\\}$
$\displaystyle+\sum_{i=1}^{r}[(\nabla_{X}h_{i}^{l})(Y,Z)-(\nabla_{Y}h_{i}^{l})(X,Z)$
$\displaystyle+\sum_{j=1}^{r}\\{\tau_{ji}(X)h_{j}(Y,Z)-\tau_{ji}(Y)h_{j}(X,Z)\\}]N_{i},$
(2.8) $\displaystyle R(X,Y)PZ=R^{*}($ $\displaystyle
X,Y)PZ+\sum_{i=1}^{r}\\{h_{i}^{*}(X,PZ)A_{\xi_{i}}^{*}Y-h_{i}^{*}(Y,PZ)A_{\xi_{i}}^{*}X\\}$
$\displaystyle+\sum_{i=1}^{r}[(\nabla_{X}h_{i}^{*})(Y,PZ)-(\nabla_{Y}h_{i}^{*})(X,PZ)$
$\displaystyle+\sum_{j=1}^{r}\\{\tau_{ij}(Y)h_{j}^{*}(X,PZ)-\tau_{ij}(X)h_{j}^{*}(Y,PZ)\\}]\xi_{i},$
(2.9)

for all $X,Y,Z\in\Gamma(TM)$. An old dimensional smooth manifold
$(\overline{M},\overline{g})$ is called a contact metric manifold [5] if there
exist a (1,1) -tensor field $\overline{\phi}$, a vector field $\zeta$, called
the characteristic vector field, and its 1-form $\eta$ satisfying

$\displaystyle\overline{\phi}^{2}X$
$\displaystyle=-X+\eta(X)\zeta,\;\;\overline{\phi}\zeta=0,\;\;\;\eta\circ\overline{\phi}=0,\;\;\;\eta(\zeta)=1,$
(2.10) $\displaystyle\overline{g}(\overline{\phi}X,\overline{\phi}Y)$
$\displaystyle=\overline{g}(X,Y)-\eta(X)\eta(Y),\;\;\;\eta(X)=\overline{g}(\zeta,X),$
(2.11) $\displaystyle
d\eta(X,Y)=\overline{g}(\overline{\phi}X,Y),\;\;\;\forall\,X,Y\in\Gamma(T\overline{M}).$
(2.12)

Then, the set $(\overline{\phi},\eta,\zeta,\overline{g})$ is called a contact
metric structure on $\overline{M}$. Furthermore, $\overline{M}$ has a normal
contact structure [5] if $N_{\overline{\phi}}+2d\eta\otimes\zeta=0$, where
$N_{\overline{\phi}}$ is the Nijenhuis tensor field of $\overline{\phi}$. A
normal contact metric manifold is called Sasakian [5] for which we have

$\displaystyle(\overline{\nabla}_{X}\overline{\phi})(Y)=\overline{g}(X,Y)\zeta-\eta(Y)X,\;\;\;\forall\,X,Y\in\Gamma(T\overline{M}),$
(2.13)

where $\overline{\nabla}$ is a metric connection on $\overline{M}$. A Sasakian
manifold $\overline{M}=(\overline{M},\overline{\phi},\zeta,\eta,\overline{g})$
is called an indefinite Sasakian manifold [5] if $(\overline{M},\overline{g})$
is a semi-Riemannian manifold of index $\nu(>0)$. Replacing $Y$ by $\zeta$ in
(2.13), and using (2.10), we get

$\displaystyle\overline{\nabla}_{X}\zeta=-\overline{\phi}X,\;\;\;\,\forall\,X\in\Gamma(T\overline{M}).$
(2.14)

A plane section $\pi$ in $T_{x}\overline{M}$ of a Sasakian manifold
$\overline{M}$ is called a $\overline{\phi}$-section if it is spanned by a
unit vector $X$ orthogonal to $\zeta$ and $\overline{\phi}X$, where $X$ is a
non-null vector field on $\overline{M}$. The sectional curvature
$K(X,\overline{\phi}X)$ of a $\overline{\phi}$-section is called a
$\overline{\phi}$-sectional curvature. If $\overline{M}$ has a
$\overline{\phi}$-sectional curvature $c$ which does not depend on the
$\overline{\phi}$-section at each point, then, $c$ is constant in
$\overline{M}$ and $\overline{M}$ is called a Sasakian space form, denoted by
$\overline{M}(c)$. Moreover, the curvature tensor $\overline{R}$ of
$\overline{M}$ satisfies (see [5] for more detailes)

