# On the geometry of null hypersurfaces of indefinite complex contact
manifolds

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

We study the geometry of null hypersurfaces in indefinite complex contact
manifolds. We prove several classification results for a variety of well-known
null hypersurfaces, including the totally umbilic, totally screen umbilic, and
the screen conformal ones. Furthermore, a characterisation of the ambient
space is given in case the underlying null hypersurface is totally contact
umbilic, totally contact screen umbilic or contact screen conformal, i.e. we
have proved that the ambient complex contact manifold must be a space of
constant $GH$-sectional curvature of $-3$.

###### Key words and phrases:

Null hypersurfaces, Totally umbilic null hypersurfaces, Complex contact
manifolds

###### 2010 Mathematics Subject Classification:

Primary 53C25; Secondary 53C40, 53C50

##  1\. Introduction

On any semi-Riemannian manifold there is a natural existence of null
(lightlike) subspaces. In 1996, Duggal-Bejancu published a book [3] on the
null geometry of submanifolds which fiiled an important missing part in the
general theory of submanifolds. This book was later updated by Duggal-Sahin in
[4], by collecting most of the new discoveries in the area since the first
publiccation. Away from these two book, many researchers have investigated the
geometry of null subspaces of semi-Riemannian manifolds. On the other hand, in
about the same tine as in the book [3], Kupeli [7] introduces the theory of
null geometry in a relatively different way. The main tool in his approach was
the consideration of a factor bundle which is isomorphic to the screen
distribution used by the authors in [3]. In Chapter 8 of [4], the authors
introduces the geometry of null submanifolds of indefinite quaternion Kaehler
manifolds. Therein, the authors study the geometry of real null hypersurfaces,
the structure of null submanifolds, both, of indefinite quaternion Kaehler
manifolds and show that a quaternion null submanifold is always totally
geodesic. This result implies that the study of null submanifolds, other than
quaternion null submanifolds, is interesting. Then, they deal with the
geometry of screen real submanifolds in detail. As a generalization of real
null hypersurfaces of quaternion Kaehler manifolds, they introduced $QR$-null
submanifolds. Furthermaore, they show that the class of $QR$-null submanifolds
does not include quaternion null submanifolds and screen real submanifolds.
They also introduced and studied the geometry of screen $QR$-null and screen
$CR$-null submanifolds as generalizations of quaternion null submanifolds and
screen real submanifolds, and provided examples for each class of null
submanifolds of indefinite quaternion Kaehler manifolds.

Despite all the above contributions, we remark that the null geometry of
submanifolds of indefinite Sasakian 3-structure manifolds, as well as
indefinite complex contact manifold have not yet been studied. In [1], the
geometry of complex contact manifolds in the Riemannian sense is done in which
the foundations on such manifiolds is given, from structures to their
curvatures. The objective of this paper is to introduce the geometry of null
hypersurfaces of indefinite complex contact manifolds. Several
characterisation results are proved. In fact, we show that normal indefinite
complex contact manifolds do not admit any totally umbilic, totally screen
umbilic as well as screen conformal null hypersurfaces tangent to the vertical
distribution. The notions of totally contact umbilic, totally contact screen
umbilic and contact screen conformal are explored in details. In particular we
prove that an indefinite complex contact manifold of constant $GH$-sectional
curvature different from $-3$ does not admit any totally contact umbilic,
totally contact screen umbilic and contact screen conformal null
hypersurfaces, tangent to the characteristic subbundle.

The paper is arranged as follows; In Section 2, we quote some basic notions on
complex contact manifolds as well as null hypersurfaces needed in the rest the
paper. In Section 3, we prove several non-existence results and in Section 4,
we focus on contact umbilic, contact screen umbilic and contact screen
conformal null hypersurface in indefinite complex contact space forms.

##  2\. Preliminaries

A complex contact manifold is a complex manifold, $\overline{M}$, of odd
complex dimension $(2n+1)$ together with an open covering
$\\{\mathcal{O}_{i}\\}$ by coordinate neighbourhoods such that: (1) On each
$\mathcal{O}_{i}$ there is a holomorphic 1-form $\theta_{i}$ such that
$\theta_{i}\wedge(d\theta_{i})^{n}\neq 0$. (2) On
$\mathcal{O}_{i}\cup\mathcal{O}_{j}\neq\emptyset$ there is a non-vanishing
holomorphic function $f_{\alpha\beta}$ such that
$\theta_{i}=f_{\ij}\theta_{j}$ (see [1, 2] for more details). Furthermore, the
subspaces $\\{X\in T_{m}\mathcal{O}_{i}:\theta_{i}(X)=0\\}$ defines a non-
integrable holomorphic subbundle $\mathcal{H}$ of complex dimension $2n$
called the complex contact subbundle or horizontal subbundle. The quotient
$L=T\overline{M}/\mathcal{H}$ is a complex line bundle over $\overline{M}$ [2,
p. 49]. Some well-known examples of complex contact metric manifolds include
the complex Heisenberg group $H_{\mathbb{C}}$ and the odd-dimensional complex
projective space, see [1, 2] for more details on these manifolds. Define a
local section $U$ of $T\overline{M}$, i.e., a section of $T\mathcal{O}$, by
$du(U,X)=0$, for every $X\in\mathcal{H}$, $u(U)=1$ and $v(U)=0$. Such local
sections then define a global subbundle $\mathcal{V}$ by
$\mathcal{V}|_{\mathcal{O}}=\mathrm{Span}\\{U,JU\\}$. Then, we have
$T\overline{M}=\mathcal{H}\perp\mathcal{V}$ and we denote the projection map
to $\mathcal{H}$ by $p:T\overline{M}\longrightarrow\mathcal{H}$. The subbundle
$\mathcal{V}$ is called the vertical subbundle or characteristic subbundle. On
the other hand, if $\overline{M}$ is a complex manifold with almost complex
structure $J$, Hermitian metric $\overline{g}$ and open covering by coordinate
neighbourhoods $\\{\mathcal{O}_{i}\\}$, $\overline{M}$ is called a complex
almost contact metric manifold if it satisfies the following two conditions:
(1) On each $\mathcal{O}_{i}$, there exists 1-forms $u_{i}$, $v_{i}=u_{i}\circ
J$, with orthogonal dual vector fields $U_{i}$ and $V_{i}=-JU_{i}$, and
(1,1)-tensor fields $G_{i}$ and $H_{i}=G_{i}J$ such that

$\displaystyle\;\;\;\;\;\;\;\;H^{2}_{i}=G^{2}_{i}=-I+u_{i}\otimes
U_{i}+v_{i}\otimes V_{i},$ (2.1)
$\displaystyle\overline{g}(G_{i}X,Y)=-\overline{g}(X,G_{i}Y),\;\;\;\;\overline{g}(U_{i},X)=u_{i}(X),$
(2.2)
$\displaystyle\;\;\;\;\;\;\;\;G_{i}J=-JG_{i},\;\;\;G_{i}U=0,\;\;\;u_{i}(U)=1,$
(2.3)

for all $X,Y\in\Gamma(T\overline{M})$. (2) On the overlaps
$\mathcal{O}_{i}\cap\mathcal{O}_{j}\neq\emptyset$, the above tensors transform
as $u_{j}=au_{i}-bv_{i}$, $v_{j}=bu_{i}+av_{i}$, $G_{j}=aG_{i}-bH_{i}$ and
$H_{j}=bG_{i}+aH_{i}$, for some functions $a$, $b$ defined on the overlaps
with $a^{2}+b^{2}=1$.

It is obvious that $H_{i}$ also anticommutes with $J$ and is skew-symmetric
with respect to $\overline{g}$ and that $G_{i}$ and $H_{i}$ annihilate both
$U$ and $V$. Furthermore, the local contact form $\theta$ is $u-iv$ to within
a nonvanishing complex-valued function multiple (see [1]). Moreover, given a
complex contact manifold, a complex almost contact metric structure can be
chosen such that

$\displaystyle du(X,Y)$ $\displaystyle=\overline{g}(X,GY)+(\sigma\wedge
v)(X,Y),$ $\displaystyle\mbox{and}\;\;\;\;dv(X,Y)$
$\displaystyle=\overline{g}(X,HY)-(\sigma\wedge u)(X,Y),$

for all $X,Y\in\Gamma(T\overline{M})$, for some 1-form $\sigma$. In this case
we say that $\overline{M}$ has a complex contact metric structure
$(u,v,U,V,G,H,\overline{g})$ [1, 2]. In this case
$\sigma(X)=\overline{g}(\overline{\nabla}_{X}U,V)$, where $\overline{\nabla}$
denotes the Levi-Civita connection on $\overline{M}$. We refer to a complex
contact manifold with a complex almost contact metric structure satisfying
these conditions as a complex contact metric manifold [1, 2].

Next, for a complex contact metric structure [1, p. 237] defined local tensor
fields $h_{U}$ and $h_{V}$ by

$\displaystyle h_{U}=\frac{1}{2}\mathrm{sym}(\pounds_{U}G)\circ
p\;\;\mbox{and}\;\;\;h_{V}=\frac{1}{2}\mathrm{sym}(\pounds_{V}H)\circ p,$

where $\mathrm{sym}$ denotes the symmetric part; $h_{U}$ anticommutes with
$G$, $h_{V}$ anticommutes with $H$, and

$\displaystyle\overline{\nabla}_{X}U=-GX-Gh_{U}X+\sigma(X)V,$ (2.4) and
$\displaystyle\overline{\nabla}_{X}V=-HX-Hh_{V}X-\sigma(X)U.$ (2.5)

In view of (2.4) and (2.5) one easily sees that the integral surfaces of
$\mathcal{V}$ are totally geodesic submanifolds. Furthermore, the associated
metric $\overline{g}$ is projectable with respect to the foliation induced by
the integrable subbundle $\mathcal{V}$ if and only if $h_{U}$ and $h_{V}$
vanish (see [1] for more details). Consider the tensor fields $S$ and $T$
given by

$\displaystyle S$
$\displaystyle(X,Y)=[G,G](X,Y)+2\overline{g}(X,GY)U-2\overline{g}(X,HY)V$
$\displaystyle\;\;\;\;+2\\{v(Y)HX-v(X)HY\\}+\sigma(GY)HX-\sigma(GX)HY$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\sigma(X)GHY-\sigma(Y)GHX,$
$\displaystyle\mbox{and}\;\;\;T$
$\displaystyle(X,Y)=[H,H](X,Y)-2\overline{g}(X,GY)U+2\overline{g}(X,HY)V$
$\displaystyle\;\;\;\;+2\\{u(Y)GX-u(X)GY\\}+\sigma(HX)GY-\sigma(HY)GX$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\sigma(X)GHY-\sigma(Y)GHX,$

for all $X,Y\in\Gamma(T\overline{M})$. In the above, $[G,G]$ and $[H,H]$
denotes the Nijenhuis tensors of $G$ and $H$, respectively. Then, a complex
contact metric structure is normal [1, p. 251] if $S(X,Y)=T(X,Y)=0$, for all
$X,Y\in\Gamma(\mathcal{H})$ and $S(U,X)=T(V,X)=0$, for all
$X\in\Gamma(T\overline{M})$. An important consequence of normality is that
$h_{U}=0$, for every $U\in\Gamma(\mathcal{V})$, see [1, p. 251]. Moreover, on
a normal complex contact manifold, $\overline{\nabla}J$, $\overline{\nabla}G$
and $\overline{\nabla}H$ satisfies the relations (see [1, p. 252] for more
details).

