# New classes of null hypersurfaces in indefinite Sasakian space-forms

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

We introduce two classes of null hypersurfaces of an indefinite Sasakian
manifold, $(\overline{M},\overline{\phi},\zeta,\eta)$, tangent to the
characteristic vector field $\zeta$, called; contact screen conformal and
contact screen umbilic null hypersurfaces. These hypersurfaces come in to fill
the existing gap in screen conformal and screen totally umbilic null
hypersurfaces. We prove that such hypersurfaces are contained in indefinite
Sasakian space forms of constant $\overline{\phi}$-sectional curvature of
$-3$.

###### Key words and phrases:

Null hypersurfaces, Contact screen conformal hypersurfaces, Contact screen
umbilic null hypersurfaces

###### 2010 Mathematics Subject Classification:

Primary 53C25; Secondary 53C40, 53C50

##  1\. Introduction

The theory of non-degenerate submanifolds [2, 3, 12] of Riemannian or semi-
Riemannian manifolds is one of the most important topics of differential
geometry. But the theory of null submanifolds of semi-Riemannian manifolds is
relatively new and in a developing stage, with a lot of applications to
mathematical physics (general relativity and electromagnetism). The geometry
of null submanifolds becomes more difficult and is completely different from
that of non-degenerate submanifolds. Such difficulty stems from a non-trivial
intersection of the normal bundle of a null submanifold and its tangent
bundle. In 1996, Duggal and Bejancu published their work [5] on null
submanifolds of semi-Riemannian manifolds and indefinite Kaehler manifolds.
This was later, in 2009, updated to [6] by Duggal and Sahin, to include the
geometry of indefinite almost contact manifolds. In particular, the geometry
of null submanifolds of indefinite Sasakian manifolds is in Chapter 7.

Recently several authors have studied the geometry of null submanifolds of
indefinite Sasakian manifolds [7, 8, 9, 10, 11]. In particular, [8] proves
that there exist no null hypersurfaces of an indefinite Sasakian space form of
constant $\overline{\phi}$-sectional curvature different from $1$ (see Theorem
3.3 and Corollary 1). On the other hand, [9, 10, 11] considers totally
umbilic, screen totally umbilic, $\eta$-totally umbilic, screen conformal and
invariant null hypersurfaces in indefinite Sasakian manifolds. In this paper,
we prove that all the above null hypersurfaces are non-existent in indefinite
Sasakian manifolds, (see Theorem 3.4), if they are assumed to be tangent to
the structure vector field, and that there exist other null hypersurfaces in
indefinite Sasakian space forms of constant $\overline{\phi}$-sectional
curvature different from $1$ (see Theorems 4.3 and 4.8).

The rest of the paper is arranged as follows; in Section 2 we quote the basic
notions required in the rest of the paper. In Section 3, we prove a non-
existence result of null hypersurfaces, tangent to the structure vector field.
Finally, in Section 4, we introduce two classes of null hypersurfaces in
indefinite Sasakian manifolds.

##  2\. Preliminaries

A $(2n+1)$-dimensional differentiable manifold $\overline{M}$ is said to have
an almost contact structure $(\overline{\phi},\zeta,\eta)$ if it admits a
vector field $\zeta$, a 1-form $\eta$ and a field $\overline{\phi}$ of
endomorphism of the tangent vector space satisfying

$\displaystyle\overline{\phi}^{2}=-I+\eta\otimes\zeta,\;\;\;\eta(\zeta)=1.$
(2.1)

It follows that $\overline{\phi}\zeta=0$, $\eta\circ\overline{\phi}=0$ and
$\mathrm{rank}(\overline{\phi})=2n$. Then, the manifold $\overline{M}$, with a
$(\overline{\phi},\zeta,\eta)$-structure is called an almost contact manifold
[2]. From the point of view of differential geometry it is desirable to define
a metric on a paracompact manifold $\overline{M}$. We say that a semi-
Riemannian metric $\overline{g}$ is an associated metric of an almost contact
structure $(\overline{\phi},\zeta,\eta)$ of $\overline{M}$ if

$\displaystyle\overline{g}(\overline{\phi}X,\overline{\phi}Y)=\overline{g}(X,Y)-\eta(X)\eta(Y),\;\;\;\forall\,X,Y\in\Gamma(T\overline{M}).$
(2.2)

Here, and in the rest of this paper, $\Gamma(\Xi)$ denotes the smooth sections
of a vector bundle $\Xi$. Then, $(\overline{M},\overline{g})$, with
$\overline{g}$ satisfying (2.2), is called an almost contact metric manifold
with a $(\overline{\phi},\zeta,\eta)$-structure and $\overline{g}$ is called
its compatible (or associated) metric, whose fundamental 2-form $\Omega$ is
defined by $\Omega(X,Y)=\overline{g}(X,\overline{\phi}Y)$, for any
$X,Y\in\Gamma(T\overline{M})$. Note that Since $\Omega$, also satisfies
$\eta\wedge\Omega^{n}$. A $(2n+1)$-dimensional manifold $\overline{M}$ is
called a contact manifold if there exists a 1-form $\eta$ on $\overline{M}$
such that $\eta\wedge(d\eta^{n})\neq 0$ everywhere. When $\Omega=d\eta$ (i.e.,
$\Omega$ is closed), then, $(\overline{M},\overline{\phi},\zeta,\eta)$ is
called a contact metric manifold [3]. These metrics can be constructed by the
polarization of $\Omega=d\eta$ evaluated on a local orthonormal basis of the
tangent space with respect to an arbitrary metric, on the contact subbundle
$\mathcal{D}$, such that $T\overline{M}=\mathcal{D}\perp\mathbb{R}\zeta$.
Here, and in the rest of this paper, $\mathbb{R}\zeta$ denotes the line bundle
spanned by $\zeta$. Thus, a contact metric manifold is an analogue of an
almost Kaehler manifold, in odd dimensions. The almost contact structure
$(\overline{\phi},\zeta,\eta)$ on $\overline{M}$ is said to be normal if
$\overline{\phi}$ is integrable. It is known (see Blair [3] for details) that
$\overline{M}$ has a normal contact structure if
$N_{\overline{\phi}}+2d\eta\otimes\zeta=0$, where
$N_{\overline{\phi}}=[\overline{\phi},\overline{\phi}]$ is the Nijenhuis
tensor field of $\overline{\phi}$. A normal contact metric manifold is called
a Sasakian manifold. It is well-known [3] that an almost contact metric
manifold $(\overline{M},\overline{g})$ is Sasakian if and only if

$\displaystyle(\overline{\nabla}_{X}\overline{\phi})Y=\overline{g}(X,Y)\zeta-\eta(Y)X,\;\;\;\;\forall\,X,Y\in\Gamma(T\overline{M}),$
(2.3)

where $\overline{\nabla}$ is the metric connection on $\overline{M}$.
Replacing $Y$ by $\zeta$ in (2.3), and using (2.1), we get

$\displaystyle\overline{\nabla}_{X}\zeta=-\overline{\phi}X,\;\;\;\,\forall\,X\in\Gamma(T\overline{M}).$
(2.4)

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 [3])

$\displaystyle\overline{R}(X,Y)Z=$
$\displaystyle\frac{(c+3)}{4}\\{\overline{g}(Y,Z)X-\overline{g}(X,Z)Y\\}+\frac{(c-1)}{4}\\{\eta(X)\eta(Z)Y$
$\displaystyle-\eta(Y)\eta(Z)X+\overline{g}(X,Z)\eta(Y)\zeta-\overline{g}(Y,Z)\eta(X)\zeta$
$\displaystyle+\overline{g}(\overline{\phi}Y,Z)\overline{\phi}X-\overline{g}(\overline{\phi}X,Z)\overline{\phi}Y-2\overline{g}(\overline{\phi}X,Y)\overline{\phi}Z\\},$
(2.5)

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

Let $(\overline{M},\overline{g})$ be a $(2n+1)$-dimensional semi-Riemannian
manifold with index $q$, where $0<q<2n+1$, and let $(M,g)$ be a hypersurface
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 $2n-1$ and the normal bundle $TM^{\perp}$ is a distribution of
rank 1 on $M$ [5]. Here, the fibers 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, we consider the
complementary distribution $S(TM)$ to $TM^{\perp}$ in $TM$, which is called a
screen distribution. It is well-known that $S(TM)$ is non-degenerate (see
[5]). Thus, $TM=S(TM)\perp TM^{\perp}$. 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
$tr(TM)$, called the null transversal bundle [5] of $M$ with respect to
$S(TM)$, of rank 1 over $M$ such that for any non-zero section $E$ of
$TM^{\perp}$ on a coordinate neighborhood $\mathcal{U}\subset M$, there exists
a unique section $N$ of $tr(TM)$ on $\mathcal{U}$ satisfying

$\displaystyle\overline{g}(\xi,N)=1,\;\overline{g}(N,N)=\overline{g}(N,Z)=0,\;\;\;\forall\,Z\in\Gamma(S(TM)).$
(2.6)

Consequently, we have the following decomposition of $T\overline{M}$.

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

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 [5].

$\displaystyle\overline{\nabla}_{X}Y=\nabla_{X}Y+h(X,Y)=\nabla_{X}Y+B(X,Y)N,$
(2.7)
$\displaystyle\overline{\nabla}_{X}N=-A_{N}X+\nabla_{X}^{t}N=-A_{N}X+\tau(X)N,$
(2.8)
$\displaystyle\nabla_{X}PY=\nabla^{*}_{X}PY+h^{*}(X,PY)=\nabla^{*}_{X}PY+C(X,PY)\xi,$
(2.9)
$\displaystyle\nabla_{X}\xi=-A^{*}_{\xi}X+\nabla_{X}^{*t}\xi=-A^{*}_{\xi}X-\tau(X)\xi,\;\;\;A_{\xi}^{*}\xi=0,$
(2.10)

for all $X,Y\in\Gamma(TM)$, $\xi\in\Gamma(TM^{\perp})$ and
$N\in\Gamma(tr(TM))$, 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

$\displaystyle
g(A^{*}_{\xi}X,Y)=B(X,Y)\;\;\;\mbox{and}\;\;\;g(A_{N}X,PY)=C(X,PY),$ (2.11)

for any $X,Y\in\Gamma(TM)$. It follows from (2.10) and (2.11) that

$\displaystyle B(X,\xi)=0,\;\;\;\;\forall\,X\in\Gamma(TM).$ (2.12)

Moreover, we have

$\displaystyle\overline{g}(A^{*}_{\xi}X,N)=0\;\;\;\mbox{and}\;\;\;\overline{g}(A_{N}X,N)=0,$
(2.13)

for all $X\in\Gamma(TM)$. From relations (2.13), we 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 $\theta=i^{*}\vartheta$ to
be its restriction on $M$, where $i:M\rightarrow\overline{M}$ is the inclusion
map. Then it is easy to show that

$\displaystyle(\nabla_{X}g)(Y,Z)=B(X,Y)\theta(Z)+B(X,Z)\theta(Y),$ (2.14)

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 [5, 6]).