$\displaystyle 4\overline{R}(X,Y)Z$
$\displaystyle=(c+3)\\{\overline{g}(Y,Z)X-\overline{g}(X,Z)Y\\}+(c-1)\\{\eta(X)\eta(Z)Y$
$\displaystyle-\eta(Y)\eta(Z)X+\overline{g}(X,Z)\eta(Y)\zeta-\overline{g}(Y,Z)\eta(X)\zeta+\overline{g}(\overline{\phi}Y,Z)\overline{\phi}X$
$\displaystyle-\overline{g}(\overline{\phi}X,Z)\overline{\phi}Y-2\overline{g}(\overline{\phi}X,Y)\overline{\phi}Z\\},\;\;\forall\,X,Y,Z\in\Gamma(T\overline{M}).$
(2.15)

Let $(M,g)$ be a null submanifold of an indefinite Sasakian manifold
$(\overline{M},\overline{g})$. If the characteristic vector field $\zeta$ is
tangent to $M$, then it is obvious that $\zeta$ does not belong to
$\mathrm{Rad}\,TM$. This enables one to choose a screen distribution $S(TM)$
which contains $\zeta$. This implies that if $\zeta$ is tangent to $M$, then
it belongs to $S(TM)$ (see Calin [2] for more details). Let $M$ be a
coisotropic null submanifold of an indefinite Sasakian $\overline{M}$.
Furthermore, we assume that $\overline{\phi}\mathrm{Rad}\,TM$ and
$\overline{\phi}l\mathrm{tr}(TM)$ are subbundles of $S(TM)$. It follows that
$S(TM)=\\{\overline{\phi}\mathrm{Rad}\,TM\oplus\overline{\phi}l\mathrm{tr}(TM)\\}\perp
D_{0}\perp\mathbb{R}\zeta$, where $D_{0}$ is a non-degenerate almost complex
distribution with respect to $\overline{\phi}$, and $\mathbb{R}\zeta$ is a
line bundle spanned by $\zeta$. Then, we have $TM=D\oplus
D^{\prime}\perp\mathbb{R}\zeta$, where
$D=\mathrm{Rad}\,TM\perp\overline{\phi}\mathrm{Rad}\,TM\perp D_{0}$ and
$D^{\prime}=\overline{\phi}l\mathrm{tr}(TM)$. Consider local null vector
fields $U_{i}$,$V_{i}$, for each $i\in\\{1,\ldots,r\\}$, and their 1-forms
$u_{i}$, $v_{i}$ defined by

$\displaystyle U_{i}$
$\displaystyle=-\overline{\phi}N_{i},\;\;\;\;\;\;\;\;\;\;\;\;\;V_{i}=-\overline{\phi}\xi_{i},$
(2.16) $\displaystyle u_{i}(X)$
$\displaystyle=g(X,V_{i}),\;\;\;\;v_{i}(X)=g(X,U_{i}).$ (2.17)

Let $S$ be the projection morphism of $TM$ onto $D$. Then, for any
$X\in\Gamma(TM)$,

$\displaystyle\overline{\phi}X=\phi X+\sum_{i=1}^{r}u_{i}(X)N_{i},$ (2.18)

where $\phi$ is a tensor field of type (1,1) globally defined on $M$ by
$\phi=\overline{\phi}\circ S$. By a direct calculation using (2.1)–(2.4),
(2.16)–(2.18), we derive

$\displaystyle h_{j}^{l}(X,U_{i})$
$\displaystyle=h_{i}^{*}(X,V_{j}),\;\;\;h_{j}^{l}(X,V_{i})=h_{i}^{l}(X,V_{j}),$
(2.19) $\displaystyle\nabla_{X}U_{i}$ $\displaystyle=\phi
A_{N_{i}}X+\sum_{j=1}^{r}\tau_{ij}(X)U_{j}-\theta_{i}(X)\zeta,$ (2.20)
$\displaystyle\nabla_{X}V_{i}$ $\displaystyle=\phi
A_{\xi_{i}}^{*}X-\sum_{j=1}^{r}\tau_{ji}(X)V_{j}+\sum_{j=1}^{r}h_{j}^{l}(X,\xi_{i})U_{j},$
(2.21)

for all $X,\in\Gamma(TM)$. On the other hand, using (2.14), (2.1)–(2.4), we
have

$\displaystyle
h_{i}^{l}(X,\zeta)=-u_{i}(X),\;\;\;\;\;h_{i}^{*}(X,\zeta)=-v_{i}(X),$ (2.22)

for all $X\in\Gamma(TM)$.