$\displaystyle\overline{g}((\overline{\nabla}_{X}J)Y,Z)$
$\displaystyle=u(X)\\{d\sigma(Z,GY)-2\overline{g}(HY,Z)\\}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+v(X)\\{d\sigma(Z,HY)+\overline{g}(GY,Z)\\},$
(2.6) $\displaystyle\overline{g}((\overline{\nabla}_{X}G)Y,Z)$
$\displaystyle=\sigma(X)\overline{g}(HY,Z)+v(X)d\sigma(GZ,GY)$
$\displaystyle\;\;\;\;-2v(X)\overline{g}(HGY,Z)-u(Y)\overline{g}(X,Z)-v(Y)\overline{g}(JY,Z)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+u(Z)\overline{g}(X,Y)+v(Z)\overline{g}(JX,Y),$
(2.7) $\displaystyle\overline{g}((\overline{\nabla}_{X}H)Y,Z)$
$\displaystyle=-\sigma(X)\overline{g}(GY,Z)-u(X)d\sigma(HZ,HY)$
$\displaystyle\;\;\;\;-2u(X)\overline{g}(GHY,Z)-v(Y)\overline{g}(X,Z)+u(Y)\overline{g}(JY,Z)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+v(Z)\overline{g}(X,Y)-u(Z)\overline{g}(JX,Y),$
(2.8)

for all $X,Y,Z\in\Gamma(T\overline{M})$.

For a unit vector $X\in\mathcal{H}_{m}$, the plane in $T_{m}\overline{M}$
spanned by $X$ and $Y=aGX+bHX$, $a,b\in\mathbb{R}$, $a^{2}+b^{2}=1$, is called
a $GH$-plane section, and its sectional curvature, $K(X,Y)$, the
$GH$-sectional curvature of the plane section. For a given vector $X$,
$K(X,Y)$ is independent of the vector $Y$ in the plane of $GX$ and $HX$ if and
only if $K(X,GX)=K(X,HX)$ and $\overline{g}(\overline{R}(X,GX)HX,X)=0$. Here,
$\overline{R}$ denotes the curvature tensor of $\overline{M}$. Let
$\overline{M}$ be a normal complex contact metric manifold; if the
$GH$-sectional curvature is independent of the choice of $GH$-section at each
point, it is constant on the manifold, and we say that $\overline{M}$ is a
complex contact space form [1, p. 253]. Moreover, the curvature tensor
$\overline{R}$ satisfies

$\displaystyle\overline{R}(X,Y)Z$
$\displaystyle=\frac{c+3}{4}[\overline{g}(Y,Z)X-\overline{g}(X,Z)Y+\overline{g}(Z,JY)JX-\overline{g}(Z,JX)JY$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;+2\overline{g}(X,JY)JZ]$
$\displaystyle+\frac{c-1}{4}[-\\{u(Y)u(Z)+v(Y)v(Z)\\}X+\\{u(X)u(Z)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;+v(X)v(Z)\\}Y+2u\wedge
v(Z,Y)JX-2u\wedge v(Z,X)JY$ $\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;+4u\wedge
v(X,Y)JZ+\overline{g}(Z,GY)GX-\overline{g}(Z,GX)GY$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;+2\overline{g}(X,GY)GZ+\overline{g}(Z,HY)HX-\overline{g}(Z,HX)HY$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;+2\overline{g}(X,HY)HZ+\\{-u(X)\overline{g}(Y,Z)+u(Y)\overline{g}(X,Z)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;+v(X)\overline{g}(JY,Z)-v(Y)\overline{g}(JX,Z)+2v(Z)\overline{g}(X,JY)\\}U$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;+\\{-v(X)\overline{g}(Y,Z)+v(Y)\overline{g}(X,Z)-u(X)\overline{g}(JY,Z)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;+u(Y)\overline{g}(JX,Z)-2u(Z)\overline{g}(X,JY)\\}V]$
$\displaystyle-\frac{3}{4}(d\sigma(U,V)+c+1)[\\{v(X)u\wedge v(Z,Y)-v(Y)u\wedge
v(Z,X)$ $\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;+2v(Z)u\wedge
v(X,Y)\\}U-\\{u(X)u\wedge v(Z,Y)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;-u(Y)u\wedge v(Z,X)+2u(Z)v(X,Y)\\}V],$
(2.9)

for all $X,Y,Z\in\Gamma(T\overline{M})$.

Let $(\overline{M},\overline{g})$ be a $(m+2)$-dimensional semi-Riemannian
manifold with index $q$, where $0<q<(m+2)$, and consider a null hypersurface
$(M,g)$ of $\overline{M}$. Let $g$ be the induced tensor field by
$\overline{g}$ on $M$. Then, $M$ is called a null hypersurface of
$\overline{M}$ if $g$ is of constant rank $m$ and the normal bundle
$TM^{\perp}$ is a distribution of rank 1 on $M$ [3]. Here, the fibres of the
vector bundle $TM^{\perp}$ are defined as $T_{x}M^{\perp}=\\{Y_{x}\in
T_{x}\overline{M}:\overline{g}_{x}(X_{x},Y_{x})=0,\;\forall\,X_{x}\in
T_{x}M\\}$, for any $x\in M$. Let $M$ be a null hypersurface, and consider the
complementary distribution $S(TM)$ to $TM^{\perp}$ in $TM$, which is called a
screen distribution [3]. It is well-known that $S(TM)$ is non-degenerate (see
[3]). Therefore, we have the decomposition

$\displaystyle TM=S(TM)\perp TM^{\perp}.$ (2.10)

As $S(TM)$ is non-degenerate with respect to $\overline{g}$, we have
$T\overline{M}=S(TM)\perp S(TM)^{\perp}$, where $S(TM)^{\perp}$ is the
complementary vector bundle to $S(TM)$ in $T\overline{M}|_{M}$. Let $(M,g)$ be
a null hypersurface of $(\overline{M},\overline{g})$ . Then, there exists a
unique vector bundle $\mathrm{tr}(TM)$, called the null transversal bundle [3]
of $M$ with respect to $S(TM)$, of rank 1 over $M$ such that for any non-zero
section $\xi$ of $TM^{\perp}$ on a coordinate neighbourhood
$\mathcal{U}\subset M$, there exists a unique section $N$ of $\mathrm{tr}(TM)$
on $\mathcal{U}$ satisfying $\overline{g}(\xi,N)=1$,
$\overline{g}(N,N)=\overline{g}(N,Z)=0$, for all $Z\in\Gamma(S(TM))$.
Consequently, we have the following decomposition of $T\overline{M}$.

$\displaystyle
T\overline{M}|_{M}=S(TM)\perp\\{TM^{\perp}\oplus\mathrm{tr}(TM)\\}=TM\oplus
tr(TM).$

Let $\nabla$ and $\nabla^{*}$ denote the induced connections on $M$ and
$S(TM)$, respectively, and $P$ be the projection of $TM$ onto $S(TM)$, then
the local Gauss-Weingarten equations of $M$ and $S(TM)$ are the following [3].

$\displaystyle\overline{\nabla}_{X}Y$
$\displaystyle=\nabla_{X}Y+B(X,Y)N,\;\;\;\;\;\overline{\nabla}_{X}N=-A_{N}X+\tau(X)N,$
(2.11) $\displaystyle\nabla_{X}PY$
$\displaystyle=\nabla^{*}_{X}PY+C(X,PY)\xi,\;\;\;\nabla_{X}\xi=-A^{*}_{\xi}X-\tau(X)\xi,$
(2.12)

for all $X,Y\in\Gamma(TM)$, $\xi\in\Gamma(TM^{\perp})$ and
$N\in\Gamma(\mathrm{tr}(TM))$, where $\overline{\nabla}$ is the Levi-Civita
connection on $\overline{M}$. In the above setting, $B$ is the local second
fundamental form of $M$ and $C$ is the local second fundamental form on
$S(TM)$. $A_{N}$ and $A^{*}_{\xi}$ are the shape operators on $TM$ and $S(TM)$
respectively, while $\tau$ is a 1-form on $TM$. The above shape operators are
related to their local fundamental forms by $g(A^{*}_{\xi}X,Y)=B(X,Y)$,
$g(A_{N}X,PY)=C(X,PY)$, for any $X,Y\in\Gamma(TM)$. It follows easily that
$B(X,\xi)=0$, for all $X\in\Gamma(TM)$. Moreover, we have
$\overline{g}(A^{*}_{\xi}X,N)=0$, $\overline{g}(A_{N}X,N)=0$, for all
$X\in\Gamma(TM)$. Thus, we immediately notice that $A_{\xi}^{*}$ and $A_{N}$
are both screen-valued operators. Let
$\vartheta=\overline{g}(N,\boldsymbol{\cdot})$ be a 1-form metrically
equivalent to $N$ defined on $\overline{M}$. Take $\eta=i^{*}\vartheta$ to be
its restriction on $M$, where $i:M\rightarrow\overline{M}$ is the inclusion
map. Then it is easy to show that

$\displaystyle(\nabla_{X}g)(Y,Z)=B(X,Y)\eta(Z)+B(X,Z)\eta(Y),$ (2.13)

for all $X,Y,Z\in\Gamma(TM)$. Consequently, $\nabla$ is generally not a metric
connection with respect to $g$. However, the induced connection $\nabla^{*}$
on $S(TM)$ is a metric connection. Denote by $\overline{R}$ and $R$ the
curvature tensors of the connection $\overline{\nabla}$ on $\overline{M}$ and
the induced linear connections $\nabla$, respectively. Using the Gauss-
Weingarten formulae, we obtain the following Gauss-Codazzi equations for $M$
and $S(TM)$ (see details in [3, 4]).

