# Remarks on screen integrable null hypersurfaces in Lorentzian manifolds

Samuel Ssekajja  *  School of Mathematics
University of Witwatersrand
Private Bag 3, Wits 20150
South Africa<EMAIL_ADDRESS>
###### Abstract.

In the present paper, we show that the geometry of a screen integrable null
hypersurface can be generated from an isometric immersion of a leaf of its
screen distribution into the ambient space. We prove, under certain geometric
conditions, that such immersions are contained in semi-Euclidean spheres or
hyperbolic spaces, and the underlying null hypersurfaces are necessarily
umbilic and screen totally umbilic. Where necessary, examples have been given
to illustrate the main ideas.

###### Key words and phrases:

Screen integrable null hypersurfaces, Normal curvature tensor of a leaf

###### 2010 Mathematics Subject Classification:

Primary 53C25; Secondary 53C40, 53C50

##  1\. Introduction

A study of null submanifolds in semi-Riemannian manifolds was introduced by
Duggal-Benjancu [4] and later updated by Duggal-Sahin [5]. In the above books,
the authors laid a foundation for research on null geometry by constracting
their structural equations, among other results. In fact, they introduced a
non-degenerate screen distribution to construct a null transversal vector
bundle which is non-intersecting to its null tangent bundle and developed
local geometry of null curves, hypersurfaces and submanifolds. Other pioneers
of the theory include D. N. Kupeli [12]–whose approach is purely intrinsic
compared to that of [4, 5]. Since then, many researchers including but not
limited to; [1, 3, 7, 8], have researched on null submanifolds and many
interesting results have been obtained. Null hypersurfaces appears in general
relativity as models of different types of black hole horizons (see [4, 5] for
details) and their theory is quite fundamental to modern mathematical physics.

Among the most studied null hypersurfaces are those with an integrable screen
distribution, and they are commonly known us screen integrable null
hypersurfaces. They include the well-known screen conformal ones, among
others. It was shown in [6], that all screen integrable null hypersurfaces are
locally isometric to $\mathcal{C}_{\xi}\times M^{*}$, where
$\mathcal{C}_{\xi}$ is a null curve tangent to the normal bundle of the
hypersurface and $M^{*}$ is a leaf of its screen distribution. In particular,
[4] proves that a null cone of an $(n+2)$-dimensional Lorentzian space
$\mathbb{R}_{1}^{n+2}$ is screen conformal, satisfying the above structure,
with $M^{*}\cong\mathbb{S}^{n}$. Under some geometric conditions on the
ambient space, Duggal-Sahin [5] also proves that a screen conformal Einstein
null hypersurface is locally a triple product $\mathcal{C}_{\xi}\times
M^{*}_{\alpha}\times M^{*}_{\beta}$, where $M^{*}_{\alpha}$ and
$M^{*}_{\beta}$ are some leaves of its screen distribution (see Theorem 2.5.17
of [5] for more details). In the book [4], Duggal and Bejancu tries to
understand the geometry of a null hypersurface $M$ from a leaf $M^{*}$ of its
screen distribution as an immersion in the ambient space. He, in fact, shows
that an umbilic leaf in the ambient space implies that the underlying null
hypersurface is umbilic too (see Proposition 5.1 of [4, p. 107]). A natural
question then arises; Which other geometric information, about the null
hypersurface, can be derived from the geometry of an isometric immersion of a
leaf of its screen distribution into the umbient space?

The main aim of this paper is to give some solutions to the above question by
studying null hypersurfaces of Lorentzian spaces. Consequently, we prove two
main theorems in that line; Theorems 3.6 and 3.10. The paper is arranged as
follows; In Section 2, we quote some basic notions needed in the rest of the
paper. In Section 3, we prove several characterization results.