##  3\. Contact screen conformal submanifolds

On a null hypersurface of a semi-Riemannian manifold, the screen and null
shape operators $A_{\xi}^{*}$ and $A_{N}$, respectively, where
$\xi\in\Gamma(TM^{\perp})$ and $N\in\Gamma(tr(TM))$, are both screen-valued
operators. Due to this fact, it is always possible to link the two operators
via a non-vanishing smooth function to give rise to a class of hypersurfaces
called; screen conformal null hypersurfaces (see [5, Definition 2.2.1]). A
similar consideration is done for half null submanifolds (see [5, Definition
4.4.1]). However, this cannot be considered for a general null submanifold due
to the fact that the shape operators $A_{N_{i}}$, for all
$i\in\\{1\ldots,r\\}$, are generally not screen-valued. For the above reason,
Duggal-Sahin [5] defined a certain type of screen conformality for a
coisotropic submanifold as follows;

######  Definition 3.1 ([5]).

A coisotropic null submanifold $(M,g,S(TM))$ of a semi-Riemannian manifold
$(\overline{M},\overline{g})$ is called a screen locally conformal submanifold
if the fundamental forms $h_{i}^{*}$ of $S(TM)$ are conformally related to the
corresponding null fundamental forms $h_{i}^{l}$ of $M$ by

$\displaystyle
h_{i}^{*}(X,PY)=\varphi_{i}h_{i}^{l}(X,Y),\;\;\;\forall\,X,Y\in\Gamma(TM),\;\;i\in\\{i,\ldots,r\\},$
(3.1)

where the $\varphi_{i}^{\prime}$s are smooth functions on the neighbourhood
$\mathcal{U}$ of $M$.

It is then proved, in Theorem 5.2.3 of [5, p. 204], that any screen
distribution satisfying Definition 3.1 is integrable. Further still, the
definition is extended to $r$-null submanifolds in which similar conclusions
on the integrability of $S(TM)$ are reached (see Theorem 5.2.6 of [5, p.
219]).

######  Example 3.2 ([5]).

Let us consider the coisotropic submanifold
$x_{2}=(x_{3}^{2}+x_{5}^{2})^{1/2}$, $x_{4}=x_{1}$, $x_{3},x_{5}>0$ of
$\overline{M}=(\mathbb{R}_{2}^{5},\overline{g})$, where $\mathbb{R}_{2}^{5}$
is a semi-Euclidean space of signature $(-,-,+,+.+)$ with respect to the
canonical basis $(\partial x_{1},\partial x_{2},\partial x_{3},\partial
x_{4},\partial x_{5})$. Then, it is easy to check that
$S(TM)=\mathrm{Span}\\{X\\}$,
$\mathrm{Rad}\,TM=\mathrm{Span}\\{\xi_{1},\xi_{2}\\}$ and
$l\mathrm{tr}(TM)=\mathrm{Span}\\{N_{1},N_{2}\\}$, where
$X=x_{5}\partial{x_{2}}+x_{2}\partial x_{5}$,
$\xi_{1}=\partial_{1}+\partial_{4}$, $\xi_{2}=x_{2}\partial
x_{2}+x_{3}\partial x_{3}+x_{5}\partial_{5}$, $N_{1}=(1/2)(-\partial
x_{1}+\partial x_{4})$ and $N_{2}=(1/2x_{3}^{2})(-x_{2}\partial
x_{2}+x_{3}\partial x_{3}-x_{5}\partial x_{5})$. Then, a direct a direct
calculation reveals that $\overline{\nabla}_{\xi_{1}}X=0$,
$\overline{\nabla}_{\xi_{2}}X=X$, $\overline{\nabla}_{\xi_{1}}\xi_{2}=0$ and
$\overline{\nabla}_{X}X=x_{2}\partial x_{2}+x_{5}\partial x_{5}$. Next, by
Gauss’ formulae, we have $\nabla_{X}X=(1/2)\xi_{2}$, $h_{1}^{*}(X,X)=0$,
$h_{1}^{*}(\xi_{1},X)=h_{2}^{*}(\xi_{1},X)=0$,
$h_{1}^{*}(\xi_{2},X)=h_{2}^{*}(\xi_{2},X)=h_{1}^{l}(\xi_{1},X)=h_{2}^{l}(\xi_{2},X)=0$,
$h_{1}^{l}=0$, $h_{2}^{l}(X,X)=-x_{3}^{2}$ and $h_{2}^{*}=1/2$. It follows
that $M$ is screen conformal with $\varphi_{1}$ arbitrary and
$\varphi_{2}=-1/2x_{3}^{2}$.