$\displaystyle\overline{g}(\overline{R}(X,Y)Z,\xi)=$
$\displaystyle(\nabla_{X}B)(Y,Z)-(\nabla_{Y}B)(X,Z)+\tau(X)B(Y,Z)$
$\displaystyle-\tau(Y)B(X,Z),$ (2.15)
$\displaystyle\overline{g}(\overline{R}(X,Y)PZ,N)=$
$\displaystyle(\nabla_{X}C)(Y,PZ)-(\nabla_{Y}C)(X,PZ)-\tau(X)C(Y,PZ)$
$\displaystyle+\tau(Y)C(X,PZ),$ (2.16)

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

$\displaystyle(\nabla_{X}B)(Y,Z)=XB(Y,Z)-B(\nabla_{X}Y,Z)-B(Y,\nabla_{X}Z),$
(2.17)

and

$\displaystyle(\nabla_{X}C)(Y,PZ)=XC(Y,PZ)-C(\nabla_{X}Y,PZ)-C(Y,\nabla^{*}_{X}PZ).$
(2.18)

A null hypersurface $(M,g)$ of a semi-Riemannian manifold
$(\overline{M},\overline{g})$ is called screen conformal [5, p. 51] if there
exist a non-vanishing smooth function $\psi$ on a neighborhood $\mathcal{U}$
in $M$ such that $A_{N}=\psi A^{*}_{\xi}$, or equivalently,

$\displaystyle C(X,PY)=\psi B(X,PY),$ (2.19)

for all $X,Y\in\Gamma(TM)$. We say that $M$ is screen homothetic if $\psi$ is
a constant function on $M$. The null hypersurface $M$ is said to be totally
umbilic [5] if

$\displaystyle B(X,Y)=\rho g(X,Y),$ (2.20)

where $\rho$ is a smooth function on a coordinate neighborhood
$\mathcal{U}\subset TM$. In case $\rho=0$, we say that $M$ is totally
geodesic. In the same line, $M$ is called screen totally umbilic if

$\displaystyle C(X,PY)=\varrho g(X,PY),$ (2.21)

where $\varrho$ is a smooth function on a coordinate neighborhood
$\mathcal{U}\subset TM$. When $\varrho=0$, we say that $M$ is screen totally
geodesic.

##  3\. A nonexistance result

Let $(M,g)$ be a null hypersurface of an indefinite Sasakian manifold
$(\overline{M},\overline{\phi},\zeta,\eta)$, which is tangent to the structure
vector field $\zeta$. That is; $\zeta\in\Gamma(TM)$. In such a case, $\zeta$
belongs to $S(TM)$ [4]. As seen in the previous section, let $\xi$ and $N$ the
metric normal and the transversal sections, respectively. Since
$(\overline{\phi},\zeta,\eta))$ is an almost contact structure and
$\overline{\phi}\xi$ is a null vector field, it follows that
$\overline{\phi}N$ is null too. Moreover,
$\overline{g}(\overline{\phi}\xi,\xi)=0$ and, thus, $\overline{\phi}\xi$ is
tangent to $TM$. Let us consider $S(TM)$ containing
$\overline{\phi}TM^{\perp}$ as a vector subbundle. Consequently, $N$ is
orthogonal to $\overline{\phi}\xi$ and we have
$\overline{g}(\overline{\phi}N,\xi)=-\overline{g}(N,\overline{\phi}\xi)=0$ and
$\overline{g}(\overline{\phi}N,N)=0$. This means that $\overline{\phi}N$ is
tangent to $TM$ and in particular, it belongs to $S(TM)$. Thus,
$\overline{\phi}tr(TM)$ is also a vector subbundle of $S(TM)$. In view of
(2.2), we have $\overline{g}(\overline{\phi}\xi,\overline{\phi}N)=1$.

It is then easy to see that
$\overline{\phi}TM^{\perp}\oplus\overline{\phi}tr(TM)$ is a non-degenerate
vector subbbundle of $S(TM)$, with 2-dimensional fibers. Since $\zeta$ is
tangent to $M$, and that
$\overline{g}(\overline{\phi}E,\zeta)=\overline{g}(\overline{\phi}N,\zeta)=0$,
then there exists a non-degenerate distribution $D_{0}$ on $TM$ such that

$\displaystyle
S(TM)=\\{\overline{\phi}TM^{\perp}\oplus\overline{\phi}tr(TM)\\}\perp
D_{0}\perp\mathbb{R}\zeta.$ (3.1)

Here, $\mathbb{R}\zeta$ denotes the line bundle spanned by $\zeta$. It is easy
to check that $D_{0}$ is an almost complex distribution with respect to
$\overline{\phi}$, i.e., $\overline{\phi}D_{0}=D_{0}$. The decomposition of
$TM$ becomes
$TM=\\{\overline{\phi}TM^{\perp}\oplus\overline{\phi}tr(TM)\\}\perp
D_{0}\perp\mathbb{R}\zeta\perp TM^{\perp}$. If we set
$D:=TM^{\perp}\perp\overline{\phi}TM^{\perp}\perp D_{0}$ and
$D^{\prime}=\overline{\phi}tr(TM)$, then

$\displaystyle TM=D\oplus D^{\prime}\perp\mathbb{R}\zeta.$ (3.2)

Here, $D$ is an almost complex distribution and $D^{\prime}$ is carried by
$\overline{\phi}$ just into the transversal bundle. Let us set

$\displaystyle
U:=-\overline{\phi}N\;\;\;\mbox{and}\;\;\;V:=-\overline{\phi}\xi.$ (3.3)

Then, from (3.2), any $X\in\Gamma(TM)$ is written as $X=RX+QX+\eta(X)\zeta$
and $QX=u(X)U$, where $R$ and $Q$ are the projection morphisms of $TM$ onto
$D$ and $D^{\prime}$, respectively, and $u$ is a differential 1-form defined
on $M$ by

$\displaystyle u(X)=g(V,X),\;\;\;\forall\,X\in\Gamma(TM).$ (3.4)

Applying $\overline{\phi}$ to $X$ and using (2.1), we get

$\displaystyle\overline{\phi}X=\phi X+u(X)N,$ (3.5)

where $\phi$ is a (1,1) tensor field defined on $M$ by $\phi
X=\overline{\phi}RX$. Furthemore, we have

$\displaystyle\phi^{2}X=-I+\eta(X)\zeta+u(X)U,\;\;\;\;u(U)=1,\;\;\;\forall\,X\in\Gamma(TM).$
(3.6)

It is easy to show that

$\displaystyle g(\phi X,\phi Y)=g(X,Y)\eta(X)\eta(Y)-u(Y)v(X)-u(X)v(Y),$ (3.7)

where $v$ is a 1-form locally defined on $M$ by

$\displaystyle v(X)=g(U,X),\;\;\;\forall\,X\in\Gamma(TM).$ (3.8)

Note that

$\displaystyle g(\phi X,Y)=-g(X,\phi Y)-u(X)\theta(Y)-u(Y)\theta(X),$ (3.9)

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

######  Lemma 3.1.

On a null hypersurface $(M,g)$ of an indefinite Sasakian manifold
$(\overline{M}(c),\overline{\phi},\zeta,\eta)$, tangent to $\zeta$, the
following important relations holds.

$\displaystyle\nabla_{X}\zeta$ $\displaystyle=-\phi X,$ (3.10) $\displaystyle
B(X,\zeta)$ $\displaystyle=-u(X),$ (3.11) $\displaystyle C(X,\zeta)$
$\displaystyle=-v(X),$ (3.12) $\displaystyle B(X,U)$ $\displaystyle=C(X,V),$
(3.13) $\displaystyle(\nabla_{X}\phi)Y$
$\displaystyle=g(X,Y)\zeta-\eta(Y)X-B(X,Y)U+u(Y)A_{N}X,$ (3.14)

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

###### Proof.

A proof uses straightforward calculations, while considering (2.3), (2.4) and
(2.7)–(2.10). ∎

We also have the following lemma.

######  Lemma 3.2.

The vector fields $U$ and $V$ in (3.3) satisfies the relations

$\displaystyle\nabla_{X}U$ $\displaystyle=\phi
A_{N}X-\theta(X)\zeta+\tau(X)U,$ (3.15)
$\displaystyle\mbox{and}\;\;\;\;\nabla_{X}V$ $\displaystyle=\phi
A^{*}_{\xi}X-\tau(X)V,\;\;\;\forall\,X\in\Gamma(TM).$ (3.16)

###### Proof.

Letting $Y=U$ in (3.14) and using the fact $\phi U=0$, we get

$\displaystyle-\phi\nabla_{X}U=v(X)\zeta-B(X,U)U+A_{N}X,$ (3.17)

for all $X\in\Gamma(TM)$. Applying $\phi$ to (3.17) and using the fact
$\phi\zeta=0$, we have

$\displaystyle-\phi^{2}\nabla_{X}U=\phi A_{N}X.$ (3.18)

Then, in view of (3.6) and (3.18), we see that

$\displaystyle\nabla_{X}U=\phi A_{N}X+\eta(\nabla_{X}U)\zeta+u(\nabla_{X}U)U.$
(3.19)

But
$\eta(\nabla_{X}U)=g(\nabla_{X}U,\zeta)=-\overline{g}(U,\overline{\nabla}_{X}\zeta)=\overline{g}(U,\overline{\phi}X)=-\theta(X)$,
in which we have used (2.4) and (3.3). On the other hand,
$u(\nabla_{X}U)=g(\nabla_{X}U,V)=\overline{g}(\overline{\nabla}_{X}U,V)=\overline{g}(\overline{\nabla}_{X}\overline{\phi}N,\overline{\phi}\xi)=\overline{g}(\overline{\phi}\,\overline{\nabla}_{X}N,\overline{\phi}\xi)=\overline{g}(\overline{\nabla}_{X}N,\xi)=\tau(X)$,
in which we have used (2.2), (3.3) and (2.8). Thus, putting all this in (3.19)
proves (3.15) of the lemma. A proof of (3.16) also follows easily from (3.14)
by putting $Y=V$, which completes the proof. ∎

In what follows, we construct a null hypersurface of an indefinite Sasakian
manifold, tangent to $\zeta$. To that end,
$(\mathbb{R}^{2n+1}_{q},\overline{\phi}_{0},\zeta,\eta,\overline{g})$ will
denote the manifold $\mathbb{R}^{2n+1}_{q}$ with its usual Sasakian structure
given by

$\displaystyle\eta=\frac{1}{2}\left(dz-\sum_{i=1}^{n}y^{i}dx^{i}\right),\;\;\;\;\zeta=2\partial
z,$ $\displaystyle\overline{g}=\eta\otimes$
$\displaystyle\eta+\frac{1}{4}\left(-\sum_{i=1}^{q/2}dx^{i}\otimes
dx^{i}+dy^{i}\otimes dy^{i}+\sum_{i=q+1}^{n}dx^{i}\otimes dx^{i}+dy^{i}\otimes
dy^{i}\right),$ $\displaystyle\overline{\phi}_{0}$
$\displaystyle\left(\sum_{i=1}^{n}(X_{i}\partial x^{i}+Y_{i}\partial
y^{i})+Z\partial z\right)=\sum_{i=1}^{n}(Y_{i}\partial x^{i}-X_{i}\partial
y^{i})+\sum_{i=1}^{n}Y_{i}y^{i}\partial z,$

where $(x^{i};y^{i};z)$ are the Cartesian coordinates. The above construction
will help in understanding the following example.

######  Example 3.3.

Let $\overline{M}=(\mathbb{R}_{2}^{7},\overline{g})$ be a semi-Euclidean
space, where $\overline{g}$ is of signature $(-,+,+,-,+,+,+)$ with respect to
canonical basis

$\\{\partial x^{1},\partial x^{2},\partial x^{3},\partial y^{1},\partial
y^{2},\partial y^{3},\partial z\\}.$

Suppose $M$ is a submanifold of $\mathbb{R}_{2}^{7}$ defined by $x^{1}=y^{3}$.
It is easy to see that a local frame of $TM$ is given by

$\displaystyle\xi=2(\partial x^{1}+\partial y^{3}+y^{1}$
$\displaystyle\partial z),\;\;\;Z_{1}=2(\partial x^{3}-\partial
y^{1}+y^{3}\partial z),$ $\displaystyle Z_{2}=2(\partial x^{2}+y^{2}\partial
z),\;\;\;Z_{3}$ $\displaystyle=2\partial y^{2},\;\;\;\;Z_{4}=\partial
x^{3}+\partial y^{1}+y^{3}\partial z,$ $\displaystyle Z_{6}$
$\displaystyle=\zeta=2\partial z.$

Hence, $TM^{\perp}=\mathrm{span}\\{\xi\\}$,
$\overline{\phi}_{0}TM^{\perp}=\mathrm{span}\\{Z_{1}\\}$. Note that
$TM^{\perp}\cap\overline{\phi}_{0}TM^{\perp}=\\{0\\}$. Next,
$\overline{\phi}_{0}Z_{2}=-Z_{3}$, which implies that
$D_{0}=\mathrm{span}\\{Z_{2},Z_{3}\\}$ is invariant with respect to
$\overline{\phi}_{0}$. By a direct calculation we see that $tr(TM)$ is spanned
by

$\displaystyle N=-\partial x^{1}+\partial y^{3}-y^{1}\partial z,$

such that $\overline{\phi}_{0}N=Z_{4}$. It is easy to see that the vector
fields $U,V$ of (3.3) are given by $U=-Z_{4}$ and $V=-Z_{1}$. Hence, $M$ is a
null hypersurface tangent to $\zeta$.