$\displaystyle\overline{g}(\overline{R}(X,Y)Z,\xi)=$
$\displaystyle(\nabla_{X}B)(Y,Z)-(\nabla_{Y}B)(X,Z)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;+\tau(X)B(Y,Z)-\tau(Y)B(X,Z),$ (2.14)
$\displaystyle\overline{g}(\overline{R}(X,Y)PZ,N)=$
$\displaystyle(\nabla_{X}C)(Y,PZ)-(\nabla_{Y}C)(X,PZ)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-\tau(X)C(Y,PZ)+\tau(Y)C(X,PZ),$
(2.15)

for all $X,Y,Z\in\Gamma(TM)$, $\xi\in\Gamma(TM^{\perp})$ and
$N\in\Gamma(\mathrm{tr}(TM))$, where $\nabla B$ and $\nabla C$ are defined as
follows;

$\displaystyle(\nabla_{X}B)(Y,Z)$
$\displaystyle=XB(Y,Z)-B(\nabla_{X}Y,Z)-B(Y,\nabla_{X}Z),$ (2.16)
$\displaystyle(\nabla_{X}C)(Y,PZ)$
$\displaystyle=XC(Y,PZ)-C(\nabla_{X}Y,PZ)-C(Y,\nabla^{*}_{X}PZ),$ (2.17)

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

##  3\. Some basic results

Let $\overline{M}:=(\overline{M},u,v,U,V,G,H,\overline{g})$ be a
$(4n+2)$-dimensional indefinite complex contact manifold, where $\overline{g}$
is a semi-Riemannian metric of index $4q$; $0<q<n$. Next, we construct an
indefinite complex contact structure from an indefinite Sasakian 3-structure.

######  Example 3.1.

Let $(\tilde{M}^{4n+3},\phi_{i},\xi_{i},\eta_{i},\overline{g})$, for all
$i=1,2,3$, be a 3-structure manifold. Ishihara and Konishi [6] proved that if
one of the contact structures, say $(\phi_{1},\xi_{1},\eta_{1},\overline{g})$,
of a Riemnnnian manifold $\tilde{M}^{4n+3}$ with a (Sasakian) 3-structure is
regular, the base manifold $\overline{M}$ of the induced fibration is a
complex contact manifold. It is easy to see that the above result is also true
for a semi-Riemannian manifold $\tilde{M}^{4n+3}_{4q}$, in which the
Riemannian metric $\overline{g}$ is replaced with a semi-Riemannian metric of
constant index $4q$, $0<q<n$, with one of its structures regular. In this
case, the base manifold becomes an indefinite complex contact manifold. Now,
let $\tilde{M}^{4n+3}_{4q}$ be an indefinite Sasakian 3-structure manifold.
The indefinite complex contact structure on the base space
$\overline{M}_{4q}^{4n+2}$ is constructed as follows. Consider the structure
$(\phi_{1},\xi_{1},\eta_{1},\overline{g})$ as above. Let
$\pi:\tilde{M}_{4q}^{4n+3}\longrightarrow\overline{M}_{4q}^{4n+2}$ be the
Boothby-Wang fibration of $\tilde{M}^{4n+3}_{4q}$ over a cosymplectic manifold
$\overline{M}^{4n+2}_{4q}$ of integral class (see [1] for details). Denoting
the horizontal lift with respect to the principal $S^{1}$ bundle connection
1-form $\eta_{1}$ by $\tilde{\pi}$. Then, $JX=\pi_{*}\phi_{1}\tilde{\pi}X$
and, in the analogous way as in [1], the projected metric form an indefinite
Kaehler structure on $\overline{M}_{4q}^{4n+2}$. For a coordinate
neighbourhood $\mathcal{U}\subset\overline{M}$ and a local cross section $s$
of $\tilde{M}^{4n+3}_{4q}$ over $\mathcal{U}$, the 1-forms $u$ and $v$ and a
tensor field $G$ defined on $\mathcal{U}$ by $u(X)\circ\pi=\eta_{2}(s_{*}X)$,
$v(X)\circ\pi=\eta_{3}(s_{*}X)$ and
$GX=\pi_{*}[\phi_{2}s_{*}X-\eta_{1}(s_{*}X)\xi_{3}+\eta_{3}(s_{*}X)\xi_{1}]$,
define the indefinite complex contact and complex almost contact structures on
$\overline{M}_{4q}^{4n+2}$.

Let us set $J_{1}:=J$, $J_{2}:=G$ and $J_{3}:=H$. Then, we have the following
lemma.

######  Lemma 3.2.

$J_{a}$’s, for all $a=1,2,3$, satisfies

$\displaystyle J_{1}J_{2}=-J_{3},\;\;\;J_{1}J_{3}=-J_{3}J_{1}=J_{2},$ (3.1)
$\displaystyle J_{3}$ $\displaystyle J_{2}=-J_{2}J_{3}=J_{1}+u\otimes
V-v\otimes U,$ (3.2) $\displaystyle J_{2}U=J_{3}$ $\displaystyle
U=J_{3}V=0,\;\;\;\overline{g}(J_{3}X,Y)=-\overline{g}(X,J_{3}Y),$ (3.3)

for all $X,Y\in\Gamma(T\overline{M})$.

###### Proof.

Using (2.3), we have $J_{1}J_{2}=JG=-GJ=-H=-J_{3}$. On the other hand,
$J_{1}J_{3}=JH=-HJ=-J_{3}J_{1}=-HJ=-GJ^{2}=G=J_{2}$. This proves (3.1). Then,
in view of (2.1) and (2.3), we have $J_{3}J_{2}=HG=GJG=-GGJ=-GH=-J_{2}J_{3}$,
and $-J_{2}J_{3}=-GH=-G^{2}J=J-(u\circ J)\otimes U-(v\circ J)\otimes
V=J_{1}+u\otimes V-v\otimes U$, proving (3.2). Note that $J_{2}U=GU=0$, by
(2.3). Also, $J_{3}U=HU=GJU=-JGU=0$. Furthermore,
$J_{3}V=HV=GJV=-GJ^{2}U=GU=0$. Finally, for any $X,Y\in\Gamma(T\overline{M})$,
we have
$\overline{g}(J_{3}X,Y)=\overline{g}(HX,Y)=\overline{g}(GJX,Y)=-\overline{g}(JX,GY)=\overline{g}(X,JGY)=-\overline{g}(X,GJY)=-\overline{g}(X,HY)=\overline{g}(X,J_{3}Y)$,
in which we have used (2.2) and (2.3), which completes the proof. ∎

Let $(M,g)$ be a null hypersurface of $\overline{M}$. Then, for each
$\xi_{x}\in\Gamma(TM^{\perp})$ at $x\in M$, we have
$\overline{g}(\xi_{x},\xi_{x})=0$. This means that $\xi_{x}\in\Gamma(T_{x}M)$.
Since $J_{1}$ is a complex structure on $\overline{M}$, we have, from (2.2),
that $\overline{g}(\xi_{x},J_{a}\xi_{x})=0$, for all $a=1,2,3$. Hence,
$J_{a}\xi_{x}$ is tangent to $M$. Thus, $J_{a}(TM^{\perp})$ is a distribution
on $M$ of rank 3, such that $J_{a}TM^{\perp}\cap TM^{\perp}=\\{0\\}$. We can,
therefore, choose a screen distribution $S(TM)$ of $M$ such that
$J_{a}(TM^{\perp})\subset S(TM)$. As the vector fields $U$ and $V$ are space-
like, we note that none of them belongs to $TM^{\perp}$ or $\mathrm{tr}(TM)$.
Therefore, we can assume that the vertical distribution
$\mathcal{V}=\mathrm{Span}\\{U,V\\}\subset S(TM)$. Then, we have
$\overline{g}(J_{a}N,\xi)=-\overline{g}(N,J_{a}\xi)=0$ and
$\overline{g}(N,J_{a}N)=0$, for all $N\in\Gamma(\mathrm{tr}(TM))$. Hence
deduce that $J_{a}N$ is tangent to $M$ and belongs to $S(TM)$. We know that
$\xi$ and $N$ are null vector fields satisfying $\overline{g}(\xi,N)=1$. Thus,
$J_{a}\xi$ and $J_{a}N$ are also null vector fields with
$\overline{g}(J_{a}\xi,J_{a}N)=1$, for all $a=1,2,3$. Otherwise, we have
$\overline{g}(J_{a}\xi,J_{b}N)=0$, for all $a\neq b$. Hence,
$J_{a}TM^{\perp}\oplus J_{a}\mathrm{tr}(TM)$ is a vector subbundle of $S(TM)$
rank of $6$. Then, there exist a non-degenerate distribution $D_{0}$ on $M$
such that

$\displaystyle S(TM)=\\{D_{1}\oplus D_{2}\\}\perp D_{0}\perp\mathcal{V},$
(3.4)

where

$\displaystyle D_{1}$ $\displaystyle=JTM^{\perp}\perp GTM^{\perp}\perp
HTM^{\perp},$ (3.5) $\displaystyle\mbox{and}\;\;\;D_{2}$
$\displaystyle=J\mathrm{tr}(TM)\perp G\mathrm{tr}(TM)\perp H\mathrm{tr}(TM).$
(3.6)

We have the following characterisation for the distribution
$D_{0}\perp\mathcal{V}$.

######  Proposition 3.3.

$D_{0}\perp\mathcal{V}$ is invariant with respect to $J_{a}$, for all
$a=1,2,3$.