However, it is very important to note that when the ambient space is an
indefinite Sasakian manifold, such screen conformal null submanifolds, tangent
to $\zeta$, i.e. $\zeta\in\Gamma(TM)$, do not exist. In fact, we have have the
following result.

######  Theorem 3.3.

There does not exist any screen locally conformal null subamnifolds
$(M,g,S(TM))$, tangent to the structure vector field $\zeta$, of an indefinite
Sasakian manifold $(\overline{M},\overline{g})$.

###### Proof.

Assume, on contrary, that $M$ is locally screen conformal, then from (2.22)
and (3.1) of Definition 3.1, we have

$\displaystyle-
v_{i}(X)=h_{i}^{*}(X,\zeta)=\varphi_{i}h_{i}^{l}(X,\zeta)=-\varphi_{i}u_{i}(X),$
(3.2)

for all $X\in\Gamma(TM)$. Setting $X=V_{j}$ in (3.2) we get
$-v_{i}(V_{j})=-\delta_{ij}=0$, for all $i,j\in\\{1,\ldots,r\\}$, which is a
contradiction. On the other hand letting $X=U_{j}$ in (3.2), we get
$-\varphi_{i}u_{i}(U_{j})=0$. As $\varphi_{i}$’s are nonzero, it follows that
$-u_{i}(U_{j})=-\delta_{ij}=0$, which is also a contradiction. Hence, $M$ can
not be locally screen conformal in an indefinite Sasakian manifold. ∎

Based on Theorem 3.3, we notice that Definition 3.1 fails for null
submanifolds of indefinite Sasakian manifolds, mainly in portions of $TM$
containing the structure vector field $\zeta$. This can be rectified by
defining the concept of screen conformality of $h_{i}^{*}$ and $h_{i}^{l}$ on
$D\oplus D^{\prime}$, instead of $TM=D\oplus D^{\prime}\perp\mathbb{R}\zeta$.
To this end, let $\tilde{P}$ be the projection morphism of $TM$ onto the
subbundle $D\oplus D^{\prime}$. It then follows easily that any
$X\in\Gamma(TM)$ can be written as $X=\tilde{P}X+\eta(X)\zeta$. Then, by a
direct calculation, we have

$\displaystyle h_{i}^{*}(\tilde{P}X,\tilde{P}PY)$
$\displaystyle=h_{i}^{*}(X,Y)-\eta(Y)h_{i}^{*}(X,\zeta)-\eta(X)h_{i}^{*}(\zeta,Y),$
(3.3)

for all $X,Y\in\Gamma(TM)$, in which we have used (2.22) to deduce that
$h_{i}^{*}(\zeta,\zeta)=-v_{i}(\zeta)=0$. We also have,

$\displaystyle h_{i}^{l}(\tilde{P}X,\tilde{P}Y)$
$\displaystyle=h_{i}^{l}(X,Y)-\eta(Y)h_{i}^{l}(X,\zeta)-\eta(X)h_{i}^{l}(\zeta,Y),$
(3.4)

for all $X,Y\in\Gamma(TM)$, in which we have used the fact that
$h_{i}^{l}(\zeta,\zeta)=-u_{i}(\zeta)=0$.