Next, we prove the following non-existence result.

######  Theorem 3.4.

Let $(M,g)$ be a null hypersuface of an indefinite Sasakian manifold
$(\overline{M},\overline{\phi},\zeta,\eta)$, tangent to the structure vector
field $\zeta$. The following are all true;

  1. (1)

$M$ is not totally umbilc or screen totally umbilic.

  2. (2)

$M$ is not $\eta$-totally umbilic.

  3. (3)

$M$ is not screen conformal.

  4. (4)

$M$ is not invariant.

  5. (5)

$S(TM)$ is not parallel.

  6. (6)

$\nabla$ is not a metric connection.

###### Proof.

Suppose that $M$ is totally umbilic in $\overline{M}$. Then, by (3.11) and
(2.20), we have $\rho g(X,\zeta)=-u(X)$, for any $X\in\Gamma(TM)$. Setting
$X=\zeta$ in this relation gives $\rho g(\zeta,\zeta)=0$, or simply $\rho=0$,
since $\zeta$ is a spacelike vector field. This means that $M$ is totally
geodesic null hypersurface. It then follows from (3.11) that $u(X)=0$, for all
$X\in\Gamma(TM)$. This is a contradiction as $u(U)=1$, which proves the first
assertion of (1). Next, let $M$ be screen totally umbilic. Then (2.21) and
(3.12) implies that $\varrho g(X,\zeta)=-v(X)$, for any $X\in\Gamma(TM)$. With
$X=\zeta$, we have $\varrho=0$ and thus, $v(X)=0$. This is a contradiction
since $v(V)=1$, which completes part (1).

In case $M$ is $\eta$-totally umbilic we see, from (4.20) of [10, p. 351],
that

$\displaystyle h(X,Y)=\lambda\\{g(X,Y)-\eta(X)\eta(Y)\\}N,$ (3.20)

for any $X,Y\in\Gamma(TM)$. Putting $Y=\zeta$ in (3.20), we get
$h(X,\zeta)=0$, for any $X\in\Gamma(TM)$. As $h(X,\zeta)=B(X,\zeta)N$, we
notice that $B(X,\zeta)=0$. Thus, (3.11) gives $u(X)=0$, for all
$X\in\Gamma(TM)$, which is a contradiction. This proves (2).

Next, let $M$ be screen conformal. Then, in view of (2.19) and (3.12), we have
$\psi B(X,\zeta)=-v(X)$. Setting $X=V$ in this relation and using (3.11), we
get $v(V)=0$. Clearly, this is a contradiction to the fact $v(V)=1$ and
therefore, $M$ is never screen conformal, proving part (3).

Now, assume that $M$ is an invariant null hypersurface. Then, by [11, p. 27],
$\overline{\phi}X\in TM$, for any $X\in\Gamma(TM)$. Note that this is
equivalent to $\overline{\phi}X=\phi X$, for all $X\in\Gamma(TM)$. Therefore,
relations (2.4) and (2.7) gives $B(X,\zeta)=0$, which, in view of (3.11),
gives $u(X)=0$. This is a contradiction too. Therefore, $M$ is never
invariant, which proves (4).

To prove (5), we note, from (2.9), that the screen distribution $S(TM)$ is
parallel if and only if $C(X,PY)=0$, for all $X,Y\in\Gamma(TM)$. Let $Y=V$ in
this relation and then compare with (3.13) to get $B(X,U)=C(X,V)=0$, for any
$X\in\Gamma(TM)$. Hence, letting $X=\zeta$ in the last relation and compare
with (3.11) gives $u(U)=0$, which is a contradiction. Finally, $\nabla$ is a
metric connection if and only if $B=0$ by (2.14). That is; $M$ is totally
geodesic. But this is not possible by part (1). This proves (6) and the
theorem is proved. ∎

######  Remark 3.5.

In [9], the authors proves that totally umbilic null hypersurfaces, of
indefinite Sasakian space forms $(\overline{M}(c),\overline{\phi},\zeta,\eta)$
and tangent to $\zeta$ exists. Moreover, they show that $c=1$ (also see the
book [6, p. 317]). But in view of Theorem 3.4 above, such hypersurfaces don’t
exist. In [10], the author study the geometry of null hypersurfaces of
indefinite Sasakian manifolds, tangent to $\zeta$. Within this paper, the
concept of $\eta$-totally umbilic is introduced (see pages 352–352), and
“Theorem 4.6” regarding such hypersurfaces is proved. Also, in [11], screen
conformal invariant null hypersurfaces are treated. But we have seen, in
Theorem 3.4, that $\eta$-totally umbilic null hypersufaces, screen conformal
and invariant null hypersurfaces, tangent to $\zeta$, do not exist in
indefinite Sasakian space forms. Therefore, in studying all null hypersurfaces
of indefinite Sasakian space forms, which are tangent to $\zeta$, equations
(3.10)–(3.13) must be treated carefully. Furthermore, the 1-forms $u,v$ in
(3.4) and (3.8) can not identically vanish on $M$ as assumed in [11, p. 28],
since their vanishing affects the normalization condition in (3.20).

######  Remark 3.6.

In “Theorem 3.3” and “Corollary 1” of [8, p. 576–577], the author proves that
there exist no null hypersurfaces of an indefinite Sasakian space form
$(\overline{M}(c),\overline{\phi},\zeta,\eta)$ if $c\neq 1$. This contradicts
the fact that totally contact umbilic null hypersurfaces exists in
$(\overline{M}(c),\overline{\phi},\zeta,\eta)$, with $c=-3$ (see [6] and
[10]). Here is the cause; first note that, the curvature tensor $R$ of a null
submanifold is not Riemannian since the induced connection, $\nabla$, on a
null hypersurface $(M,g)$ is not a metric connection (see the expression for
$\nabla g$ in (2.14)). This means that $R$ does not enjoy all the symmetries
exhibited by Riemannian curvature tensors. In fact, it is easy to show that
$g(R(X,Y)Z,W)=-g(R(Y,X)Z,W)$, but $g(R(X,Y)Z,W)\neq-g(R(X,Y)W,Z)$, for any
$X,Y,Z,W\in\Gamma(TM)$. For any null hypersurface $M$, the relation
$\overline{R}(X,Y)\xi=R(X,Y)\xi$, for any $X,Y\in\Gamma(TM)$ and
$\xi\in\Gamma(TM^{\perp})$, holds (see (3.6) of [5, p. 94]). Consequently,

$\displaystyle\overline{g}(\overline{R}(X,Y)\xi,Z)=g(R(X,Y)\xi,Z)\neq-g(R(X,Y)Z,\xi)=0.$
(3.21)

It was assumed in [8, p. 576] that all signs in (3.21) are equalities, which
lead to $c=1$, and any further deductions, such as Corollary 1. For the above
reasons, the curvature tensor of a null submanifold must be treated carefully.

Theorem 3.4 shows that most well-known null hypersurfaces do not exist in
indefinite Sasakian manifolds. In particular $B$ and $C$ can not be linked by
a non-zero conformal factor on the entire null hypersurface $M$. However, such
a conformality can be defined partially on $M$. This is the aim of the next
section.

##  4\. Contact screen conformal null hypersurfaces

In this section, we introduce two new classes of null hypersurface of
indefinite almost contact manifolds
$(\overline{M}(c),\overline{\phi},\zeta,\eta)$, tangent to $\zeta$, called;
contact screen conformal and contact screen umbilic. This is motivated by
Theorem 3.4. In particular, the absence of the classical screen conformal and
umbilic null hypersurfaces. As this fails mainly in portions of $TM$
containing $\zeta$, we restrict to a region on $TM$ without $\zeta$, and
relate $B$ and $C$ via a non-zero conformal factor for the contact screen
conformal null hypersurfaces as shown below. From the decomposition (3.2), we
have $TM=D\oplus D^{\prime}\perp\mathbb{R}\zeta$. Let $\tilde{P}$ be the
projection morphism of $TM$ onto $D\oplus D^{\prime}$. Then, any
$X\in\Gamma(TM)$ is represented as $X=\tilde{P}X+\eta(X)\zeta$. Then, as
$B(\zeta,\zeta)=0$ by (3.11), we have

$\displaystyle B(\tilde{P}X,\tilde{P}Y)=B(X,Y)+\eta(Y)u(X)+\eta(X)u(Y),$ (4.1)

for any $X,Y\in\Gamma(TM)$. Next, since $\phi\zeta=0$, we see, from (2.9),
that $C(\zeta,\zeta)=0$. Hence, using (3.12), we derive

$\displaystyle
C(\tilde{P}X,\tilde{P}PY)=C(X,PY)+\eta(Y)v(X)-\eta(X)\omega(Y),$ (4.2)

where $\omega$ is a 1-form on $M$ defined as

$\displaystyle\omega(X)=C(\zeta,PX),\;\;\;X\in\Gamma(TM).$ (4.3)

In view of (2.9), (3.10), (3.11), (3.13) and (4.3), we have

$\displaystyle\omega(\zeta)=0\;\;\;\;\;\mbox{and}\;\;\;\;\;\omega(V)=-1.$
(4.4)

Next, we define contact screen conformal null hypersurfaces.

######  Definition 4.1.

Let $(M,g)$ be a null hypersurface of an indefinite Sasakian manifold
$(\overline{M},\overline{\phi},\zeta,\eta)$, tangent to the structure vector
field $\zeta$. Then, $M$ will be called contact screen conformal null
hypersurface if there exists a non-zero smooth function $\varphi$ on a
neighborhood $\mathcal{U}\subset M$ such that

$\displaystyle C(\tilde{P}X,\tilde{P}Y)=\varphi
B(\tilde{P}X,\tilde{P}Y),\;\;\;\;\forall\,X,Y\in\Gamma(TM).$ (4.5)

Equivalently, using (4.1) and (4.2), $M$ is contact screen conformal if

$\displaystyle C(X,PY)=\varphi\\{B$
$\displaystyle(X,Y)+\eta(Y)u(X)+\eta(X)u(Y)\\}$
$\displaystyle-\eta(Y)v(X)+\eta(X)\omega(Y),$ (4.6)

for all $X,Y\in\Gamma(TM)$. We say that $M$ is contact screen homothetic if
$\varphi$ is a conatant function.

As an example, we have the following.

######  Example 4.2.

Consider the null hypersurface in Example 3.3. By (2.4) and (2.7), we have
$B(X,\zeta)=-u(X)$, for all $X\in\Gamma(TM)$. By a direct calculation, we have
$[\xi,V]=-4\zeta$, and $[\xi,X]=0$, for all other $X\in\Gamma(TM)$. Thus,
using Koszul formula [12, p. 61], we have
$\overline{g}(\overline{\nabla}_{X}Y,\xi)=0$, for all $X,Y\in\Gamma(D\oplus
D^{\prime})$ . Hence, $B=0$ on $D\oplus D^{\prime}$. On the other hand,
$[N,U]=-\zeta$ and $[N,X]=0$ for any other $X\in\Gamma(TM)$. Also, using
Koszul formula, we have $\overline{g}(\overline{\nabla}_{X}Y,N)=0$, for any
$X,Y\in\Gamma(D\oplus D^{\prime})$. Thus, $C=0$ on $D\oplus D^{\prime}$. Note
that, as $[\zeta,X]=0$ for all $X\in\Gamma(TM)$, we have
$C(\zeta,X)=C(X,\zeta)=-v(X)$. Clearly, $(M,g)$ is a contact screen conformal
null hypersurface with $\varphi$ an arbitrary function.