###### Proof.

Using (2.2) and Lemma 3.2, we gave
$\overline{g}(J_{a}X,Y)=-\overline{g}(X,J_{a}Y)$, $a=1,2,3$, for all
$X\in\Gamma(D_{0}\perp\mathcal{V})$ and $Y\in\Gamma(TM)$. Now, for
$Y=J_{a}\xi$, the last relation gives
$\overline{g}(J_{a}X,J_{a}\xi)=-\overline{g}(X,J_{a}^{2}\xi)=-\overline{g}(X,\xi)=0$,
and by Lemma 3.2, we have
$\overline{g}(J_{a}X,J_{b}\xi)=-\overline{g}(X,J_{a}J_{b}\xi)=\overline{g}(X,J_{c}\xi)=0$,
$a\neq b$, for any $Y=J_{a}\xi\in\Gamma(D_{1})$. Hence, $J_{a}X\perp D_{1}$.
Also, we have $\overline{g}(J_{a}X,\xi)=-\overline{g}(X,J_{a}\xi)=0$, which
shows that $J_{a}X\perp TM^{\perp}$. On the other hand, using (2.1), (2.2) and
Lemma 3.2, we have
$\overline{g}(J_{a}X,J_{a}N)=-\overline{g}(X,J_{a}^{2}N)=\overline{g}(X,N)=0$
and
$\overline{g}(J_{a}X,J_{b}N)=-\overline{g}(X,J_{a}J_{b}N)=-\overline{g}(X,J_{c}N)=0$,
for any $N\in\Gamma(\mathrm{tr}(TM))$. Hence, $J_{a}X\perp\\{\\{D_{1}\oplus
D_{2}\\}\perp TM^{\perp}\\}$. Finally, we have
$\overline{g}(J_{a}X,N)=-\overline{g}(X,J_{a}N)=0$, and hence
$J_{a}X\perp\\{\\{D_{1}\oplus
D_{2}\\}\perp\\{TM^{\perp}\otimes\mathrm{tr}(TM)\\}\\}$, that is
$J_{a}(D_{0}\perp\mathcal{V})=D_{0}\perp\mathcal{V}$, for all $a=1,2,3$, which
completes the proof. ∎

The decompositions of $TM$ and $T\overline{M}$ becomes;

$\displaystyle TM$ $\displaystyle=TM^{\perp}\perp\\{D_{1}\oplus D_{2}\\}\perp
D_{0}\perp\mathcal{V},$ (3.7) $\displaystyle\mbox{and}\;\;\;\;T\overline{M}$
$\displaystyle=\\{TM^{\perp}\oplus\mathrm{tr}(TM)\\}\perp\\{D_{1}\oplus
D_{2}\\}\perp D_{0}\perp\mathcal{V}.$ (3.8)

From the decompositions (3.7) and (3.8), we have the following result.

######  Proposition 3.4.

If $(M,g)$ is a null hypersurface of an indefinite complex contact manifold
$(\overline{M},\overline{g})$, then $\dim(M)\geq 13$ and
$\dim(\overline{M})\geq 14$.

Next, let us set

$\displaystyle
V_{a}=-J_{a}\xi\;\;\;\;\mbox{and}\;\;\;\;U_{a}=J_{a}N,\;\;\;\forall\,a=1,2,3.$
(3.9)

Let us consider the distribution $D=\\{TM^{\perp}\perp D_{1}\\}\perp D_{0}$,
and denote by $S,R$ the projection morphisms of $TM$ onto $D$ and $D_{2}$,
respectively. Then, any $X\in\Gamma(TM)$ can be written as

$\displaystyle X=SX+\sum_{a=1}^{3}u_{a}(X)U_{a}+u(X)U+v(X)V,$ (3.10)

where $u_{a}(X)$ are 1-forms on $M$ locally defined by

$\displaystyle u_{a}(X)=g(X,V_{a}),\;\;\;\;\forall\,a=1,2,3.$ (3.11)

Applying $J_{a}$ to (4.27) leads to

$\displaystyle J_{a}X=\phi_{a}X+u_{a}(X)N+u(X)J_{a}U+v(X)J_{a}V,$ (3.12)

where $\phi_{a}X:=JSX$. It follows from (4.29) that

$\displaystyle\phi_{a}U=\phi_{a}V=0,\;\;\;\forall\,a=1,2,3.$ (3.13)

On the other hand, from (4.28), (4.29), (2.1) and (2.2), we have
$u_{a}(\phi_{a}X)=g(\phi_{a}X,V_{a})=\overline{g}(J_{a}X,V_{a})-u(X)\overline{g}(J_{a}U,V_{a})-v(X)\overline{g}(J_{a}V,V_{a})=0$,
for any $a=1,2,3$. Also,
$u(\phi_{a}X)=g(\phi_{a}X,U)=\overline{g}(J_{a}X,U)-v(X)\overline{g}(J_{a}V,U)=-\overline{g}(X,J_{a}U)+v(X)\overline{g}(V,J_{a}U)=0$,
for any $a=1,2,3$. In a similar way, we have $v(\phi_{a}X)=0$. Therefore, we
have

$\displaystyle
u_{a}\circ\phi_{a}=0,\;\;\;u\circ\phi_{a}=v\circ\phi_{a}=0,\;\;\forall\,a=1,2,3.$
(3.14)

Applying $J$ to (4.29) and using (3.14), we have

$\displaystyle
J_{a}^{2}X=\phi_{a}^{2}X-u_{a}(X)U_{a}+u(X)J_{a}^{2}U+v(X)J_{a}^{2}V.$ (3.15)

Now, for $a=1$, we have $J_{1}=J$, and relation (3.15) gives
$J^{2}X=\phi_{1}^{2}X-u_{1}(X)U_{1}+u(X)J^{2}U+v(X)J^{2}V$. Since $J^{2}=-I$,
the previous gives

$\displaystyle\phi_{1}^{2}X=-X+u_{1}(X)U_{1}+u(X)U+v(X)V,\;\;u_{1}(U_{1})=1.$
(3.16)

On the other hand, when $a=2$, we have $J_{2}=G$. Since $J_{2}U=GU=0$ and
$J_{2}V=GV=-GJU=-HU=-J_{3}U=0$ (see Lemma 3.2), then, (3.15) gives
$G^{2}X=\phi_{2}^{2}X-u_{2}(X)U_{2}$. Now, applying (2.1) to this relation we
get

$\displaystyle\phi_{2}^{2}X=-X+u_{2}(X)U_{2}+u(X)U+v(X)V,\;\;u_{2}(U_{2})=1.$
(3.17)

In a similar way, we have

$\displaystyle\phi_{3}^{2}X=-X+u_{3}(X)U_{3}+u(X)U+v(X)V,\;\;u_{3}(U_{3})=1.$
(3.18)

Now, for all $a\neq b$, we have

$\displaystyle u_{a}(U_{b})$ $\displaystyle=0,$ (3.19)
$\displaystyle\mbox{and}\;\;\;\;\ \phi_{a}U_{b}$
$\displaystyle=J_{a}U_{b}-u_{a}(U_{b})N-u(U_{b})J_{a}U-v(U_{b})J_{a}V$
$\displaystyle=J_{a}U_{b}=-J_{a}J_{b}N=-J_{c}N=U_{c},$ (3.20)

in which we have used (4.28), (4.29) and Lemma 3.2. Furthermore, using (4.28),
(4.29) and Lemma 3.2, we have

$\displaystyle(u_{a}\circ\phi_{b})X=u_{a}(\phi_{b}X)=\overline{g}(J_{b}X,V_{a})=\overline{g}(X,J_{b}J_{a}\xi)=u_{c}(X).$
(3.21)

Finally, using (4.29), (3.14) and Lemma 3.2, we have

$\displaystyle(\phi_{a}\circ\phi_{b}-U_{a}\otimes u_{b})X$
$\displaystyle=J_{c}X-u_{c}(X)N-u(X)J_{c}U-v(X)J_{c}V$
$\displaystyle=\phi_{c}X,\;\;\;\;\forall\,a\neq b.$ (3.22)

Putting all the relations (4.28)–(3) together, we have the following result.

######  Proposition 3.5.

On a null hypersurface $(M,g)$, tangent to the characteristic subbundle
$\mathcal{V}$, of an indefinite complex contact manifold $\overline{M}$, the
following holds

$\displaystyle\phi_{a}^{2}=-I$ $\displaystyle+u_{a}\otimes U_{a}+u\otimes
U+v\otimes V,$ $\displaystyle u_{a}(U_{a})$
$\displaystyle=1,\;\;u(U)=1,\;\;v(V)=1,$ $\displaystyle u_{a}\circ\phi_{a}$
$\displaystyle=0,\;\;u\circ\phi_{a}=0,\;\;v\circ\phi_{a}=0,$
$\displaystyle\phi_{a}U_{a}$
$\displaystyle=0,\;\;\phi_{a}U=0,\;\;\phi_{a}V=0,$
$\displaystyle\;\;\;\;\;\;\;\mbox{and}\;\;\forall\;a\neq b;$ $\displaystyle
u_{a}(U_{b})$
$\displaystyle=0,\;\;\;\phi_{a}U_{b}=U_{c},\;\;u_{a}\circ\phi_{b}=u_{c}$
$\displaystyle\;\;\phi_{a}\circ\phi_{b}=\phi_{c}+u_{b}\otimes U_{a}.$

Consider non-zero vector-valued functions $\omega_{a}$, $a=1,2,3$, on a
neighbourhood $\mathcal{U}\subset M$. Let $V_{a}^{\prime}=\omega_{a}V_{a}$, it
follows that $U_{a}^{\prime}=(1/\omega_{a})U_{a}$,
$u_{a}^{\prime}=u_{a}\circ\omega_{a}$,
$v_{a}^{\prime}=v_{a}\circ(1/\omega_{a})$. Also, let $\omega_{a}U^{\prime}=U$,
$\omega_{a}V^{\prime}=V$. Then, $u^{\prime}=u\circ\omega_{a}$ and
$v^{\prime}=v\circ\omega_{a}$. Let us define $\phi_{a}^{\prime}$ by
$\omega_{a}\circ\phi^{\prime}_{a}=\phi_{a}\circ\omega_{a}$. Then, applying
$\phi^{\prime}$ from the right hand side of each side in this relation and
using the previous relations, we have