Then, we have the following definition;

######  Definition 3.4.

Let $(\overline{M},\overline{\phi},\zeta,\eta,\overline{g})$ be an indefinite
almost contact manifold. A null submanifold $(M,g,S(TM))$, tangent to the
structure vector field $\zeta$, is called contact locally screen conformal if
the fundamental forms $h_{i}^{*}$ of $S(TM)$ are conformally related to the
corresponding null fundamental forms $h_{i}^{l}$ of $M$, on the subbundle
$D\oplus D^{\prime}$ by

$\displaystyle
h_{i}^{*}(\tilde{P}X,\tilde{P}PY)=\varphi_{i}h_{i}^{l}(\tilde{P}X,\tilde{P}Y),\;\;\;\forall\,X,Y\in\Gamma(TM),\;\;i\in\\{i,\ldots,r\\},$
(3.5)

where the $\varphi_{i}^{\prime}$s are smooth functions on the neighbourhood
$\mathcal{U}$ of $M$. In view of (2.22), (3.3) and (3.4), $M$ is contact
locally screen conformal if

$\displaystyle h_{i}^{*}(X,PY)=\varphi_{i}\\{h_{i}^{l}$
$\displaystyle(X,Y)+u_{i}(X)\eta(Y)+u_{i}(Y)\eta(X)\\}$ $\displaystyle-
v_{i}(X)\eta(Y)+h_{i}^{*}(\zeta,Y)\eta(X),$ (3.6)

for all $X,Y\in\Gamma(TM)$.

It has has been established (see Theorem 5.2.3 of [5, p. 204]) that when $M$
is locally screen conformal then $S(TM)$ is integrable. However, this is not
generally true for a contact locally screen conformal null submanifold. In
fact, we have the following result.

######  Theorem 3.5.

The screen distribution $S(TM)$ of a contact locally screen conformal null
submanifold of an indefinite Sasakian manifold is integrable if and only if
$h^{*}_{i}(\zeta,PX)=-v_{i}(X)$, for all $X\in\Gamma(TM)$.

###### Proof.

In view of (3.4) and the symmetry of $h_{i}^{l}$’s, we have

$\displaystyle h_{i}^{*}(X,Y)-h_{i}^{*}(Y,X)$
$\displaystyle=v_{i}(Y)\eta(X)-v_{i}(X)\eta(Y)$
$\displaystyle+h_{i}^{*}(\zeta,Y)\eta(X)-h_{i}^{*}(\zeta,X)\eta(Y),$ (3.7)

for all $X,Y\in\Gamma(S(TM))$. Now, if $S(TM)$ is integrable then all
$A_{N_{i}}$ are symmetric on $S(TM)$ by Theorem 5.1.5 of [5]. Hence, the left
hand side of (3) vanishes which further implies that
$v_{i}(Y)\eta(X)-v_{i}(X)\eta(Y)+h_{i}^{*}(\zeta,Y)\eta(X)-h_{i}^{*}(\zeta,X)\eta(Y)=0$,
for all $X,Y\in\Gamma(TM)$. Letting $Y=\zeta$ in this relation we get
$-v_{i}(X)+h_{i}^{*}(\zeta,\zeta)\eta(X)-h_{i}^{*}(\zeta,X)=0$. But, by
(2.22), we see that $h_{i}^{*}(\zeta,\zeta)=-v_{i}(\zeta)=0$. Hence, we have
$h_{i}^{*}(\zeta,X)=-v_{i}(X)$. The converse is obvious, which completes the
proof. ∎

######  Theorem 3.6.

Let $(M,g)$ be a contact screen conformal coisotropic null submanifold,
tangent to $\zeta$, of an indefinite Sasakian space form $\overline{M}(c)$.
Then, $c=-3$. Moreover, the functions $\varphi_{i}$, $i\in\\{i,\ldots,r\\}$,
satisfies the differential equations

$\displaystyle(\xi_{j}\varphi_{i})h_{i}^{l}(V_{i},U_{i})-\sum_{k=1}^{r}[\varphi_{k}\tau_{ik}(\xi_{j})+\varphi_{i}\tau_{ki}(\xi_{j})]h_{k}^{l}(V_{i},U_{i})=0,$
(3.8)

and the curvature tensor of $(M,g)$ takes the form

$\displaystyle R(X,Y)Z$
$\displaystyle=-\sum_{i=1}^{r}\\{h_{i}^{l}(X,Z)A_{N_{i}}Y-h_{i}^{l}(Y,Z)A_{N_{i}}X\\}-\eta(X)\eta(Z)Y$
$\displaystyle+\eta(Y)\eta(Z)X-\overline{g}(X,Z)\eta(Y)\zeta+\overline{g}(Y,Z)\eta(X)\zeta-\overline{g}(\overline{\phi}Y,Z)\phi
X$ $\displaystyle+\overline{g}(\overline{\phi}X,Z)\phi
Y+2\overline{g}(\overline{\phi}X,Y)\phi Z,\;\;\;\forall\,X,Y,Z\in\Gamma(TM).$
(3.9)