Next, let $(M,g)$ be a null hypersurface of an indefinite Sasakian space-form.
Then, using (2), (2.15), (2.16), (3.4) and (3.8), we have, for any
$X,Y,Z\in\Gamma(TM)$,

$\displaystyle(\nabla_{X}$ $\displaystyle
B)(Y,Z)-(\nabla_{Y}B)(X,Z)=\tau(Y)B(X,Z)-\tau(X)B(Y,Z)$
$\displaystyle+\frac{(c-1)}{4}\\{\overline{g}(\overline{\phi}Y,Z)u(X)-\overline{g}(\overline{\phi}X,Z)u(Y)-2\overline{g}(\overline{\phi}X,Y)u(Z)\\},$
(4.7)

and also

$\displaystyle(\nabla_{X}C)(Y,PZ)-(\nabla_{Y}C)$
$\displaystyle(X,PZ)=\tau(X)C(Y,PZ)-\tau(Y)C(X,PZ)$
$\displaystyle+\frac{(c+3)}{4}\\{\overline{g}(Y,PZ)\theta(X)-\overline{g}(X,PZ)\theta(Y)\\}$
$\displaystyle+\frac{(c-1)}{4}\\{\eta(X)\eta(PZ)\theta(Y)-\eta(Y)\eta(PZ)\theta(X)$
$\displaystyle+\overline{g}(\overline{\phi}Y,PZ)v(X)-\overline{g}(\overline{\phi}X,PZ)v(Y)$
$\displaystyle-2\overline{g}(\overline{\phi}X,Y)v(PZ)\\}.$ (4.8)

In what follows, we prove the following result.

######  Theorem 4.3.

Let $(M,g)$ be a contact screen conformal null hypersurface of an indefinite
Sasakian space form $(\overline{M}(c),\overline{\phi},\zeta,\eta)$. Then,
$c=-3$ (that is; $\overline{M}$ is a space of constant
$\overline{\phi}$-sectional curvature $-3$) and $\omega(U)=C(\zeta,U)=0$.
Moreover, if $B(U,V)$ is nonzero, then $\varphi$ satisfies the differential
equation $\xi\varphi-2\varphi\tau(\xi)=0$.

###### Proof.

From (4.1), (2.10), (2.14), (3.10) and (3.11), we derive

$\displaystyle(\nabla_{X}C)$
$\displaystyle(Y,PZ)=(X\varphi)\\{B(Y,Z)+\eta(Z)u(Y)+\eta(Y)u(Z)\\}$
$\displaystyle+\varphi\\{(\nabla_{X}B)(Y,Z)-g(Z,\phi
X)u(Y)+B(X,V)\eta(Z)\theta(Y)$
$\displaystyle+\eta(Z)g(\nabla_{X}V,Y)-u(X)u(Z)\theta(Y)-g(Y,\phi X)u(Z)$
$\displaystyle+\eta(Y)g(\nabla_{X}V,Z)\\}+g(Z,\phi
X)v(Y)-B(X,U)\eta(Z)\theta(Y)$
$\displaystyle-\eta(Z)g(\nabla_{X}U,Y)-u(X)\theta(Y)\omega(Z)-g(Y,\phi
X)\omega(Z)$
$\displaystyle+\eta(Y)X\omega(Z)-\eta(Y)\omega(\nabla^{*}_{X}PZ),\;\;\;\forall\,X,Y,Z\in\Gamma(TM).$
(4.9)

Interchanging $X$ and $Y$ in (4) and subtracting the two relations gives

$\displaystyle(\nabla_{X}C)$ $\displaystyle(Y,PZ)-(\nabla_{Y}C)(X,PZ)$
$\displaystyle=(X\varphi)\\{B(Y,Z)+\eta(Z)u(Y)+\eta(Y)u(Z)\\}-(Y\varphi)\\{B(X,Z)$
$\displaystyle+\eta(Z)u(X)+\eta(X)u(Z)\\}+\varphi\\{(\nabla_{X}B)(Y,Z)-(\nabla_{Y}B)(X,Z)$
$\displaystyle-g(Z,\phi X)u(Y)+g(Z,\phi Y)u(X)+B(X,V)\theta(Y)\eta(Z)$
$\displaystyle-B(Y,V)\theta(X)\eta(Z)+\eta(Z)g(\nabla_{X}V,Y)-\eta(Z)g(\nabla_{Y}V,X)$
$\displaystyle-u(X)\theta(Y)u(Z)+u(Y)\theta(X)u(Z)-g(Y,\phi X)u(Z)$
$\displaystyle+g(X,\phi
Y)u(Z)+\eta(Y)g(\nabla_{X}V,Z)-\eta(X)g(\nabla_{Y}V,Z)\\}$
$\displaystyle+g(Z,\phi X)v(Y)-g(Z,\phi Y)v(X)-B(X,U)\theta(Y)\eta(Z)$
$\displaystyle+B(Y,U)\theta(X)\eta(Z)+\eta(Z)g(\nabla_{Y}U,X)-\eta(Z)g(\nabla_{X}U,Y)$
$\displaystyle-u(X)\theta(Y)\omega(Z)+u(Y)\theta(X)\omega(Z)-g(Y,\phi
X)\omega(Z)$ $\displaystyle+g(X,\phi
Y)\omega(Z)+\eta(Y)X\omega(Z)-\eta(X)Y\omega(Z)$
$\displaystyle-\eta(Y)\omega(\nabla^{*}_{X}PZ)+\eta(X)\omega(\nabla_{Y}^{*}PZ).$
(4.10)

In view of Lemma 3.2 and (3.9), we simplify (4) to

$\displaystyle(\nabla_{X}C)$ $\displaystyle(Y,PZ)-(\nabla_{Y}C)(X,PZ)$
$\displaystyle=(X\varphi)\\{B(Y,Z)+\eta(Z)u(Y)+\eta(Y)u(Z)\\}-(Y\varphi)\\{B(X,Z)$
$\displaystyle+\eta(Z)u(X)+\eta(X)u(Z)\\}+\varphi\\{(\nabla_{X}B)(Y,Z)-(\nabla_{Y}B)(X,Z)$
$\displaystyle-g(Z,\phi X)u(Y)+g(Z,\phi Y)u(X)+\eta(Z)B(Y,\phi X)$
$\displaystyle-\eta(Z)B(X,\phi Y)+\eta(Z)u(X)\tau(Y)-\eta(Z)u(X)\tau(X)$
$\displaystyle-u(X)\theta(Y)u(Z)+u(Y)\theta(X)u(Z)-g(Y,\phi X)u(Z)$
$\displaystyle+g(X,\phi Y)u(Z)-\eta(Y)\theta(Z)B(X,V)-\eta(Y)B(X,\phi Z)$
$\displaystyle-\eta(Y)u(Z)\tau(X)+\eta(X)\theta(Z)B(Y,V)+\eta(X)B(Y,\phi Z)$
$\displaystyle+\eta(X)u(Z)\tau(Y)\\}+g(Z,\phi X)v(Y)-g(Z,\phi Y)v(X)$
$\displaystyle-B(X,U)\theta(Y)\eta(Z)+B(Y,U)\theta(X)\eta(Z)+\eta(Z)\theta(Y)C(X,V)$
$\displaystyle-\eta(Z)\theta(X)C(Y,V)+\eta(Z)C(X,P\phi Y)-\eta(Z)C(Y,P\phi X)$
$\displaystyle+\eta(Z)\eta(Y)\theta(X)-\eta(Z)\eta(X)\theta(Y)+\eta(Z)v(X)\tau(Y)$
$\displaystyle-\eta(Z)v(Y)\tau(X)-u(X)\theta(Y)\omega(Z)+u(Y)\theta(X)\omega(Z)$
$\displaystyle-g(Y,\phi X)\omega(Z)+g(X,\phi Y)\omega(Z)+\eta(Y)X\omega(Z)$
$\displaystyle-\eta(X)Y\omega(Z)-\eta(Y)\omega(\nabla^{*}_{X}PZ)+\eta(X)\omega(\nabla_{Y}^{*}PZ).$
(4.11)

Then, putting (4), (4.8), (4) and (4.1), we get

$\displaystyle(X\varphi)\\{B(Y,Z)+\eta(Z)u(Y)+\eta(Y)u(Z)\\}-(Y\varphi)\\{B(X,Z)$
$\displaystyle+\eta(Z)u(X)+\eta(X)u(Z)\\}+\varphi\\{2B(X,Z)\tau(Y)-2B(Y,Z)\tau(X)$
$\displaystyle+((c-1)/4)\\{\overline{g}(\overline{\phi}Y,Z)u(X)-\overline{g}(\overline{\phi}X,Z)u(Y)-2\overline{g}(\overline{\phi}X,Y)u(Z)\\}$
$\displaystyle-\eta(Z)u(Y)\tau(X)-2\eta(Y)u(Z)\tau(X)+\eta(Z)u(X)\tau(Y)$
$\displaystyle+2\eta(X)u(Z)\tau(Y)-g(Z,\phi X)u(Y)+g(Z,\phi
Y)u(X)+\eta(Z)B(Y,\phi X)$ $\displaystyle-\eta(Z)B(X,\phi
Y)+\eta(Z)u(X)\tau(Y)-\eta(Z)u(X)\tau(X)$
$\displaystyle-u(X)\theta(Y)u(Z)+u(Y)\theta(X)u(Z)-g(Y,\phi X)u(Z)$
$\displaystyle+g(X,\phi Y)u(Z)-\eta(Y)\theta(Z)B(X,V)-\eta(Y)B(X,\phi Z)$
$\displaystyle+\eta(X)\theta(Z)B(Y,V)+\eta(X)B(Y,\phi
Z)+\eta(Z)\theta(Y)B(X,V)$
$\displaystyle-\eta(X)\theta(X)B(Y,V)+\eta(Z)\eta(Y)\theta(X)+\eta(Z)B(X,\phi
Y)$ $\displaystyle-\eta(Z)B(Y,\phi X)\\}+g(Z,\phi X)v(Y)-g(Z,\phi Y)v(X)$
$\displaystyle-\eta(Y)\theta(Y)\eta(X)-B(X,U)\theta(Y)\eta(Z)+B(Y,U)\theta(X)\eta(Z)$
$\displaystyle+\eta(Z)\eta(Y)\theta(X)-\eta(Z)\eta(X)\theta(Y)+\eta(Z)v(X)\tau(Y)$
$\displaystyle-\eta(Z)v(Y)\tau(X)-u(X)\theta(Y)\omega(Z)+u(Y)\theta(X)\omega(Z)$
$\displaystyle-g(Y,\phi X)\omega(Z)+g(X,\phi Y)\omega(Z)+\eta(Y)X\omega(Z)$
$\displaystyle-\eta(X)Y\omega(Z)-\eta(Y)\omega(\nabla^{*}_{X}PZ)+\eta(X)\omega(\nabla_{Y}^{*}PZ)$
$\displaystyle+\eta(Z)\eta(X)\omega(\phi Y)-\eta(Z)\eta(Y)\omega(\phi
X)-\eta(Y)\omega(Z)\tau(X)$
$\displaystyle+\eta(X)\omega(Z)\tau(Y)-((c+3)/4)\\{\overline{g}(Y,PZ)\theta(X)-\overline{g}(X,PZ)\theta(Y)\\}$
$\displaystyle-((c-1)/4)\\{\eta(X)\eta(PZ)\theta(Y)-\eta(Y)\eta(PZ)\theta(X)$
$\displaystyle+\overline{g}(\overline{\phi}Y,PZ)v(X)-\overline{g}(\overline{\phi}X,PZ)v(Y)-2\overline{g}(\overline{\phi}X,Y)v(PZ)\\}=0,$
(4.12)

for any $X,Y,Z\in\Gamma(TM)$. Next, setting $X=\xi$ in (4), we get

$\displaystyle(\xi\varphi)$
$\displaystyle\\{B(Y,Z)+\eta(Z)u(Y)+\eta(Y)u(Z)\\}-\varphi\\{2B(Y,Z)\tau(\xi)$
$\displaystyle+(3(c-1)/4)u(Y)u(Z)-\eta(Z)u(Y)\tau(\xi)-2\eta(Y)u(Z)\tau(\xi)$
$\displaystyle+3u(Y)u(Z)+\eta(Y)\eta(Z)\\}-u(Z)v(Y)+B(Y,U)\eta(Y)$
$\displaystyle-\eta(Z)v(Y)\tau(\xi)+2u(Y)\omega(Z)+\eta(Y)\xi\omega(Z)$
$\displaystyle-\eta(Y)\omega(\nabla^{*}_{\xi}PZ)-\eta(Y)\omega(Z)\tau(\xi)-((c+3)/4)g(Y,PZ)$
$\displaystyle-((c-1)/4)\\{u(PZ)v(Y)+2u(Y)v(PZ)\\}=0.$ (4.13)