##  2\. Preliminaries

Suppose $M$ is an $(n+1)$-dimensional smooth manifold and
$F:M\longrightarrow\overline{M}$ a smooth mapping such that each point $x\in
M$ has an open neighborhood $\mathcal{U}$ for which $F$ restricted to
$\mathcal{U}$ is one-to-one and $F^{-1}:F(\mathcal{U})\longrightarrow M$ are
smooth. Then, we say that $F(M)$ is an immersed hypersurface of
$\overline{M}$. If this condition globally holds, then $F(M)$ is called an
embedded hypersurface of $\overline{M}$, which we assume in this paper. The
embedded hypersurface has a natural manifold structure inherited from the
manifold structure on $\overline{M}$ via the embedding mapping. At each point
$F(x)$ of $F(M)$, the tangent space is naturally identified with an
$(n+1)$-dimensional subspace $T_{F(x)}M$ of the tangent space
$T_{F(x)}\overline{M}$. The embedding $F$ induces, in general, a symmetric
tensor field, say $g$, on $F(M)$ such that
$g(X,Y)|_{x}=\overline{g}(F_{*}X,F_{*}Y)|_{F(x)}$, for all $X,Y\in T_{x}M$.
Here, $F_{*}$ is the differential map of $F$ defined by
$F_{*}:T_{x}M\longrightarrow T_{F(x)}\overline{M}$ and
$(F_{x}X)\omega=X(\omega\circ F)$, for an arbitrary smooth function $\omega$
in a neighborhood of $F(x)$ of $F(M)$. Henceforth, we write $M$ and $x$
instead of $F(M)$ and $F(x)$. Due to the causal character of three categories
(spacelike, timelike and lightlike) of the vector fields of $\overline{M}$,
there are three types of hypersurfaces $M$, namely, Riemannian, semi-
Riemannian and null (or lightlike) and $g$ is a non-degenerate or a degenerate
symmetric tensor field on $M$ according as $M$ is of the first two types and
of the third type, respectively. The geometry of Riemannian or semi-Riemannian
hypersurfaces is well-known and has received a considerable attention, for
example see [10] and many more references cited therein. In the present paper,
we focus on null hypersurfaces using the approach of Duggal-Bejancu [4].

Now, let $g$ be degenerate on $M$. Then, there exists a nonzero vector field
$\xi$ on $M$ such that $g(\xi,X)=0$, for all $X\in\Gamma(TM)$. The radical or
the null space [10, p. 53] of $T_{x}M$, at each point $x\in M$, is a subspace
$\mathrm{Rad}\,T_{x}M$ defined by $\mathrm{Rad}\,T_{x}M=\\{\xi\in
T_{x}M:g_{x}(\xi,X)=0,\;\;\forall\,X\in T_{x}M\\}$, whose dimension is called
the nullity degree of $g$ and $M$ is called a null hypersurface of
$\overline{M}$. It follows that $T_{x}M^{\perp}$ is also null and satisfy
$\mathrm{Rad}\,T_{x}M=T_{x}M\,\cap\,T_{x}M^{\perp}$. For a hypersurface $M$
$\dim(T_{x}M^{\perp})=1$, implies that $\dim(\mathrm{Rad}\,T_{x}M)=1$ and
$\mathrm{Rad}\,T_{x}M=T_{x}M^{\perp}$. We call $\mathrm{Rad}\,TM$ a radical
(null) distribution of $M$. Thus, for a null hypersurface $M$, $TM$ and
$TM^{\perp}$ have a nontrivial intersection and their sum is not the whole of
tangent bundle space $T\overline{M}$ . In other words, a vector of
$T_{x}\overline{M}$ cannot be decomposed uniquely into a component tangent to
$T_{x}M$ and a component of $T_{x}M^{\perp}$. Therefore, the standard text-
book definition of the second fundamental form and the Gauss-Weingarten
formulas do not work, in the usual way, for the null case.