###### Proof.

For all $X,Y,Z\in\Gamma(D\oplus D^{\prime})$, relations (2.22) and (3.5) leads
to

$\displaystyle(\nabla_{X}$ $\displaystyle
h_{i}^{*})(Y,PZ)=(X\varphi_{i})h_{i}^{l}(Y,PZ)+\varphi_{i}\\{Xh_{i}^{l}(Y,PZ)-h_{i}^{l}(\tilde{P}\nabla_{X}Y,PZ)$
$\displaystyle-
h_{i}^{l}(Y,\tilde{P}\nabla_{X}^{*}PZ)\\}-h_{i}^{*}(\zeta,PZ)\eta(\nabla_{X}Y)+v_{i}(Y)\eta(\nabla_{X}^{*}PZ).$
(3.10)

On the other hand, using (3), we have

$\displaystyle(\nabla_{X}h_{i}^{l})(Y,PZ)$
$\displaystyle=Xh_{i}^{l}(Y,PZ)-h_{i}^{l}(\tilde{P}\nabla_{X}Y,PZ)-h_{i}^{l}(Y,\tilde{P}\nabla_{X}^{*}PZ)$
$\displaystyle+u_{i}(PZ)\eta(\nabla_{X}Y)+u_{i}(Y)\eta(\nabla_{X}PZ),$ (3.11)

for all $X,Y,Z\in\Gamma(D\oplus D^{\prime})$. Using (3) and (3), we derive

$\displaystyle(\nabla_{X}h_{i}^{*})(Y,PZ)$
$\displaystyle=(X\varphi_{i})h_{i}^{l}(Y,PZ)+\varphi_{i}\\{(\nabla_{X}h_{i}^{l})(Y,PZ)$
$\displaystyle-u_{i}(PZ)\eta(\nabla_{X}Y)-u_{j}(Y)\eta(\nabla_{X}^{*}PZ)\\}$
$\displaystyle-
h_{i}^{*}(\zeta,PZ)\eta(\nabla_{X}Y)+v_{i}(Y)\eta(\nabla_{X}^{*}PZ).$ (3.12)

Interchanging $X$ and $Y$ in (3), subtracting the two relations and then use
(2.8) and (2.9), we get

$\displaystyle\overline{g}(\overline{R}$
$\displaystyle(X,Y)PZ,N_{i})-\varphi_{i}\overline{g}(\overline{R}(X,Y)PZ,\xi_{i})$
$\displaystyle=(X\varphi_{i})h_{i}^{l}(Y,PZ)-(Y\varphi_{i})h_{i}^{l}(X,PZ)+\varphi_{i}\\{u_{i}(PZ)\eta(\nabla_{Y}X)$
$\displaystyle-
u_{i}(PZ)\eta(\nabla_{X}Y)+u_{i}(X)\eta(\nabla_{Y}^{*}PZ)-u_{i}(Y)\eta(\nabla_{X}^{*}PZ)\\}$
$\displaystyle+h_{i}^{*}(\zeta,PZ)\eta(\nabla_{Y}X)-h_{i}^{*}(\zeta,PZ)\eta(\nabla_{X}Y)+v_{i}(Y)\eta(\nabla_{X}^{*}PZ)$
$\displaystyle-
v_{i}(X)\eta(\nabla_{Y}^{*}PZ)+\sum_{j=1}^{r}\varphi_{j}\\{h_{j}^{l}(X,PZ)\tau_{ij}(Y)-h_{j}^{l}(Y,PZ)\tau_{ij}(X)\\}$
$\displaystyle-\sum_{j=1}^{r}\varphi_{i}\\{h_{j}(Y,PZ)\tau_{ji}(X)-h_{j}^{l}(X,PZ)\tau_{ji}(Y)\\}.$
(3.13)