Putting $Y=V$ and $Z=U$ in (4) gives

$\displaystyle\\{\xi\varphi-2\varphi\tau(\xi)\\}B(V,U)-\frac{1}{2}(c+3)=0.$
(4.14)

On the other hand, putting $Y=U$ and $Z=V$ in (4) gives

$\displaystyle\\{\xi\varphi-2\varphi\tau(\xi)\\}B(U,V)-\frac{3}{4}(c+3)=0.$
(4.15)

Also, putting $Y=Z=U$ in (4), we get

$\displaystyle\\{\xi\varphi-2\varphi\tau(\xi)\\}B(U,U)+\frac{3}{4}\varphi(c+3)+2\omega(U)=0.$
(4.16)

Finally, putting $Y=Z=V$ leads to

$\displaystyle\\{\xi\varphi-2\varphi\tau(\xi)\\}B(V,V)=0.$ (4.17)

From (4.14) and (4.15), we get

$\displaystyle
c=-3\;\;\;\;\mbox{and}\;\;\;\;\\{\xi\varphi-2\varphi\tau(\xi)\\}B(V,U)=0.$
(4.18)

As $M$ is contact screen conformal, (3.13) and (4.1) gives $B(U,U)=\varphi
B(V,U)$. Therefore, considering this relation in (4.16) together with (4.18),
we get $\omega(U)=0$. Note that, by (3.13) and (4.1), we have $B(V,U)=\varphi
B(V,V)$. Therefore, if $B(V,U)\neq 0$, then $B(U,U)\neq 0$ and $B(V,V)\neq 0$,
which all implies that $\xi\varphi-2\varphi\tau(\xi)=0$. This completes the
proof. ∎

As a consequence of Theorem 4.3, we have the following.

######  Corollary 4.4.

There exist no contact screen conformal null hypersurfaces of an indefinite
Sasakian space form $(\overline{M}(c\neq-3),\overline{\phi},\zeta,\eta)$ with
$\zeta\in\Gamma(TM)$.

######  Example 4.5.

In view of Example 4.2, both $B$ and $C$ vanishes on $D\oplus D^{\prime}$.
Hence, $(M,g)$ is a contact screen conformal null hypersurface with $\varphi$
arbitrary, and $\mathbb{R}_{2}^{7}$ is a space of constant
$\overline{\phi}$-sectional curvature $-3$.

######  Remark 4.6.

Geometrically, the proportionality of $B$ to the metric tensor $g$ on $D\oplus
D^{\prime}$ is equivalent to $(M,g)$ being totally contact umbilic in
$\overline{M}$. In fact, let $B=\lambda\otimes g$ on $D\oplus D^{\prime}$,
where $\lambda$ is a smooth function on $\mathcal{U}\subset M$. Then, by
(4.1), we have $\lambda
g(\tilde{P}X,\tilde{P}Y)=B(X,Y)+\eta(Y)u(X)+\eta(X)u(Y)$, for any
$X,Y\in\Gamma(TM)$. Thus, using $X=\tilde{P}X+\eta(X)\xi$ to this relation, we
get

$\displaystyle
B(X,Y)=\lambda\\{g(X,Y)-\eta(X)\eta(Y)\\}-\eta(Y)u(X)-\eta(X)u(Y).$ (4.19)

Clearly, (4.19) is the defining condition for $(M,g)$ to be totally contact
umbilic [6] null hypersurface. It has already been established that for such a
null hypersurface in an indefinite Sasakian space form
$(\overline{M}(c),\overline{\phi},\zeta,\eta)$, then $c=-3$ (see [10] for
details). On the other hand, we know that $C$ is generally not symmetric on
$S(TM)$. In fact, by (2.9), we have $C(X,Y)-C(Y,X)=\theta([X,Y])$, for any
$X,Y\in\Gamma(S(TM))$. Thus, $C$ is symmetric if and only if $S(TM)$ is an
integrable distribution on $M$. For this reason, if the screen local second
fundamental form $C$ is proportional to $g$ on $D\oplus D^{\prime}$, we get it
symmetric on $D\oplus D^{\prime}$ but not on $S(TM)$, and therefore, we do not
recover the equation for a totally contact screen umbilic null hypersurface
(see [10]) for which it is assumed to be symmetric on $S(TM)$. In fact, if
$C=\gamma\otimes g$ on $D\otimes D^{\prime}$, we see, from (4.2), that

$\displaystyle
C(X,PY)=\gamma\\{g(X,Y)-\eta(X)\eta(Y)\\}-\eta(Y)v(X)+\eta(X)\omega(Y),$
(4.20)

for any $X,Y\in\Gamma(TM)$. It follows from (4.20) and (3.12) that; if $C$ is
symmetric on $S(TM)$ then $\omega(Y)=C(\zeta,Y)=C(Y,\zeta)=-v(Y)$, for any
$Y\in\Gamma(S(TM))$, and thus the relation of totally contact screen umbilic
is recovered. In such a case, it has been proved that $c=-3$ and, in fact,
$(M,g)$ is totally contact screen geodesic, i.e., $\gamma=0$ (see [10]).
Therefore, in general, the proportionality of $C$ to $g$ on $D\oplus
D^{\prime}$ does not imply totally contact screen umbilic considered in [10].
Based on the above, we have the following general definition of a contact
screen umbilic null hypersurfaces.

######  Definition 4.7.

Let $(M,g)$ be a null hypersurface of an indefinite Sasakian manifold
$(\overline{M},\overline{\phi},\zeta,\eta)$, tangent to the structure vector
field $\zeta$. Then, $M$ will be called contact screen umbilic null
hypersurface if there exists a non-zero smooth function $\gamma$ on a
neighborhood $\mathcal{U}\subset M$ such that

$\displaystyle C(\tilde{P}X,\tilde{P}PY)=\gamma
g(\tilde{P}X,\tilde{P}Y),\;\;\;\;\forall\,X,Y\in\Gamma(TM).$ (4.21)

Equivalently, using (4.2) and (4.3), $M$ is contact screen umbilic if

$\displaystyle
C(X,PY)=\gamma\\{g(X,Y)-\eta(X)\eta(Y)\\}-\eta(Y)v(X)+\eta(X)\omega(Y),$
(4.22)

for all $X,Y\in\Gamma(TM)$. We say that $(M,g)$ is contact screen geodesic if
$\gamma=0$.

Accordingly, we have the following result.

######  Theorem 4.8.

Let $(M,g)$ be a contact screen umbilic null hypersurface of an indefinite
Sasakian space form $(\overline{M}(c),\overline{\phi},\zeta,\eta)$. Then,
$c=-3$ (that is; $\overline{M}$ is a space of constant
$\overline{\phi}$-sectional curvature $-3$). Moreover, $\gamma$ satisfies the
relations;

$\displaystyle\gamma^{2}-2\omega(U)=0,\;\;\;\gamma
B(V,V)=0\;\;\;\mbox{and}\;\;\;\xi\gamma-\gamma\tau(\xi)=0.$

Furthermore, if $C$ is symmetric on $S(TM)$, then $M$ is contact screen
geodesic.

###### Proof.

Using (4.22), (3.10), (3.11), (3.12), (2.14), we derive

$\displaystyle(\nabla_{X}$ $\displaystyle
C)(Y,PZ)-(\nabla_{Y}C)(X,PZ)=(X\gamma)\\{g(Y,Z)-\eta(Y)\eta(Z)\\}$
$\displaystyle-(Y\gamma)\\{g(X,Z)-\eta(X)\eta(Z)\\}+\gamma\\{B(X,PZ)\theta(Y)-B(Y,PZ)\theta(X)$
$\displaystyle+u(X)\theta(Y)\eta(Z)-u(Y)\theta(X)\eta(Z)+g(Y,\phi
X)\eta(Z)-g(X,\phi Y)\eta(Z)$ $\displaystyle+\eta(Y)g(Z,\phi
X)-\eta(X)g(Z,\phi Y)\\}+g(Z,\phi X)v(Y)-g(Z,\phi Y)v(X)$
$\displaystyle+\eta(Z)\theta(X)B(Y,U)-\eta(Z)\theta(Y)B(X,U)+\eta(Z)g(\nabla_{Y}U,X)$
$\displaystyle-\eta(Z)g(\nabla_{X}U,Y)+u(Y)\theta(X)\omega(Z)-u(X)\theta(Y)\omega(Z)$
$\displaystyle+g(X,\phi Y)\omega(Z)-g(Y,\phi
X)\omega(Z)+\eta(Y)X\omega(Z)-\eta(X)Y\omega(Z)$
$\displaystyle+\eta(X)\omega(\nabla_{Y}PZ)-\eta(Y)\omega(\nabla_{X}PZ),$
(4.23)

for all $X,Y,Z\in\Gamma(TM)$. Substituting (4) in (4.8) and using (4.22), we
get

$\displaystyle(X$
$\displaystyle\gamma)\\{g(Y,Z)-\eta(Y)\eta(Z)\\}-(Y\gamma)\\{g(X,Z)-\eta(X)\eta(Z)\\}$
$\displaystyle+\gamma\\{B(X,PZ)\theta(Y)-B(Y,PZ)\theta(X)+g(X,PX)\tau(Y)-g(Y,PZ)\tau(X)$
$\displaystyle+u(X)\theta(Y)\eta(Z)-u(Y)\theta(X)\eta(Z)+g(Y,\phi
X)\eta(Z)-g(X,\phi Y)\eta(Z)$ $\displaystyle+\eta(Y)g(Z,\phi
X)-\eta(X)g(Z,\phi Y)+\eta(Y)\eta(Z)\tau(X)-\eta(X)\eta(Z)\tau(Y)\\}$
$\displaystyle+g(Z,\phi X)v(Y)-g(Z,\phi Y)v(X)+\eta(Z)\theta(X)B(Y,U)$
$\displaystyle-\eta(Z)\theta(Y)B(X,U)+\eta(Z)g(\nabla_{Y}U,X)-\eta(Z)g(\nabla_{X}U,Y)$
$\displaystyle+u(Y)\theta(X)\omega(Z)-u(X)\theta(Y)\omega(Z)+g(X,\phi
Y)\omega(Z)$ $\displaystyle-g(Y,\phi
X)\omega(Z)+\eta(Y)X\omega(Z)-\eta(X)Y\omega(Z)+\eta(X)\omega(\nabla_{Y}PZ)$
$\displaystyle-\eta(Y)\omega(\nabla_{X}PZ)+\eta(Z)v(Y)\tau(X)-\eta(Z)v(X)\tau(Y)$
$\displaystyle+\eta(X)\omega(Z)\tau(Y)-\eta(Y)\omega(Z)\tau(X)-((c+3)/4)\\{\overline{g}(Y,PZ)\theta(X)$
$\displaystyle-\overline{g}(X,PZ)\theta(Y)\\}-((c-1)/4)\\{\eta(X)\eta(PZ)\theta(Y)-\eta(Y)\eta(PZ)\theta(X)$
$\displaystyle+\overline{g}(\overline{\phi}Y,PZ)v(X)-\overline{g}(\overline{\phi}X,PZ)v(Y)-2\overline{g}(\overline{\phi}X,Y)v(PZ)\\}=0.$
(4.24)