$\displaystyle\omega_{a}\circ{\phi_{a}^{\prime}}^{2}$
$\displaystyle=\phi_{a}\circ(\omega_{a}\circ\phi_{a}^{{}^{\prime}})=\phi_{a}^{2}\circ\omega_{a}$
$\displaystyle=(-I+u_{a}\otimes U_{a}+u\otimes U+v\otimes V)\circ\omega_{a}$
$\displaystyle=-\omega_{a}+u_{a}^{\prime}\otimes U_{a}+u^{\prime}\otimes
U+v^{\prime}\otimes V$ $\displaystyle=\omega_{a}\circ(-I+u_{a}^{\prime}\otimes
U_{a}^{\prime}+u^{\prime}\otimes U^{\prime}+v^{\prime}\otimes V^{\prime}),$

from which we get ${\phi_{a}^{\prime}}^{2}=-I+u_{a}^{\prime}\otimes
U_{a}^{\prime}+u^{\prime}\otimes U^{\prime}+v^{\prime}\otimes V^{\prime}$,
since $\omega_{a}\neq 0$, for each $a=1,2,3$. Furthermore,
$\omega_{a}\circ\phi_{a}^{\prime}U_{a}^{\prime}=\phi_{a}\circ\omega_{a}U_{a}^{\prime}=\phi_{a}U_{a}=0$,
which implies that $\phi_{a}^{\prime}U_{a}^{\prime}=0$. On the other hand,
$\omega_{a}\circ\phi_{a}^{\prime}U^{\prime}=\phi_{a}\circ\omega_{a}U^{\prime}=\phi_{a}U=0$.
This implies that $\phi_{a}^{\prime}U^{\prime}=0$. In same way, we have
$\phi_{a}^{\prime}V^{\prime}=0$. Note that
$u_{a}^{\prime}\circ\phi_{a}^{\prime}=u^{\prime}\circ\phi_{a}^{\prime}=v^{\prime}\circ\phi_{a}^{\prime}=0$.
From these calculations, we have the following corollary.

######  Corollary 3.6.

The induced structure $(\phi_{a},U_{a},u_{a},U,u,V,v)$, for each $a=1,2,3$, on
$(M,g)$ is not unique.

Now, let $\lambda_{a}$ be an eigenvalue of $\phi_{a}$, with respect to
eigenvector $\zeta_{a}$, for all $a=1,2,3$. Then,
$\phi_{a}\zeta_{a}=\lambda_{a}\zeta_{a}$. Applying $\phi_{a}$ to the previous
relation and then using Proposition 3.5, we get
$\lambda_{a}^{2}\zeta_{a}=\phi_{a}^{2}\zeta_{a}=-\zeta_{a}+u_{a}(\zeta_{a})U_{a}+u(\zeta_{a})U+v(\zeta_{a})V$.
Applying $\phi_{a}$ to this relation gives $(\lambda_{a}^{2}+1)\lambda_{a}=0$.
Therefore, we have the following corollary for each $\phi_{a}$.

######  Corollary 3.7.

The eigenvalues of $\phi_{a}$, for all $a=1,2,3$, are $0$, $-i$ and $i$.

Furthermore, in view of Proposition 3.5 and the definition of an almost
contact $3$-structure (see [8, p. 325]), we have the following result.

######  Corollary 3.8.

Under the assumptions of Proposition 3.5, with $\xi$ and $N$ globally defined
on $M$, $({\phi_{a}}_{|\mathcal{V}^{\perp}},U_{a},u_{a})$ defines an almost
contact $3$-structure on the complement $\mathcal{V}^{\perp}$ of the vertical
distribution in $TM$.

######  Lemma 3.9.

Let $(M,g)$ be a null hypersurface of an indefinite complex contact manifold
$\overline{M}$. Then, we have

$\displaystyle B(X,U)$ $\displaystyle=-g(X,V_{2})-\overline{g}(h_{U}X,V_{2}),$
(3.23) $\displaystyle C(X,U)$
$\displaystyle=-g(X,U_{2})-\overline{g}(h_{U}X,U_{2}),$ (3.24) $\displaystyle
B(X,V)$ $\displaystyle=-g(X,V_{3})-\overline{g}(h_{V}X,V_{3}),$ (3.25)
$\displaystyle C(X,V)$ $\displaystyle=-g(X,U_{3})-\overline{g}(h_{V}X,U_{3}),$
(3.26) $\displaystyle\mbox{and}\;\;\;\sigma(X)$
$\displaystyle=g(\nabla_{X}U,V),\;\;\;\forall\,X\in\Gamma(TM).$ (3.27)

###### Proof.

In view of (2.4) and first relation of (2.11), we have

$\displaystyle\nabla_{X}U+B(X,U)N=-GX-Gh_{U}X+\sigma(X)V,$ (3.28)

for any $X\in\Gamma(TM)$. The inner product of (3.28) with $\xi$ and $N$ in
turns gives

$\displaystyle B(X,U)$
$\displaystyle=-\overline{g}(GX,\xi)-\overline{g}(Gh_{U}X,\xi)$ (3.29)
$\displaystyle\mbox{and}\;\;\;\overline{g}(\nabla_{X}U,N)$
$\displaystyle=-\overline{g}(GX,N)-\overline{g}(Gh_{U}X,N).$ (3.30)

Then, applying (2.2), we get
$B(X,U)=\overline{g}(X,G\xi)+\overline{g}(h_{U}X,G\xi)$, which implies (3.23).
On the other hand, (2.2), (2.12) and (3.30) gives
$C(X,U)=\overline{g}(X,GN)+\overline{g}(h_{U}X,GN)$, which proves (3.24).
Relations (3.25) and (3.26) follows easily as in (3.23) and (3.24), while
considering (2.5), (2.11), (2.12) and (2.2). Finally, (3.27) follows from
(3.28), (2.2) and (2.3), which completes the proof. ∎

A null hypersurface $(M,g)$ of a semi-Riemannian manifold
$(\overline{M},\overline{g})$ is called; totally umbilic [3, p. 106] if and
only if, on each coordinate neighbourhood $\mathcal{U}$ of $M$ there exist a
smooth function $\rho$ such that $A_{\xi}^{*}X=\rho PX$, or equivalently,
$B(X,PY)=\rho g(X,Y)$, for all $X,Y\in\Gamma(TM)$. In case $\rho=0$, we say
that $M$ is totally geodesic otherwise it is proper totally umbilic. In the
same line, $M$ is screen totally umbilic [3, p. 109] if and only if, on each
coordinate neighbourhood $\mathcal{U}$ of $M$ there exist a smooth function
$\varrho$ such that $A_{N}X=\varrho PX$, or equivalently, $C(X,PY)=\varrho
g(X,Y)$, for all $X,Y\in\Gamma(TM)$. In case $\varrho=0$, we say that $M$ is
screen totally geodesic otherwise it is proper screen totally umbilic.
Furthermore, $M$ is screen locally conformal [4, p. 179] if and only if, on
any coordinate neighbourhood $\mathcal{U}$ there exist a non-vanishing smooth
function $\psi$ such that $A_{N}X=\psi A_{\xi}^{*}X$, or equivalently,
$C(X,PY)=\psi B(X,Y)$, for any $X,Y\in\Gamma(TM)$. The conformality is said to
be global if $\mathcal{U}=M$. In the sequel, by screen conformal we shall mean
screen locally conformal.

Next, in view of Lemma 3.9, we have the following characterization result.

######  Theorem 3.10.

A normal indefinite complex contact manifold $\overline{M}$ does not admit any
totally umbilic, screen totally umbilic or screen conformal null hypersurface
$(M,g)$, tangent to the characteristic subbundle $\mathcal{V}$.

###### Proof.

As $\overline{M}$ is normal, we have $h_{U}=0$, for any
$U\in\Gamma(\mathcal{V})$. Now, assume that $M$ is totally umbilic, then
(3.23) and (3.25) implies $\rho g(X,U)=-g(X,V_{2})$ and $\rho
g(X,V)=-g(X,V_{3})$, for all $X\in\Gamma(TM)$ and $U,V\in\Gamma(\mathcal{V})$.
Setting $X=U_{2}$ and $X=U_{3}$ in the first and second of the previous
relations and noting that $U_{2},U_{3}\perp\mathcal{V}$ (see decomposition
3.4), we, respectively, get $-g(U_{2},V_{2})=0$ and $-g(U_{3},V_{3})=0$, which
are both contradictions. Therefore, $M$ is never totally umbilic. On the other
hand, if $M$ is screen totally geodesic, (3.24) and (3.26) leads to $\varrho
g(X,U)=-g(X,U_{2})$ and $\varrho g(X,V)=-g(X,U_{3})$, for all
$X\in\Gamma(TM)$. Letting $X=V_{2}$ and $X=V_{3}$ in the first and second
relations gives $-g(V_{2},U_{2}=0$ and $-g(V_{3},U_{3})=0$, which are both
contradictions. Hence, $M$ is never totally umbilic in $\overline{M}$.
Finally, assume that $M$ is screen conformal, then (3.4) and (3.24) leads to
$\psi g(X,V_{2})=g(X,U_{2})$, while (3.25) and (3.26) gives $\psi
g(X,V_{3})=g(X,U_{3})$, for all $X\in\Gamma(TM)$. Setting $X=V_{2}$ in the
first one and $X=V_{3}$ in the second, while noting that $V_{2}$ and $V_{3}$
are both null vector fields, gives $g(V_{2},U_{2})=0$ and $g(V_{3},U_{3})=0$.
These are contradictions, and hence $M$ is never screen conformal in
$\overline{M}$, which completes the proof. ∎

According to [3, p. 89], $S(TM)$ is parallel with respect to $\nabla$ if
$\nabla_{X}PY\in\Gamma(S(TM))$, for all $X,Y\in\Gamma(TM)$. It then follows
from (2.12) that $C(X,PY)=0$, for all $X,Y\in\Gamma(TM)$. That is, $S(TM)$ is
totally geodesic. Furthermore, we see from (2.13) that the induced connection
is a metric connection if and only if $B=0$, i.e., $M$ is totally geodesic. In
fact, assume that $\nabla$ is a metric connection, then (2.13) implies that
$B(X,Z)\eta(Y)+B(X,Y)\eta(Z)=0$, for all $X,Y,Z\in\Gamma(TM)$, since $\nabla
g=0$. Hence, setting $Z=\xi$ in this relation and using the fact that
$B(X,\xi)=0$, for all $X\in\Gamma(TM)$ and $\xi\in\Gamma(TM^{\perp})$, we get
$B(X,Y)=0$, for all $X,Y\in\Gamma(TM)$. The converse is obvious. The normal
distribution $TM^{\perp}$ is said to be killing if $\pounds_{\xi}g=0$, for all
$\xi\in\Gamma(TM^{\perp})$. Here, $\pounds_{\xi}$ denotes the usual Lie
derivative with respect to $\xi$. By a simple calculation, while using (2.13)
and the second relation of (2.12), this is equivalent to $2B=0$, i.e. $M$
totally geodesic.