Then, applying (2) to (3) and then let $X=\xi_{k}$, we derive

$\displaystyle\frac{c+3}{4}$ $\displaystyle
g(Y,PZ)\delta_{ik}+\frac{c-1}{4}\\{u_{k}(PZ)v_{i}(Y)+2u_{k}(Y)v_{i}(PZ)\\}$
$\displaystyle\frac{c-1}{4}\varphi_{i}\\{u_{k}(PZ)u_{i}(Y)+2u_{k}(Y)u_{i}(PZ)=(\xi_{k}\varphi_{i})h_{i}^{l}(Y,PZ)$
$\displaystyle+\varphi_{i}\\{u_{i}(PZ)u_{k}(Y)-u_{i}(PZ)\eta(\nabla_{\xi_{k}}Y)-u_{i}(Y)\eta(\nabla^{*}_{\xi_{k}}PZ)\\}$
$\displaystyle+h_{i}^{*}(\zeta,PZ)u_{k}(Y)-h_{i}^{*}(\zeta,PZ)\eta(\nabla_{\xi_{k}}Y)+v_{i}(Y)\eta(\nabla_{\xi_{k}}^{*}PZ)$
$\displaystyle-\sum_{j=1}^{r}\varphi_{j}h_{j}^{l}(Y,PZ)\tau_{ij}(\xi_{k})-\sum_{j=1}^{r}\varphi_{i}h_{j}^{l}(Y,PZ)\tau_{ji}(\xi_{k}),$
(3.14)

for all $Y,Z\in\Gamma(D\oplus D^{\prime})$. Interchanging $j$ and $k$ in (3)
and then substitute $Y=V_{\ell}$ and $PZ=U_{\ell}$, we get

$\displaystyle\frac{c+3}{4}\delta_{ij}$
$\displaystyle+\frac{c-1}{4}\delta_{i\ell}\delta_{j\ell}=(\xi_{j}\varphi_{i})h_{i}^{l}(V_{\ell},U_{\ell})-\varphi_{i}\eta(\nabla_{\xi_{j}}V_{\ell})\delta_{i\ell}$
$\displaystyle-
h_{i}^{*}(\zeta,U_{\ell})\eta(\nabla_{\xi_{j}}V_{\ell})+\eta(\nabla_{\xi_{j}}U_{\ell})\delta_{i\ell}-\sum_{k=1}^{r}\varphi_{k}h_{k}^{l}(V_{\ell},U_{\ell})\tau_{ik}(\xi_{j})$
$\displaystyle-\sum_{k=1}^{r}\varphi_{i}h_{k}^{l}(V_{\ell},U_{\ell})\tau_{ki}(\xi_{j}).$
(3.15)

But, in view of (2.20) and (2.21), we have

$\displaystyle\eta(\nabla_{\xi_{j}}U_{\ell})$ $\displaystyle=\eta(\phi
A_{N_{\ell}}\xi_{j})+\sum_{k=1}^{r}\tau_{\ell
k}(\xi_{j})\eta(U_{k})-\theta_{\ell}(\xi_{j})\eta(\zeta)=-\delta_{\ell j},$
(3.16) $\displaystyle\eta(\nabla_{\xi_{j}}V_{\ell})$ $\displaystyle=\eta(\phi
A^{*}_{\xi_{\ell}}\xi_{j})-\sum_{k=1}^{r}\tau_{k\ell}(\xi_{j})\eta(V_{k})+\sum_{k=1}^{r}h_{k}^{l}(\xi_{j},\xi_{\ell})\eta(U_{k})=0.$
(3.17)

Replacing (3.16) and (3.17) in (3), we get

$\displaystyle\frac{c+3}{4}\\{\delta_{ij}+\delta_{\ell i}\delta_{\ell j}\\}$
$\displaystyle=(\xi_{j}\varphi_{i})h_{i}^{l}(V_{\ell},U_{\ell})-\sum_{k=1}^{r}\varphi_{k}h_{k}^{l}(V_{\ell},U_{\ell})\tau_{ik}(\xi_{j})$
$\displaystyle-\sum_{k=1}^{r}\varphi_{i}h_{k}^{l}(V_{\ell},U_{\ell})\tau_{ki}(\xi_{j}).$
(3.18)