Letting $X=\xi$ in (4) leads to

$\displaystyle(\xi\gamma)\\{g(Y,Z)-\eta(Y)\eta(Z)\\}+\gamma\\{-B(Y,PZ)-g(Y,PZ)\tau(\xi)-u(Y)\eta(Z)$
$\displaystyle-\eta(Y)u(Z)+\eta(Y)\eta(Z)\tau(\xi)\\}-u(Z)v(Y)+\eta(Z)B(Y,U)-\eta(Z)g(\nabla_{\xi}U,Y)$
$\displaystyle+2u(Y)\omega(Z)+\eta(Y)\xi\omega(Z)-\eta(Y)\omega(\nabla_{\xi}PZ)+\eta(Z)v(Y)\tau(\xi)-\eta(Y)\omega(Z)\tau(\xi)$
$\displaystyle-((c+3)/4)g(Y,PZ)-((c-1)/4)\\{-\eta(Y)\eta(PZ)+u(PZ)v(Y)$
$\displaystyle+2u(Y)v(PZ)\\}=0,$ (4.25)

for any $Y,Z\in\Gamma(TM)$. Setting $Y=Z=U$ in (4) gives $-\gamma
B(U,U)+2\omega(U)=0$. But using (3.13) and (4.22), we have $B(U,U)=\gamma$.
Thus,

$\displaystyle-\gamma^{2}+2\omega(U)=0.$ (4.26)

On the other hand, putting $Y=V$ and $Z=U$ in (4), leads to

$\displaystyle\xi\gamma-\gamma\tau(\xi)-\frac{1}{2}(c+3)=0.$ (4.27)

With $Y=U$ and $Z=V$, (4) gives

$\displaystyle\xi\gamma-\gamma\tau(\xi)-\frac{3}{4}(c+3)=0.$ (4.28)

From (4.28) and (4.27), we have $c=-3$ and $\xi\gamma-\gamma\tau(\xi)=0$. Note
that $Y=Z=V$ gives $\gamma B(V,V)=0$. When $C$ is symmetric, we see from
(3.12) that $\omega(U)=C(\zeta,U)=C(U,\zeta)=-v(U)=0$. Thus, (4.26) gives
$-\gamma^{2}=0$ or simply $\gamma=0$, which shows that $(M,g)$ is contact
screen geodesic. Hence, the theorem is proved. ∎

From Theorem 4.8, the following hold.

######  Corollary 4.9.

There exist no contact screen umbilic null hypersurfaces of an indefinite
Sasakian space form $(\overline{M}(c\neq-3),\overline{\phi},\zeta,\eta)$ with
$\zeta\in\Gamma(TM)$.

######  Example 4.10.

The null hypersurface of Example 4.2 is contact screen geodesic, i.e.,
$\gamma=0$, and $\mathbb{R}_{2}^{7}$ is a space of constant
$\overline{\phi}$-sectional curvature $-3$.

## References

  * [1] C. Atindogbé, Scalar curvature on lightlike hypersurfaces, Balkan Society of Geometers, Applied Sciences, 11(2009), 9–18.
  * [2] D. E. Blair, Contact Manifolds in Riemannian Geometry, Lecture Notes in Math., No: 509, Springer Verlag, 1967.
  * [3] D. E. Blair, Riemannian Geometry of Contact and Symplectic manifolds, PM 203, Birkhauser, 2002.
  * [4] C. Calin, Contribution to geometry of CR-submanifolds, thesi. University of Iasi. Iasi, Romania, 1998.
  * [5] K. L. Duggal and A. Bejancu, Lightlike submanifolds of semi-Riemannian manifolds and applications, Kluwer Academic Publishers, 1996.
  * [6] K. L. Duggal and B. Sahin, Differential geometry of lightlike submanifolds. Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2010.
  * [7] D. H. Jin, Ascreen lightlike hypersurfaces of an indefinite Sasakian manifold, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. Volume 20, 1(2013), 25–35.
  * [8] D. H. Jin, Geometry of lightlike hypersurfaces of an indefinite sasakian manifold, Indian J. Pure Appl. Math., 41(4) (2010), 569–581.
  * [9] T. H. Kang, S. D. Jung, B. H. Kim, H. K. Pak, J. S. Pak, Lightlike hypersurfaces of indefinite Sasakian manifolds. Indian J. Pure Appl. Math. 34 (2003), No. 9, 1369–1380.
  * [10] F. Massamba, Totally contact umbilical lightlike hypersurfaces of indefinite Sasakian manifolds, KodaiMath.J. 31, (2008), 338–358.
  * [11] F. Massamba, Screen conformal invariant lightlike hypersurfaces of indefinite sasakian sapce forms, African Diaspora Journal of Mathematics, Volume 14, Number 2, (2012), 22–37.
  * [12] B. O’Neill, Semi-Riemannian Geometry, with Applications to Relativity. New York: Academic Press (1983).
  * [13] D. N. Kupeli, Singular semi-Riemannian geometry, Mathematics and Its Applications, Vol. 366, Kluwer Academic Publishers, 1996.


#  Patchy landscapes promote stability of small groups

Gianni Jacucci  Laboratoire Kastler Brossel, Sorbonne Université, Ecole
Normale Supérieure-Paris Sciences et Lettres (PSL) Research University, Centre
National de la Recherche Scientifique (CNRS) UMR 8552, Collège de France, 24
rue Lhomond, 75005 Paris, France  To whom correspondence should be addressed.
E-mail<EMAIL_ADDRESS>Davide Breoni  Institut für Theoretische
Physik II—Weiche Materie, Heinrich-Heine-Universität Düsseldorf,
Universitätsstraße 1, D-40225 Düsseldorf, Germany  Sandrine Heijnen
Department of Chemistry, University College London, 20 Gordon Street, London
WC1H 0AJ, United Kingdom  José Palomo  Laboratoire de Physique de l’Ecole
Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université
Paris Cité, F-75005 Paris, France  Philip Jones  Department of Physics $\&$
Astronomy, University College London, Gower Street, London WC1E 6BT, United
Kingdom  Hartmut Löwen  Institut für Theoretische Physik II—Weiche Materie,
Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, D-40225
Düsseldorf, Germany  Giorgio Volpe  Department of Chemistry, University
College London, 20 Gordon Street, London WC1H 0AJ, United Kingdom  Sylvain
Gigan  Laboratoire Kastler Brossel, Sorbonne Université, Ecole Normale
Supérieure-Paris Sciences et Lettres (PSL) Research University, Centre
National de la Recherche Scientifique (CNRS) UMR 8552, Collège de France, 24
rue Lhomond, 75005 Paris, France

###### Abstract

Group formation and coordination are fundamental characteristics of living
systems, essential for performing tasks and ensuring survival. Interactions
between individuals play a key role in group formation, and the impact of
resource distributions is a vibrant area of research. Using active particles
in a tuneable optical environment as a model system, we demonstrate that
heterogeneous energy source distributions result in smaller, more stable
groups with reduced individual exchange between clusters compared to
homogeneous conditions. Reduced group sizes can be beneficial to optimise
resources in heterogeneous environments and to control information flow within
populations. Devoid of biological complications, our system provides insights
into the importance of patchy landscapes in ecological dynamics and holds
implications for refining swarm intelligence algorithms and enhancing crowd
control techniques.

## Introduction

Group living is widespread across different levels of biological organisation
[1] and confers many advantages to individuals, such as increased survival
from predators [2, 3, 4], enhanced foraging efficiency [5, 6, 7], or improved
communication and decision making [8, 9, 10]. The formation, maintenance, and
disbanding of groups depend not only on the interactions among individuals but
also on the characteristics of the surrounding environment, such as spatial or
temporal heterogeneity [11]. For instance, while large group sizes are
advantageous for foraging in homogeneous environments, the presence of a
patchy landscape of resources compels individuals to adopt smaller group
formations as an adaptive strategy to ensure sufficient resources for all
members [12, 13, 14, 15]. However, the challenge of simultaneously monitoring
multiple individuals in natural ecosystems poses limits to our understanding
of the dynamics of such groups, particularly regarding their durability and
time stability [16, 17].

Recently, active colloids have emerged as a useful tool for studying
collective behaviour in living systems [18, 19]. These artificial systems,
unlike natural ones [20, 21, 22, 23], have the advantage of evolving on short
timescales. Moreover, they are tuneable, and their small sizes make data
acquisition more convenient. Previous studies have investigated the role of
individual interactions in group formation, such as those due to attraction
[24, 25, 26], repulsion [27], alignment [28, 29], reorientation [30, 31], or
vision [32, 33] and communication [34, 35]. The role of physical features in
the environment has also been explored, such as the presence of obstacles [36,
37, 38, 39, 40, 41] or of a disordered potential [42]. However, the impact
that a heterogeneous distribution of energy sources has on collective
phenomena is not clear yet. Indeed, while work exists that studied the impact
of linear gradients in the energy source [43, 44, 45], natural landscapes of
resources hardly follow linear gradients and the impact of patchiness on group
dynamics is largely unknown.

Here, we use phototactic active colloids moving in a spatially complex
distribution of resources, generated optically, to investigate the role of
patchy landscapes in the dynamics of group formation and cohesion. Our results
demonstrate that patchiness restricts the size of groups and increases their
stability by decreasing the exchange of individuals compared to a homogeneous
landscape.

## Results

Figure 1: Active particles in complex energy landscapes. a) Trajectory of an
active particle in a spatially complex distribution of energy, with high light
intensities in green. The particle self-propels, avoiding the most intense
illumination. The trajectory displays the particle’s centre position over 45
minutes. b) As the particles prefer to stay in the dark areas, the group
(defined as comprising more than two individuals) size appears to be
determined by the spatial characteristics of the
environment—$\sigma_{\text{e}}$, i.e. the average size of the energy patches.
The particles belonging to a group are coloured in pink while the isolated
particles and the particles in pairs are in grey.

As phototactic active particles, we used monodispersed silica colloids
(diameter $d=$4.77\pm 0.20\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) with a
$90\text{\,}\mathrm{nm}$ carbon half-coating (Methods). These particles self-
propel in a water-2,6-lutidine critical mixture when exposed to light. The
propulsion arises from the critical mixture’s local demixing, induced by the
carbon heating upon light absorption. [46, 47, 27, 43, 44, 45, 48].

We created an energy source landscape with controllable spatial heterogeneity
by shining a laser on an optical diffuser, generating a diffraction pattern of
randomly distributed light patches, whose intensity follows a negative
exponential probability distribution [49, 50].

Figure 2: Statistics of single particle trajectories on the energy landscape.
A complex energy landscape
($\sigma_{\text{e}}=$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) affects the
spatial occupancy and statistical features of particle trajectories. a)
Histogram comparing the distribution of the light intensity in the locations
visited by the particles with the intensity distribution of the energy
landscape. b) Particles tend to move perpendicular to the gradient of the
illumination ($\alpha=\pi/2$). $\alpha$, illustrated in the inset, is the
angle between the velocity of the particle (v) and the intensity gradient
($\mathbf{\nabla}I$). c) A lower mean squared displacement reflects avoidance
of high-energy areas compared to a more homogeneous environment (Gaussian
illumination, $\sigma_{\text{e}}\to\infty$). The inset shows the long-time
diffusive behaviour. Shaded areas indicate standard deviations on the mean
squared displacement. Data was obtained from averaging at least 20
trajectories, each lasting 45 minutes.