Putting all the above details to Theorem 3.10, we have the following.

######  Corollary 3.11.

In view of Theorem 3.10, we see that

  1. (1)

$S(TM)$ is never parallel,

  2. (2)

$\nabla$ is never a metric connection,

  3. (3)

$TM^{\perp}$ is never a killing distribution,

on any null hypersurface of a normal indefinite complex contact manifold
$\overline{M}$.

######  Lemma 3.12.

For a null hypersurface $(M,g)$ of a normal indefinite complex contact
manifold $\overline{M}$, the following holds

$\displaystyle C(X,V_{1})=B(X,$ $\displaystyle
U_{1})-u(X)d\sigma(\xi,U_{2})-v(X)d\sigma(\xi,U_{3}),$ (3.31) $\displaystyle
C(X,V_{2})$ $\displaystyle=B(X,U_{2})+v(X)d\sigma(V_{2},U_{2}),$ (3.32)
$\displaystyle\mbox{and}\;\;\;\;\;C(X,$ $\displaystyle
V_{3})=B(X,U_{3})-u(X)d\sigma(V_{3},U_{3}),$ (3.33)

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

###### Proof.

Setting $Y=N$ and $Z=\xi$ in (2.6) and applying Lemma 3.2, we get

$\displaystyle\overline{g}((\overline{\nabla}_{X}J)N,\xi)$
$\displaystyle=u(X)d\sigma(\xi,GN)+v(X)d\sigma(\xi,HN)$
$\displaystyle=-u(X)d\sigma(\xi,U_{2})-v(X)d\sigma(\xi,U_{3}),$ (3.34)

for all $X\in\Gamma(T\overline{M})$. With the help of (3.9), the left hand
side of (3) gives

$\displaystyle\overline{g}((\overline{\nabla}_{X}J)N,\xi)$
$\displaystyle=\overline{g}(\overline{\nabla}_{X}JN,\xi)-\overline{g}(J\overline{\nabla}_{X}N,\xi)$
$\displaystyle=-\overline{g}(\overline{\nabla}_{X}U_{1},\xi)-\overline{g}(\overline{\nabla}_{X}N,V_{1}).$
(3.35)

Now, for all $X\in\Gamma(TM)$, (3.35) simplifies to

$\displaystyle\overline{g}((\overline{\nabla}_{X}J)N,\xi)=-B(X,U_{1})+C(X,V_{1}).$
(3.36)

Thus, (3.31) follows from (3.36) and (3). On the other hand, setting $Y=N$ and
$Z=\xi$ in (2.7) and using Lemma 3.2, we get

$\displaystyle\overline{g}((\overline{\nabla}_{X}G)N,\xi)=v(X)d\sigma(G\xi,GN)=v(X)d\sigma(V_{2},U_{2}),$
(3.37)

for all $X\in\Gamma(T\overline{M})$. Simplifying the left hand of (3.37), we
see that

$\displaystyle\overline{g}((\overline{\nabla}_{X}G)N,\xi)$
$\displaystyle=\overline{g}(\overline{\nabla}_{X}GN,\xi)-\overline{g}(G\overline{\nabla}_{X}N,\xi)$
$\displaystyle=-\overline{g}(\overline{\nabla}_{X}U_{2},\xi)-\overline{g}(\overline{\nabla}_{X}N,V_{2}),$
(3.38)

for $X\in\Gamma(T\overline{M})$, in which we have used (2.2). For all
$X\in\Gamma(TM)$, (3.38), (2.11) and (2.12) gives

$\displaystyle\overline{g}((\overline{\nabla}_{X}G)N,\xi)=-B(X,U_{2})+C(X,V_{2}).$
(3.39)

Then, (3.32) follows immediately from (3.37) and (3.39). Lastly, relation
(3.33) follows by similar calculations, while using (2.8), (2.11) and (2.12).
∎

With the aid of Lemma 3.12, we have the following result.

######  Theorem 3.13.

The transversal bundle $\mathrm{tr}(TM)$ of a null hypersurface $(M,g)$,
tangent to the characteristic subbundle $\mathcal{V}$, of a normal indefinite
complex contact manifold $\overline{M}$ is never a killing distribution.

###### Proof.

Suppose that $\mathrm{tr}(TM)$ is a killing distribution. Then,
$\pounds_{N}\overline{g}=0$, for all $N\in\Gamma(\mathrm{tr}(TM))$. This
implies that

$\displaystyle\overline{g}(\overline{\nabla}_{X}N,Y)+\overline{g}(X,\overline{\nabla}_{Y}N)=0,$
(3.40)

for all $X,Y\in\Gamma(T\overline{M})$. For all $X,Y\in\Gamma(TM)$, (3.40) and
(2.11) gives

$\displaystyle C(X,PY)+C(Y,PX)=\tau(X)\eta(Y)+\tau(Y)\eta(X).$ (3.41)

Setting $Y=U$ in (3.41) and using (3.24), we get
$C(U,PX)=g(X,U_{2})+\tau(U)\eta(X)$. And, for $X=V_{2}$ in the previous
relation we have $C(U,V_{2})=1$. On the other hand, (3.32) and (3.23) implies
that $C(U,V_{2})=B(U,U_{2})=B(U_{2},U)=-g(U_{2},V_{2})=-1$, which contradicts
the previous relation. Hence, the transversal bundle is never a killing a
distribution. ∎

##  4\. Main results

We have seen (see Theorem 3.10) that an indefinite complex contact manifold
does not admit any totally umbilic, screen totally umbilic or screen conformal
null hypersurface, tangent to the characteristic subbundle $\mathcal{V}$. It
is easy to see that the well-known definitions of the above mentioned null
hypersurfaces fails in portions of $TM$ which includes the vector fields $V,U$
spanning $\mathcal{V}$. However, these definitions can be confined to
$\mathcal{V}^{\perp}:=TM^{\perp}\perp\\{D_{1}\oplus D_{2}\\}\perp D_{0}$,
which is the complementary distribution of $\mathcal{V}$ in $TM$ (see
decomposition (3.7) of $TM$). Such considerations give rise to totally contact
umbilic, totally contact screen umbilic and contact screen conformal null
hypersurfaces (see [5]) in case the ambient manifold is an indefinite Sasakian
manifold. In the same way, if we let $\tilde{P}$ be the projection morphism of
$TM$ onto $TM^{\perp}\perp\\{D_{1}\oplus D_{2}\\}\perp D_{0}$, then each
$X\in\Gamma(TM)$ can be written as

$\displaystyle X=\tilde{P}X+u(X)U+v(X)V,$ (4.1)

where $u(X)=g(U,X)$ and $v(X)=g(V,X)$. Hence, we have the following
definition.

######  Definition 4.1.

Let $(M,g)$ be a null hypersurface, tangent to the characteristic subbundle
$\mathcal{V}$, of an indefinite complex contact manifold $\overline{M}$. Then,

  1. (1)

$M$ is totally contact umbilic if and only if on each coordinate neighbourhood
$\mathcal{U}|_{\mathcal{V}^{\perp}}$ of $M$, there exists a smooth function
$\beta$ such that $B=\beta\otimes g$ on $\mathcal{V}^{\perp}$ or equivalently

$\displaystyle B(\tilde{P}X,\tilde{P}Y)=\beta
g(\tilde{P}X,\tilde{P}Y),\;\;\;\forall\,X,Y\in\Gamma(TM).$ (4.2)

  2. (2)

$M$ is totally contact screen umbilic if and only if on each coordinate
neighbourhood $\mathcal{U}|_{\mathcal{V}^{\perp}}$ of $M$, there exists a
smooth function $\mu$ such that $B=\beta\otimes g$ on $\mathcal{V}^{\perp}$ or
equivalently

$\displaystyle C(\tilde{P}X,P\tilde{P}Y)=\mu
g(\tilde{P}X,\tilde{P}Y),\;\;\;\forall\,X,Y\in\Gamma(TM).$ (4.3)

In case $\mu=0$, then $M$ is called totally contact screen geodesic.

  3. (3)

$M$ is contact screen conformal if and only if on each coordinate
neighbourhood $\mathcal{U}|_{\mathcal{V}^{\perp}}$ of $M$, there exists a
nonzero smooth function $\varphi$ such that $C=\varphi\otimes B$ on
$\mathcal{V}^{\perp}$ or equivalently

$\displaystyle C(\tilde{P}X,P\tilde{P}Y)=\varphi
B(\tilde{P}X,\tilde{P}Y),\;\;\;\forall\,X,Y\in\Gamma(TM),$ (4.4)

and $M$ is contact screen homothetic if $\varphi$ is a constant function.