Setting $i=\ell$ in (3.18), we get

$\displaystyle\frac{c+3}{2}\delta_{ij}$
$\displaystyle=(\xi_{j}\varphi_{i})h_{i}^{l}(V_{i},U_{i})-\sum_{k=1}^{r}\varphi_{k}h_{k}^{l}(V_{i},U_{i})\tau_{ik}(\xi_{j})$
$\displaystyle-\sum_{k=1}^{r}\varphi_{i}h_{k}^{l}(V_{i},U_{i})\tau_{ki}(\xi_{j}).$
(3.19)

On the other hand, if we set $Y=U_{\ell}$ and $PZ=V_{\ell}$ in (3), and then
following the simplifications in (3)-(3.19), we have

$\displaystyle\frac{3}{4}(c+3)\delta_{ij}$
$\displaystyle=(\xi_{j}\varphi_{i})h_{i}^{l}(V_{i},U_{i})-\sum_{k=1}^{r}\varphi_{k}h_{k}^{l}(V_{i},U_{i})\tau_{ik}(\xi_{j})$
$\displaystyle-\sum_{k=1}^{r}\varphi_{i}h_{k}^{l}(V_{i},U_{i})\tau_{ki}(\xi_{j}).$
(3.20)

Then, from (3.19) and (3.20), we have $\frac{c+3}{4}=0$ or simply $c=-3$.
Moreover, we also have

$(\xi_{j}\varphi_{i})h_{i}^{l}(V_{i},U_{i})-\sum_{k=1}^{r}\varphi_{k}h_{k}^{l}(V_{i},U_{i})\tau_{ik}(\xi_{j})-\sum_{k=1}^{r}\varphi_{i}h_{k}^{l}(V_{i},U_{i})\tau_{ki}(\xi_{j})=0,$

which proves (3.8). Finally, (3.6) follows from (2.8) and (2) with $c=-3$,
which completes the proof. ∎

The following result also follows from Theorem 3.6.

######  Corollary 3.7.

There does not exist any contact screen conformal coisotropic submanfold,
tangent to $\zeta$, of an indefinite Sasakian space form
$\overline{M}(c\neq-3)$.

## References

  * [1] C. Atindogbé, Scalar curvature on lightlike hypersurfaces, Balkan Society of Geometers, Geometry Balkan Press 2009, Applied Sciences, 11 (2009), 9-18.
  * [2] C. Calin, Contributions to geometry of CR-submanifold, Ph.D. Thesis, University of Iasi (Romania), 1998.
  * [3] J. Dong and X. Liu, Totally Umbilical Lightlike Hypersurfaces in Robertson-Walker Spacetimes, Hindawi Publishing Corporation, Volume 2014, Article ID 974695, 10 pages.
  * [4] K. L. Duggal and A. Bejancu, Lightlike submanifolds of semi-Riemannian manifolds and applications, Mathematics and Its Applications, Kluwer Academic Publishers, 1996.
  * [5] K. L. Duggal and B. Sahin, Differential geometry of lightlike submanifolds. Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2010.
  * [6] K. L. Duggal and D. H. Jin, Null curves and hypersurfaces of semi-Riemannian manifolds, World Scientific, 2007.
  * [7] K. L. Duggal and B. Sahin, Lightlike CR hypersurfaces of indefinite Kaehler manifolds, Acta Applicandae Mathematicae 31 (1993) 171-190.
  * [8] D. H. Jin, Ascreen lightlike hypersurfaces of an indefinite Sasakian manifold. J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. Volume 20, No. 1 (2013), 25-35.
  * [9] D. H. Jin, Geometry of lightlike hypersurfaces of an indefinite sasakian manifold, Indian J. Pure Appl. Math., 41(4)(2010), 569-581.
  * [10] D. H. Jin, Special lightlike hypersurfaces of indefinite Kaehler manifolds, Filomat, 30(2016), no. 7, 1919–1930.
  * [11] F. Massamba and S. Ssekajja, Quasi generalized CR-lightlike submanifolds of indefinite nearly Sasakian manifolds, Arab. J. Math. 5 (2016) 87-101.
  * [12] D. N. Kupeli, Singular semi-Riemannian geometry, Mathematics and Its Applications, Vol. 366, Kluwer Academic Publishers, 1996.