The energy density of the landscape and the typical size of its patches
($\sigma_{\text{e}}$) can be precisely controlled by optical means—Figure S1
and Methods for details. $\sigma_{\text{e}}$ is defined as the full width at
half maximum of the landscape’s autocorrelation. Figure 1a shows a typical
example of a particle trajectory in a complex illumination. What can be
qualitatively observed is that an active particle tends to navigate in the
energy landscape by spending most of its time in the low-illumination areas—in
other words, it performs negative phototaxis. This property of the individual
trajectories has a direct consequence on group formation, with a group being
defined as a collection of more than two particles in contact. As shown in
Figure 1b, in a heterogeneous energy landscape ($\sigma_{\text{e}}\simeq
d=$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$), active particles tend to form
smaller groups compared to the case of a homogeneous illumination
($\sigma_{\text{e}}\to\infty$). Under both illumination conditions, the
average local energy density
($I=$0.1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{W}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{-2}$$)
was kept the same by matching the incoming power and envelope size Figure S2.

Figure 3: Modeling the dynamics of active particles in complex energy
landscapes. A torque and spatial heterogeneities in the energy landscape
determine the dynamics of the active particles. Simulated trajectories of
individual active particles with a) different torque values ($\Omega$) for a
fixed $\sigma_{\text{e}}=$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, and b)
environment heterogeneity ($\sigma_{\text{e}}$), for a fixed $\Omega=6$. c) In
a complex environment
($\sigma_{\text{e}}=$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$), increasing
the torque causes the particles to avoid high-energy areas. d) In the presence
of a torque, particles move more perpendicular to the gradient
($\alpha=\pi/2$). e) The difference in the effective diffusion coefficients
between complex and Gaussian illuminations observed in the experiments is most
accurately described by $\Omega=0.6$. f) Role of the heterogeneity of the
energy landscape ($\sigma_{\text{e}}$) on the effective diffusion coefficient
($D_{\text{eff}}(\sigma_{e})$) for a fixed value of the torque ($\Omega=0.6$).
The effective diffusion coefficient, normalised to its value for a Gaussian
illumination, is minimised when the spatial variation of the landscape is
comparable to the size of the particles. All simulations averaged 150
particles over 50 minutes each. The standard deviation of the data is
represented in e) as shaded areas for experiments and vertical bars for
numerics, comparable with the size of the scatter plot points. In f), the
errors match the scatter point size.

In order to explain the phenomena illustrated in Figure 1, we first need to
better understand the statistics of the dynamics of the individual particles
on the underlying energy landscape.  Figure 2a shows quantitatively that
particles perform negative phototaxis. By comparing the light intensity at
locations visited by the particles along their trajectories with the overall
energy landscape, we can observe that particles tend to avoid the most intense
light regions. Delving deeper, Figure 2b shows that as active particles
navigate the darker, low-energy channels of the light landscape, they tend to
align perpendicular to the light intensity gradient. This is due to their
continuous interactions with the two-dimensional gradient, channelling their
trajectories between high-energy patches and causing them to move
perpendicular to the gradient—i.e. along low-intensity areas. This is
evidenced by the peak at the angle $\alpha=\pi/2$ in Figure 2b. Here, $\alpha$
represents the angle between the particle’s instantaneous velocity and the
local intensity gradient, as illustrated in the inset of Figure 2b. This
result extends beyond observations in one-dimensional light gradients, where
previous findings associated $\alpha=\pi$ ($\alpha=0$) with negative
(positive) phototaxis [45].

A complex energy landscape determines not only the preferential orientation of
the active colloids but also their speed. Both in homogeneous energy
landscapes [43, 44] and in one-dimensional light gradients [45] a linear
increase of the velocity with the light intensity was observed after a given
activation energy. This is also reproduced in two-dimensional gradients when
the illumination has a Gaussian profile ($\sigma_{\text{e}}\to\infty$ in
Figure S3a). In this case the local value of the gradient is relatively low,
making it comparable to the previously reported linear cases also in terms of
orientation with respect to the gradient ($\alpha=\pi$ in Figure S4a) [43, 44,
45]. However, when the heterogeneity of the landscape is comparable to the
size of the particles
($\sigma_{\text{e}}=$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ in Figure S3b)
the velocity does not show a monotonic increase with the light intensity. This
results from local values of the light gradient which are much stronger than
in the previous cases, giving rise to a complex dependency of the speed of the
particle on the energy landscape. Active particles do indeed experience a
torque proportional to the light gradient, which, in turn, is proportional to
the local light intensity. This torque causes them to change direction and
reduce speed near high-energy patches [43, 44, 45]. The aligning torque is a
consequence of breaking the axial symmetry of the velocity field around the
particle [51]. The torque-induced reorientation and avoidance of high-energy
zones reduce the mean squared displacement of active particles in a complex
landscape  when compared to a Gaussian illumination, as seen in Figure 2c. The
mean squared displacement, despite the decrease, exhibits a superdiffusive
transition followed by a long-time return to typical diffusion, as for typical
active colloids in both scenarios [19].

To disentangle the role of the torque (the dimensionless parameter $\Omega$,
Figure 3a) exerted by the field gradient and the heterogeneity of the energy
landscape ($\sigma_{\text{e}}$, Figure 3b) on the microscale dynamics, we
employed a particle-based model that includes an aligning torque proportional
to the gradient of the optical field (Methods) [43, 44, 45]. Figure 3c
confirms how the presence of a torque prevents particles from accessing the
high energy regions, with a cutoff in energy that decreases as the torque
increases. The emergence of this behaviour coincides also with a preferential
orientation of the particles’ velocity perpendicular to the gradient of the
illumination, whereas at $\Omega=0$ the velocity shows a relatively weak
dependence on the gradient (Figure 3d). Our model further validates that under
a two-dimensional Gaussian illumination, the preferred orientation relative to
the gradient ($\alpha=\pi$ in Figure S4b) is governed by the effect of the
torque.  Additionally, as depicted in Figure 3e, the discrepancy in the long-
term behavior of the mean squared displacement between
$\sigma_{\text{e}}=$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$\sigma_{\text{e}}\to\infty$ in Figure 2 can be attributed to the torque. In a
complex illumination, an increase in torque restricts access to regions of
higher energy, thereby reducing the mean squared displacement.  Importantly,
altering the heterogeneity of the environment (and thus the gradient value)
while keeping the torque constant also results in a lower mean squared
displacement, as shown in Figure 3f (represented as effective diffusion
coefficient). This finding highlights the significant influence of
environmental heterogeneity on particle dynamics— i.e. when
$\sigma_{\text{e}}$ is on a similar length scale to the particle size, it
significantly affects its long-time diffusion coefficient. Conversely, in the
asymptotic cases where energy variations occur on scales much smaller or
larger than the particle, differences on long-time diffusivity are less
marked.

To study the impact of the differences in individual trajectories on the
collective behaviour of active systems we performed experiments with a higher
concentration of particles ($\approx 4\%$, compared to $\approx 0.1\%$ in
Figure 2). When the concentration of active particles increases, they are more
likely to collide with each other, slowing down and forming groups. A group
lasts until the propulsion direction of one of the individuals points outside
the cluster and its speed is such that it can escape the group by winning over
the short-range attraction among particles [27, 26].

Figure 4: Group formation analysis. Comparison of group formation in energy
landscapes with different spatial characteristics, i.e. different sizes of the
energy patches ($\sigma_{\text{e}}$).  a) Experimental trajectories showing an
example of group aggregation and fragmentation in a complex landscape with
$\sigma_{\text{e}}=$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. When the
patchiness of the landscape is comparable to the dimension of the particles,
it reduces both the rate of individual exchange between them and the group
size, b) and c), respectively. In both panels of c), the insets show the
distribution of sizes for different spatial heterogeneities. In b), the error
bars represent the standard deviation of the data.

A heterogeneous energy landscape plays a key role in the formation of groups.
As illustrated in Figure 4a, particles tend to aggregate, and eventually
separate, in the dark regions of the optical landscape where their motility is
lower. In complex energy landscapes, groups exhibit greater durability, with
the average rate of aggregation and fragmentation events minimised when the
spatial heterogeneity matches the individual particle scale (Figure 4b). The
torque does indeed play a dual role in maintaining group stability. Firstly,
it makes encounters between individuals more difficult by causing particles to
linger in low energy regions for longer periods of time, which reduces the
rate of aggregation. Secondly, once groups have formed, the torque orients
particles toward areas of low energy, causing them to face inward the cluster
and reducing the likelihood of fragmentation. The importance of the torque is
further confirmed by Figure S5a, where decreasing it leads to i) an overall
decrease of the rates and ii) a smaller difference in rates between
homogeneous and complex energy landscapes.

The patchiness of the landscape also has a significant impact on group size,
as demonstrated in Figure 4c. As individuals display avoidance behaviour
towards high energy regions, groups tend to grow until they fill the space
confined between adjacent high-energy patches. The data presented in Figure 4c
support this qualitative relationship, revealing that complex energy
landscapes with heterogeneity comparable to the particle size lead to a
reduction in the group dimension. The interplay between the dimension of
energy patches and the group size is strongly influenced by the torque exerted
by the gradient of resources. Decreasing the torque provides colloids with
more freedom to move, enabling larger groups to form, as depicted in Figure
S5a.

Notably, the differences in group dynamics depicted in Figure 4 cannot be
solely attributed to a lower average motility of active particles in complex
landscapes. Even when the mean squared displacement in a homogeneous energy
landscape is reduced to match that in patchy ones (as shown in Figure S5b),
disparities persist in both group dimension and stability. This further
demonstrates that the spatial complexity of the energy landscape plays a
critical role in shaping group dynamics.

## Discussion

In summary, our study investigated how heterogeneous energy landscapes affect
group formation using active particles in optical fields as a model system. We
found that by manipulating the spatial complexity of the landscape, we can
control the size of groups and the exchange of individuals between them. Our
findings offer important biological perspectives. While smaller groups
optimise resource allocation in patchy environments, their increased stability
and reduced individual exchange might compromise genetic diversity. Yet, such
limited interchange could also curtail the spread of diseases and parasites
among groups and the transmission of antibiotic resistance in bacteria [52].
As our experimental approach provides a versatile and precise method to
modulate the spatial dynamics of the energy landscape using light—with a
natural extension to explore its temporal properties—it could be instrumental
in studying search strategies in the presence of limited resources. Its
applicability could span various contexts, ranging from animals engaged in
foraging or migration to guiding robots through complex search-and-rescue
missions.

## Methods

### Materials

Glass capillaries were purchased from CM Scientific (5005-050 and 5001-050 for
single particle and cluster experiments, respectively). The lutidine was
purchased and used as received: 2,6-lutidine ($\geq$ 99%, Sigma-Aldrich).
Deionised (DI) water ($\geq$
$18\text{\,}\mathrm{M}\Omega\cdot\mathrm{c}\mathrm{m}$) was collected from a
Milli-Q purification system. Aqueous colloidal dispersions (5% w/v) of silica
(SiO2) colloids, used for sample preparation, were purchased from
Microparticles GmbH. Additionally, carbon rods measuring
$300\text{\,}\mathrm{mm}$ in length and $6.15\text{\,}\mathrm{mm}$ in diameter
were procured from Agar Scientific and subsequently cut to a length of
$50\text{\,}\mathrm{mm}$ and tapered for the coating of Janus particles
through sputtering.