By direct calculations using (2.6), (2.7), (2.8), (2.11) and (2.12), we have

$\displaystyle g((\nabla_{X}\phi_{1})Y,V_{1})$
$\displaystyle=-B(X,Y)+u_{1}(Y)C(X,V_{1}),$ (4.5) $\displaystyle
g((\nabla_{X}\phi_{2})Y,V_{2})$
$\displaystyle=-B(X,Y)+u_{2}(Y)C(X,V_{2})+\sigma(X)u_{1}(Y),$ (4.6)
$\displaystyle g((\nabla_{X}\phi_{3})Y,V_{3})$
$\displaystyle=-B(X,Y)+u_{3}(Y)C(X,V_{3})+\sigma(X)u_{1}(Y),$ (4.7)

for all $X,Y\in\Gamma(\mathcal{V}^{\perp})$. Moreover, $u_{1}$, $u_{2}$ and
$u_{3}$ satisfies the following relations

$\displaystyle(\nabla_{X}u_{1})(Y)$
$\displaystyle=-B(X,\phi_{1}Y)-u_{1}(Y)\tau(X),$ (4.8)
$\displaystyle(\nabla_{X}u_{2})(Y)$
$\displaystyle=-B(X,\phi_{2}Y)-u_{2}(Y)\tau(X)+\sigma(X)u_{3}(Y),$ (4.9)
$\displaystyle(\nabla_{X}u_{3})(Y)$
$\displaystyle=-B(X,\phi_{3}Y)-u_{3}(Y)\tau(X)-\sigma(X)u_{2}(Y),$ (4.10)

for all $X,Y\in\Gamma(\mathcal{V}^{\perp})$. It then follows from (4.5), (4.6)
and (4.7) that $B(X,U_{a})=C(X,V_{a})$ and
$g((\nabla_{X}\phi_{a})Y,V_{a})=-B(X,Y)$, for all
$X\in\Gamma(\mathcal{V}^{\perp})$ and $Y\in\Gamma(D)$. Therefore, we have the
following result.

######  Theorem 4.2.

Let $\overline{M}$ be a normal indefinite complex contact manifold, and
$(M,g)$ a null hypersurface of $\overline{M}$, tangent to the characteristic
subbundle $\mathcal{V}$. Then,

  1. (1)

$M$ is totally contact geodesic if and only if $C(X,V_{a})=0$ and
$(\nabla_{X}\phi_{a})Y=0$, for all $X\in\Gamma(\mathcal{V}^{\perp})$ and
$Y\in\Gamma(D)$.

  2. (2)

$M$ is totally contact screen geodesic if and only if $B(X,U_{a})=0$ and
$\nabla_{X}PY\notin\Gamma(TM^{\perp})$, for all
$X\in\Gamma(\mathcal{V}^{\perp})$ and $Y\in\Gamma(D_{2}\perp D_{0})$.

In view of (2) and (2.14), we have

$\displaystyle\;\frac{c+3}{4}[\overline{g}(Z,JY)u_{1}(X)-\overline{g}(Z,JX)u_{1}(Y)+2\overline{g}(X,JY)u_{1}(Z)]$
$\displaystyle+\frac{c-1}{4}[\overline{g}(Z,GY)u_{2}(X)-\overline{g}(Z,GX)u_{2}(Y)+2\overline{g}(X,GY)u_{2}(Z)$
$\displaystyle+\overline{g}(Z,HY)u_{3}(X)-\overline{g}(Z,HX)u_{3}(Y)+2\overline{g}(X,HY)u_{3}(Z)]$
$\displaystyle=(\nabla_{X}B)(Y,Z)-(\nabla_{Y}B)(X,Z)+\tau(X)B(Y,Z)-\tau(Y)B(X,Z),$
(4.11)

for all $X,Y,Z\in\Gamma(\mathcal{V}^{\perp})$. On the other hand, using (2)
and (2.15), we get

$\displaystyle\frac{c+3}{4}$
$\displaystyle[\overline{g}(Y,Z)\eta(X)-\overline{g}(X,Z)\eta(Y)+\overline{g}(Z,JY)v_{1}(X)$
$\displaystyle-\overline{g}(Z,JX)v_{1}(Y)+2\overline{g}(X,JY)v_{1}(Z)]+\frac{c-1}{4}[\overline{g}(Z,GY)v_{2}(X)$
$\displaystyle-\overline{g}(Z,GX)v_{2}(Y)+2\overline{g}(X,GY)v_{2}(Z)+\overline{g}(Z,HY)v_{3}(X)$
$\displaystyle-\overline{g}(Z,HX)v_{3}(Y)+2\overline{g}(X,HY)v_{3}(Z)]=(\nabla_{X}C)(Y,PZ)$
$\displaystyle-(\nabla_{Y}C)(X,PZ)-\tau(X)C(Y,PZ)+\tau(Y)C(X,PZ),$ (4.12)

for all $X,Y,Z\in\Gamma(\mathcal{V}^{\perp})$.

######  Lemma 4.3.

Let $(M,g)$ be a null hypersurface, tangent to the characteristic subbundle
$\mathcal{V}$, of a normal indefinite complex contact manifold $\overline{M}$.
Then, if

  1. (1)

$M$ is totally contact umbilic, we have

$\displaystyle(\nabla_{X}B)(Y,Z)-(\nabla_{Y}B)(X,Z)=(X\beta)g(Y,Z)-(Y\beta)g(X,Z)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;+\beta^{2}[g(X,Z)\eta(Y)-g(Y,Z)\eta(X)]+u_{2}(Z)\overline{g}(GX,Y)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;-u_{2}(Z)\overline{g}(GY,X)+u_{3}(Z)\overline{g}(HX,Y)-u_{3}(Z)\overline{g}(HY,X)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;+u_{2}(Y)\overline{g}(GX,Z)-u_{2}(X)\overline{g}(GY,Z)+u_{3}(Y)\overline{g}(HX,Z)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;-u_{3}(X)\overline{g}(HY,Z),$ (4.13)

  2. (2)

$M$ is totally contact screen umbilic, we have

$\displaystyle(\nabla_{X}C)(Y,PZ)-(\nabla_{Y}C)(X,PZ)=(X\mu)g(Y,PZ)-(Y\mu)g(X,PZ)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;+\mu[B(X,Z)\eta(Y)-B(Y,Z)\eta(X)]-C(U,PZ)[\overline{g}(GX,Y)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;-\overline{g}(GY,X)]-C(V,PZ)[\overline{g}(HX,Y)-\overline{g}(HY,X)]+v_{2}(Y)\overline{g}(GX,Z)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;-v_{2}(X)\overline{g}(GY,Z)+v_{3}(Y)\overline{g}(HX,Z)-v_{3}(X)\overline{g}(HY,Z),$
(4.14)

for all $X,Y,Z\in\Gamma(\mathcal{V}^{\perp})$.

###### Proof.

Using (2.16), (4.1), (4.2) and Lemma 3.9, we derive

$\displaystyle(\nabla_{X}B)(Y,Z)$
$\displaystyle=X(B(Y,Z))-B(\tilde{P}\nabla_{X}Y,Z)-B(Y,\tilde{P}\nabla_{X}Z)$
$\displaystyle\;\;\;\;+u(\nabla_{X}Y)u_{2}(Z)+v(\nabla_{X}Y)u_{3}(Z)+u(\nabla_{X}Z)u_{2}(Y)$
$\displaystyle\;\;\;\;+v(\nabla_{X}Z)u_{3}(Y)$
$\displaystyle=(X\beta)g(Y,Z)+\beta[Xg(Y,Z)-g(\nabla_{X}Y,Z)-g(Y,\nabla_{X}Z)]$
$\displaystyle\;\;\;\;+u(\nabla_{X}Y)u_{2}(Z)+v(\nabla_{X}Y)u_{3}(Z)+u(\nabla_{X}Z)u_{2}(Y)$
$\displaystyle\;\;\;\;+v(\nabla_{X}Z)u_{3}(Y),$ (4.15)

for all $X,Y,Z\in\Gamma(\mathcal{V}^{\perp})$. On the other hand, using
(2.13), (2.4) and (2.11), we derive

$\displaystyle
u(\nabla_{X}Y)=g(\nabla_{X}Y,U)=-(\nabla_{X}g)(Y,U)-g(Y,\nabla_{X}U)=\overline{g}(GX,Y).$
(4.16)

In the same way, we have

$\displaystyle
v(\nabla_{X}Y)=g(\nabla_{X}Y,V)=-(\nabla_{X}g)(Y,V)-g(Y,\nabla_{X}V)=\overline{g}(HX,Y).$
(4.17)

In view of (4), (4.16), (4.17), (2.13) and (4.2), we have

$\displaystyle(\nabla_{X}B)(Y,Z)$
$\displaystyle=(X\beta)g(Y,Z)+\beta^{2}[g(X,Z)\eta(Y)+g(X,Y)\eta(Z)]$
$\displaystyle\;\;\;\;\;+u_{2}(Z)\overline{g}(GX,Y)+u_{3}(Z)\overline{g}(HX,Y)+u_{2}(Y)\overline{g}(GX,Z)$
$\displaystyle\;\;\;\;\;+u_{3}(Y)\overline{g}(HX,Z),$ (4.18)

for all $X,Y,Z\in\Gamma(\mathcal{V}^{\perp})$. Then, relation (1) follows from
(4). Furthermore, using (2.17), (4.1), (4.3), (2.4), (2.5) and the relation
$X=PX+\eta(X)\xi$, for all $X\in\Gamma(TM)$, we derive

$\displaystyle(\nabla_{X}C)(Y,PZ)$
$\displaystyle=(X\mu)g(Y,PZ)+\mu[(\nabla_{X}g)(Y,Z)-g(\nabla_{X}\xi,Y)\eta(Z)]$
$\displaystyle\;\;\;\;-C(U,PZ)\overline{g}(GX,Y)-C(V,PZ)\overline{g}(HX,Y)$
$\displaystyle\;\;\;\;+v_{2}(Y)\overline{g}(GX,Z)+V_{3}(Y)\overline{g}(HX,PZ),$
(4.19)

for all $X,Y,Z\in\Gamma(\mathcal{V}^{\perp})$. Then, applying (2.12) and
(2.13) to (4), we get

$\displaystyle(\nabla_{X}C)(Y,PZ)$
$\displaystyle=(X\mu)g(Y,PZ)+\mu[B(Y,Z)\eta(X)+2B(X,Y)\eta(Z)]$
$\displaystyle\;\;\;\;-C(U,PZ)\overline{g}(GX,Y)-C(V,PZ)\overline{g}(HX,Y)$
$\displaystyle\;\;\;\;+v_{2}(Y)\overline{g}(GX,Z)+V_{3}(Y)\overline{g}(HX,PZ),$
(4.20)

for all $X,Y,Z\in\Gamma(\mathcal{V}^{\perp})$. Finally, (2) follows from (4),
which completes the proof. ∎

######  Theorem 4.4.

Let $\overline{M}$ be a normal indefinite complex contact manifold, and
$(M,g)$ a totally contact umbilic or totally contact screen umbilic null
hypersurface of $\overline{M}$, tangent to the characteristic subbundle
$\mathcal{V}$. Then, $c=-3$, that is; $\overline{M}$ is a space of constant
$GH$-sectional curvature $-3$.

###### Proof.

Assume that $M$ is totally contact umbilic. Then, letting $X=\xi$ in (1) of
Lemma 4.3, we get

$\displaystyle(\nabla_{\xi}B)(Y,Z)-(\nabla_{Y}B)(\xi,Z)$
$\displaystyle=[\xi\beta-\beta]g(Y,Z)$
$\displaystyle-3u_{2}(Y)u_{2}(Z)-3u_{3}(Y)u_{3}(Z),$ (4.21)

for all $X,Y,Z\in\Gamma(\mathcal{V}^{\perp})$. Then letting $X=\xi$ in (4) and
then using (4), we get

$\displaystyle[\xi\beta+\beta\tau(\xi)-\beta^{2}]g(Y,Z)=\frac{3}{4}(c+3)\sum_{a=1}^{3}u_{a}(Y)u_{a}(Z),$
(4.22)

for all $X,Y,Z\in\Gamma(\mathcal{V}^{\perp})$. Letting $Y=Z=U_{a}$, $a=1,2,3$,
we get $c=-3$. Moreover, $\xi\beta+\beta\tau(\xi)-\beta^{2}=0$. On the other
hand, if $M$ is totally contact screen umbilic, we let $X=\xi$ in (2) and get

$\displaystyle(\nabla_{\xi}C)(Y,PZ)$
$\displaystyle-(\nabla_{Y}C)(\xi,PZ)=(\xi\mu)g(Y,PZ)-\mu B(Y,Z)$
$\displaystyle\;\;+2u_{2}(Y)C(U,PZ)+2u_{3}(Y)C(V,PZ)$
$\displaystyle\;\;\;\;-v_{2}(Y)u_{2}(Z)-v_{3}(Y)u_{3}(PZ),$ (4.23)

for all $X,Y,Z\in\Gamma(\mathcal{V}^{\perp})$. Using (4) in (4), with $X=\xi$,
we get

$\displaystyle[\xi\mu-\mu\tau(\xi)]g(Y,PZ)-\mu
B(Y,Z)=-2u_{2}(Y)C(U,PZ)-2u_{3}(Y)C(V,PZ)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;+\frac{c+3}{4}[g(Y,Z)-u_{1}(Z)v_{1}(Y)+2u_{1}(Y)v_{1}(Z)+v_{2}(Y)u_{2}(Z)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+v_{3}(Y)u_{3}(Z)]+\frac{c-1}{2}[u_{2}(Y)v_{2}(Z)+u_{3}(Y)v_{3}(Z)],$
(4.24)

for all $X,Y,Z\in\Gamma(\mathcal{V}^{\perp})$. Letting $Y=V_{a}$ and
$Z=U_{a}$, $a=1,2,3$, in (4) and noting, from Lemma 3.12 and (4.3), that
$B(V_{a},U_{a})=C(V_{a},V_{a})=0$, we get

$\displaystyle\xi\mu-\mu\tau(\xi)=\frac{1}{2}(c+3).$ (4.25)

On the other hand, using Lemmas 3.9 and 3.12, we get
$C(U,V_{2})=B(U,U_{2})=B(U_{2},U)=-1$ and
$C(V,V_{3})=B(V,U_{3})=B(U_{3},V)=-1$. Thus, letting $Y=U_{a}$ and $Z=V_{a}$,
$a=1,2,3$, in (4), we get

$\displaystyle\xi\mu-\mu\tau(\xi)=\frac{3}{4}(c+3).$ (4.26)

Therefore, from (4.25) and (4.26), we get $c=-3$ and $\xi\mu-\mu\tau(\xi)=0$,
which completes the proof. ∎

From Theorem 4.4, we have the following.

######  Corollary 4.5.

A normal indefinite complex contact manifold $\overline{M}$ with $c\neq-3$
does not admit any totally contact umbilic or totally contact screen umbilic
null hypersurface $(M,g)$, tangent to the characteristic subbundle
$\mathcal{V}$.

In view of Lemma 3.12 and Theorem 4.4, we also have the following.

######  Corollary 4.6.

Any null hypersurface of $\overline{M}$ which is both totally umbilic and
contact screen umbilic is totally contact geodesic and contact screen
geodesic, that is $\beta=\mu=0$.

Unlike $B$, the local second fundamental form $C$ is not, generally, symmetric
on $S(TM)$. In fact, by a direct calculation, we have
$C(X,Y)-C(Y,X)=\eta([X,Y])$, for all $X,Y\in\Gamma(S(TM))$. It the follows
from the above relation that $C$ is symmetric on $S(TM)$ if and only if
$S(TM)$ is integrable. Now, assume that $S(TM)$ is integrable, then

$\displaystyle C(U,PZ)=C(PZ,U)=-v_{2}(PZ),$ (4.27) and $\displaystyle
C(V,PZ)=C(PZ,V)=-v_{3}(PZ),$ (4.28)

for any $Z\in\Gamma(TM)$, in which we have used Lemma 3.9. On the other hand,
for a totally contact screen umbilic null hypersurface $M$, we have seen that
$c=-3$ and $\xi\mu-\mu\tau(\xi)=0$. Considering these relations in (4),
together with (4.27) and (4.28), we get

$\displaystyle-\mu
B(Y,PZ)=0,\;\;\;\forall\,Y,Z\in\Gamma(\mathcal{V}^{\perp}).$ (4.29)

Setting $Y=Z=U_{2}$ in (4.29), and then apply Lemma 3.12 and (4.3), we get
$0=-\mu B(U_{2},U_{2})=-\mu C(U_{2},V_{2})=-\mu^{2}$. Hence, from Theorem 4.4,
we have the following corollary.

######  Corollary 4.7.

Under the assumptions of Theorem 4.4, any screen integrable totally contact
screen umbilic nullhypersurface of $\overline{M}$ is totally contact screen
geodesic, that is $\mu=0$.

By direct calculations, while using (2.16), (2.17),(2.4), (2.5), (2.13),
(4.1), (4.4) and Lemma 3.9, we note that on a contact screen conformal null
hypersurface, tangent to $\mathcal{V}$, of a normal indefinite complex contact
manifold $\overline{M}$, we have

$\displaystyle(\nabla_{X}C)(Y,PZ)-(\nabla_{Y}C)(X,PZ)-\varphi[(\nabla_{X}B)(Y,PZ)-(\nabla_{Y}B)(X,PZ)]$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;=(X\varphi)B(Y,PZ)-(Y\varphi)B(X,PZ)+\varphi[u_{2}(X)\overline{g}(GY,PZ)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;-u_{2}(Y)\overline{g}(GX,PZ)+u_{3}(X)\overline{g}(HY,PZ)-u_{3}(Y)\overline{g}(HX,PZ)]$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;+C(U,PZ)[\overline{g}(GY,X)-\overline{g}(GX,Y)]+C(V,PZ)[\overline{g}(HY,X)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;-\overline{g}(HX,Y)]+v_{2}(Y)\overline{g}(GX,PZ)-v_{2}(X)\overline{g}(GY,PZ)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;+v_{3}(Y)\overline{g}(HX,PZ)-v_{3}(X)\overline{g}(HY,PZ),$
(4.30)

for all $X,Y,Z\in\Gamma(\mathcal{V}^{\perp})$.

######  Theorem 4.8.

Let $\overline{M}$ be a normal indefinite complex contact manifold, and
$(M,g)$ a contact screen conformal null hypersurface of $\overline{M}$,
tangent to the characteristic subbundle $\mathcal{V}$. Then, $c=-3$, i.e;
$\overline{M}$ is a space of constant $GH$-sectional curvature $-3$.

###### Proof.

Setting $X=\xi$ in (4) and then apply (2), (2.14) and (2.15), we derive

$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{c+3}{4}[g(Y,PZ)+u_{1}(Z)v_{1}(Y)+2u_{1}(Y)v_{1}(Z)]$
$\displaystyle\frac{c-1}{4}[u_{2}(Z)v_{2}(Y)+2u_{2}(Y)v_{2}(Z)+u_{3}(Z)v_{3}(Y)+2u_{3}(Y)v_{3}(Z)]$
$\displaystyle\;\;\;-\varphi[\frac{3c+9}{4}u_{1}(Z)u_{1}(Y)+\frac{3c+1}{4}\\{u_{2}(Y)u_{2}(Z)+u_{3}(Y)u_{3}(Z)\\}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;-2\tau(\xi)B(Y,Z)]=(\xi\varphi)B(Y,Z)+2u_{2}(Y)C(U,PZ)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+2u_{3}(Y)C(V,PZ)-v_{2}(Y)u_{2}(Z)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-v_{3}(Y)u_{3}(Z),$
(4.31)

for all $X,Y,Z\in\Gamma(\mathcal{V}^{\perp})$. Since, by Lemmas 3.9 and 3.12,
we have $C(U,V_{2})=B(U,U_{2})=B(U_{2},U)=-1$ and
$C(V,V_{3})=B(V,U_{3})=B(U_{3},V)=-1$. Then, letting $Y=U_{a}$ and $Z=V_{a}$,
$a=1,2,3$, in (4), we get

$\displaystyle[\xi\varphi-2\varphi\tau(\xi)]B(U_{a},V_{a})=\frac{3}{4}(c+3).$
(4.32)

On the other hand, letting $Y=V_{a}$ and $Z=U_{a}$, $a=1,2,3$, in (4), we get

$\displaystyle[\xi\varphi-2\varphi\tau{\xi}]B(V_{a},U_{a})=\frac{1}{2}(c+3).$
(4.33)

It then follows from (4.32), (4.33) and the symmetry of $B$ that $c=-3$, and
$[\xi\varphi-2\varphi\tau(\xi)]B(V_{a},U_{a})=0$, for all $a=1,2,3$, which
completes the proof. ∎

The following is an immediate consequence of Theorem 4.8.

######  Corollary 4.9.

There exist no any contact screen conformal null hypersurface of
$\overline{M}$ such that $c\neq-3$.

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