### Particles fabrication

Janus particles were synthesised from SiO2 colloids with a diameter of
$d=$4.77\pm 0.20\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. Initially, a
monolayer of colloids was deposited on a clean glass slide by evaporating a
$40\text{\,}\mathrm{\SIUnitSymbolMicro L}$ droplet containing a 2.5% w/v
dispersion of the colloids in DI water. Subsequently, an automatic carbon
coater (AGB7367A, Agar Scientific) was employed to coat the particles with a
$90\text{\,}\mathrm{nm}$ thick layer of carbon. The thickness of the carbon
layer was confirmed through atomic force microscopy (AFM) measurements. After
coating, sonication was performed to dislodge the half-coated particles in DI
water from the glass slide, facilitating their use for sample preparation. To
reduce the interaction between particles, and the sticking of the particles to
the substrate, the Janus colloids were functionalised with bovine serum
albumin (BSA, Sigma Aldrich). This was done by replacing the water solvent of
the colloidal suspension with a 1$\%$w/v water solution of BSA. Prior to
utilisation, the colloidal dispersions underwent centrifugation at 1000
relative centrifugal force (RCF) for 3 minutes, resulting in the formation of
a pellet. The supernatant was subsequently discarded and replaced with a
solution containing 28.6$\%$ w/v of water and 2,6-lutidine. This purification
procedure was repeated three times to ensure the removal of any remaining BSA
water solution from the original dispersion.

### Sample preparation

In single-particle experiments (Figure 1 and Figure 2), a suspension of Janus
particles was confined in a rectangular capillary with a width of
$700\text{\,}\mathrm{\SIUnitSymbolMicro m}$, a length of
$50\text{\,}\mathrm{mm}$, and a thickness of
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The capillary was then sealed by
applying a UV-curable glue to avoid evaporation and drifts. In the multiple-
particles systems (Figure 4), we reduced the sample’s thickness by using a
capillary with a width of $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$, a
length of $50\text{\,}\mathrm{mm}$, and a thickness of
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ to restrict the motion of the
colloids to two dimensions.

### Experimental setup

Figure S1 shows a schematic of the experimental setup. A laser (Oxxius
$532\text{\,}\mathrm{nm}$, $300\text{\,}\mathrm{mW}$ of maximum output power)
and a diffuser ($1\text{\,}\mathrm{\SIUnitSymbolDegree}$, Newport 10DKIT-C1)
are used to illuminate the Janus particles with a random optical field, also
known as speckle. The laser is directed to the sample via two mirrors (M1, M2)
and a 4f-lens configuration (L1=L2, Thorlabs, LA1461-A-ML) illuminating the
back aperture of a 10x objective (Nikon, N10X-PF, NA=0.3). By changing the
position of the diffuser and that of the sample, it is possible to control
both the dimension of the speckle grains and the size of the illumination
envelope. To create a Gaussian illumination, we removed the diffuser and one
lens (L1) to image the beam on the sample. The illumination envelope was fixed
to be close to the acquisition area of the camera ($\approx$ $200\times
200\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}$ and $\approx$ $100\times
100\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}$ in the single and multiple
particles experiments, respectively) and it was kept the same for different
patchiness of the landscape (Figure S2). The intensity of the laser—controlled
by a half-waveplate (FOCtek, WPF212H) and a polarising beam splitter (FOCtek,
BSC1204) was then adjusted to have a comparable energy density in the
different optical landscapes, i.e. different values of $\sigma_{\text{e}}$.
The active particles were imaged with a 20x objective (Nikon, N20X-PF, NA=0.5)
and a tube lens (L3, Thorlabs, LA1805-A-ML) on a camera (Basler,
acA5472-17um). A white LED (Thorlabs, MWWHF2), for simplicity not depicted in
Figure S1, coupled to a fibre (Thorlabs, M28L01) is used for imaging. The
particles’ dynamics were analysed by reconstructing their trajectories from
videos (typically of the duration of 50 minutes at 2 frame per second (fps))
using a homemade code based on the python package Trackpy [53, 54]. The
position of the sample was controlled in three dimensions using an xyz-stage
(Thorlabs, RB13M/M). The z-coordinate was adjusted with a stepper motor
(Thorlabs, ZFS25B). For all experiments, the suspension of Janus particles was
kept close to the critical temperature of the water-lutidine mixture
($T_{\text{exp}}\simeq$31\text{\,}\mathrm{\SIUnitSymbolCelsius}$$, $\Delta
T\simeq$3\text{\,}\mathrm{\SIUnitSymbolCelsius}$$) by a heater driven via a
temperature controller (Thorlabs, HT19R and TC200, respectively).

### Numerical model

We numerically implemented an Euler-Maruyama time integrator for the evolution
of the positions $\textbf{r}_{i}$ and orientations $\phi_{i}$ of each particle
$i$ in two dimensions, defined by the following equations, based on [45]:

$\displaystyle\dot{\textbf{r}}_{i}(t)$
$\displaystyle=v(\textbf{r}_{i})\textbf{n}_{i}-\frac{1}{\gamma^{\prime}}\sum_{j\neq
i}\nabla
V\left(\left|\textbf{r}_{i}-\textbf{r}_{j}\right|\right)+\sqrt{2D}\boldsymbol{\xi}_{i}(t),$
(1) $\displaystyle\dot{\phi}_{i}(t)$
$\displaystyle=\omega(\phi_{i},\textbf{r}_{i})+\sqrt{2D_{r}}\xi_{i}^{\phi}(t),$
(2)

where $\textbf{n}_{i}=(\cos(\phi),\sin(\phi))$, $\boldsymbol{\xi}_{i}$ and
$\xi_{i}^{\phi}$ represent Gaussian white noise. The translational and
rotational diffusion coefficients $D$ and $D_{\text{r}}$ are defined as

$D=\frac{k_{B}T}{\gamma^{\prime}},\quad
D_{\text{r}}=\frac{k_{B}T}{\beta^{\prime}},$ (3)

where $\gamma^{\prime}=\frac{16}{3}\pi\eta d$ and
$\beta^{\prime}=\frac{8}{7}\pi\eta d^{3}$ are respectively the translational
and rotational friction coefficient, corrected by taking into account a
distance from the substrate $s=d/2$ [41], and $\eta=$2.1\times
10^{-3}\text{\,}\mathrm{Pa}\text{\,}\mathrm{s}$$ is the viscosity of water-
lutidine. The torque $\omega$, which is defined as:

$\omega(\phi_{i},\textbf{r}_{i})=\frac{{\color[rgb]{0,0,0}\Omega
d}}{D}v(\textbf{r}_{i})\left[\nabla
v(\textbf{r}_{i})\times\textbf{n}_{i}\right]\cdot\hat{e}_{z},$ (4)

where the adimensional prefactor $\Omega$ determines whether the torque steers
the particle towards the light ($\Omega<0$) or away from it ($\Omega>0$). The
value of $\Omega$ that best fits the experimental change in the long-time mean
squared displacement is $\Omega=0.6$ (Figure 3e). $V$ is a 6-12 Lennard-Jones
potential with cutoff distance at five colloid diameters $d$ and depth
$\epsilon\simeq 8k_{\text{B}}T$. This value was chosen by fitting the
experimental cluster aggregation and fragmentation rates for a Gaussian
illumination. The intensity speckle field was generated by scaling the Fourier
transform of a white noise map to match the specific characteristics of the
optical fields observed in experiments: average intensity per unit of area and
grain size. This intensity field was then translated into a motility field
using the results from [45]. The gradient of the motility field acting on the
particle $\nabla v(\vec{r}_{i})$ was averaged over the square area where the
particle is inscribed.

### Cluster analysis

We defined the clusters by measuring all distances among particles and then
creating bonds between all the particles with distances smaller than the
cutoff radius 1.25d. Similarly to Ref. [26, 41], clusters made of 2 particles
(i.e. dimers) were disregarded in the analysis as they are unstable in time.
We determined whether a cluster survives from one timestep to the next by
confronting its particles with those of all the clusters in the next timestep.
We consider any of these clusters to be the same as a previous one if it
satisfies two properties: i) that more than half of its particles are shared
with the old cluster (i.e. the cluster has not decreased more than half in
size); ii) that at least half of the particles of the new cluster are also
present in the old one, i.e. the cluster has not grown more than double in
size. We calculated mainly two properties regarding clusters for each
experiment: the cluster size and the monomer acquisition/loss rate. The first
is simply the average number of monomers for each cluster, while the second is
defined as the number of monomer acquisition/loss events (i.e. the number of
times clusters receive/lose a single monomer during the experiment) normalised
by the total experiment time and the average number of particles.

## Data availability

All data needed to evaluate the conclusions in the paper are present in the
paper and/or the Supplementary Materials.

## Acknowledgements

The authors thank Dr. Raphaël Jeanneret and Dr. Jean Francois Allemand for
suggesting the use of bovine serum albumin to reduce particles attraction and
Alessia Gentili for insightful discussions on cluster analysis. This project
has received funding from the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement No
945298-ParisRegionFP, GJ is a fellow of the Paris Region Fellowship Programme
supported by the Paris Region. The work of HL was supported by the German
Research Foundation (project LO 418/29-1). DB and SH acknowledge support by
the EU MSCA-ITN ActiveMatter (Proposal No. 812780).

## Author Contributions

Author contributions are defined based on the CRediT (Contributor Roles
Taxonomy) and listed alphabetically. Conceptualization: SG, GJ, HL, GV. Data
curation: DB, GJ. Formal analysis: DB, GJ. Resources: SG, SH, PJ, HL, JP, GV.
Writing: GJ wrote the manuscript with the help of all authors. Funding
acquisition: SG, GJ, HL, GV.

## Conflict of interest

The authors declare that they have no competing financial interests.

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newPage

Figure S1: Experimental setup. A laser and a diffuser are used to generate a
spatially complex energy landscape for active particles. The laser is directed
to the sample via two mirrors (M1, M2) and a 4f-lens configuration (L1=L2)
illuminating the back aperture of an objective (obj. 1). Active particles are
imaged via a second objective (obj. 2), a tube lens (L3) and a camera (CMOS).
The trajectories are then reconstructed using a custom software. The speckle
grain size and envelope can be adjusted by changing the longitudinal position
of the diffuser with respect to the sample. The laser power is controlled by a
half-waveplate ($\lambda$/2) and a polarising beamsplitter (PBS).  Figure S2:
Illumination profile. a) Three representative illumination patterns for
different speckle grain sizes ($\sigma_{e}$). b) Radially averaged intensity
profile as a function of distance, based on images in a). The illumination
profiles share a common envelope, indicating that the active particles
experience the same energy density across different energy landscapes. Figure
S3: Velocity as a function of light intensity. a) In a Gaussian illumination
($\sigma_{\text{e}}\to\infty$), where the gradient is small, there is a
monotonic increase in velocity with light intensity. b) In contrast, in a
complex case ($\sigma_{\text{e}}=$5\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$)—where the local gradient is much larger and linearly dependent on the
light intensity—the speed shows a weak dependence on the intensity. This
arises as a result of the torque exerted by the gradient, causing particles to
change direction and therefore reducing their speed near high-intensity
regions. Figure S4: Orientation of an active particle in a Gaussian energy
landscape. When moving in a Gaussian energy landscape
($\sigma_{\text{e}}\to\infty$)—i.e. in the presence of a light gradient on a
scale much larger than their size—active particles tend to align
preferentially antiparallel to the global gradient of the illumination
($\alpha=\pi$). a) experiments, b) numerics.  Figure S5: Effect of torque and
lower motility on group formation. Numerical simulations show that a smaller
torque ($\Omega=0.1$ vs $\Omega=0.6$) leads to a less marked difference in
rates a) and group size b) between complex
($\sigma_{\text{e}}=$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) and Gaussian
motility fields (Gaussian). In order to investigate whether the difference in
group dynamics was solely influenced by the lower average motility of active
particles in complex illumination, simulations were conducted in a homogeneous
motility field with an identical average motility to that of the inhomogeneous
scenario—achieved by reducing the light intensity (Gaussian, low I). However,
reducing the average motility does not bridge the gap, in terms of both rates
and sizes, c) and d), respectively, between homogeneous and complex
illuminations.