To overcome the above difficulty, Duggal-Bejancu [4] introduced an approach to
null geometry, which we follow in this paper. The approach consists of fixing,
on the null hypersurface, a geometric data formed by a null section and a
screen distribution. By screen distributionon of $M$, we mean a complementary
bundle of $TM^{\perp}$ in $TM$. It is then a rank $n$ non-degenerate
distribution over $M$. In fact, there are infinitely many possibilities of
choices for such a distribution provided the hypersurface $M$ is paracompact,
but each of them is canonically isomorphic to the factor vector bundle
$TM/TM^{\perp}$ [12]. We denote by $S(TM)$ the screen distribution over $M$.
Then we have the decompostion $TM=S(TM)\perp TM^{\perp}$, where $\perp$
denotes the orthogonal direct sum. From [4] or [5], it is known that for a
null hypersurface equipped with a screen distribution, there exists a unique
rank $1$ vector subbundle $tr(TM)$ of $T\overline{M}$ over $M$, such that for
any non-zero section $\xi$ of $TM^{\perp}$ on a coordinate neighborhood
$\mathcal{U}\subset M$, there exists a unique section $N$ of $tr(TM)$ on
$\mathcal{U}$ satisfying $\overline{g}(N,\xi)=1$,
$\overline{g}(N,N)=\overline{g}(N,W)=0$, for all
$W\in\Gamma(S(TM)|_{\mathcal{U}}).$ It then follows that
$T\overline{M}|_{M}=S(TM)\perp\\{TM^{\perp}\oplus tr(TM)\\}=TM\oplus tr(TM),$
where $\oplus$ denote the direct (non-orthogal) sum. We call $tr(TM)$ a (null)
transversal vector bundle along $M$. Throughout the paper, all manifolds are
supposed to be paracompact and smooth. We denote by $\mathcal{F}(M)$ the
algebra of differentiable functions on $M$ and by $\Gamma(E)$ the
$\mathcal{F}(M)$-module of differentiabale sections of a vector bundle $E$
over $M$. We also assume that all associated structures are smooth.

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 [4]

$\displaystyle\overline{\nabla}_{X}Y$ $\displaystyle=\nabla_{X}Y+B(X,Y)N,$
(2.1) $\displaystyle\overline{\nabla}_{X}N$ $\displaystyle=-A_{N}X+\tau(X)N,$
(2.2) $\displaystyle\nabla_{X}PY$ $\displaystyle=\nabla^{*}_{X}PY+C(X,PY)\xi,$
(2.3) $\displaystyle\nabla_{X}\xi$
$\displaystyle=-A^{*}_{\xi}X-\tau(X)\xi,\;\;\;A_{\xi}^{*}\xi=0,$ (2.4)

for all $X,Y\in\Gamma(TM)$, $\xi\in\Gamma(TM^{\perp})$ and
$N\in\Gamma(tr(TM))$. 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 $X,Y\in\Gamma(TM)$. Moreover,
$\overline{g}(A^{*}_{\xi}X,N)=0$ and $\overline{g}(A_{N}X,N)=0$, for all
$X\in\Gamma(TM)$. From the above relations, we notice that $A_{\xi}^{*}$ and
$A_{N}$ are both screen-valued operators.

The null hypersurface $M$ is said to be totally umbilic [4] if $B=\rho\otimes
g$, 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
$C=\varrho\otimes g$, 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. The mean curvature vector $H$ of a null hypersurface
is transverssal to $M$, and given by
$H=\frac{1}{n}(\mathrm{trace}_{S(TM)}B)N=\frac{1}{n}(\mathrm{trace}_{S(TM)}A_{\xi}^{*})N$.
We say that $M$ is a minimal null hypersurface if $H=0$. More precisely, $M$
is minimal if $\mathrm{trace}_{S(TM)}A_{\xi}^{*}=0$ (see [4, 5] for more
details and examples).

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
$(\nabla_{X}g)(Y,Z)=B(X,Y)\eta(Z)+B(X,Z)\eta(Y)$, 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 $\overline{R}$ the curvature
tensor of the connection $\overline{\nabla}$. Using the Gauss-Weingarten
formulae (2.1)-(2.4), we obtain the following curvature relations (see details
in [4, 5]).

$\displaystyle\overline{g}(\overline{R}(X,Y)\xi,N)=$ $\displaystyle
C(Y,A_{\xi}^{*}X)-C(X,A_{\xi}^{*}Y)-2d\tau(X,Y),$ (2.5)

where $2d\tau(X,Y)=X\tau(Y)-Y\tau(X)-\tau([X,Y])$, for all
$X,Y\in\Gamma(TM)|_{\mathcal{U}}$, $\xi\in\Gamma(TM^{\perp})$ and
$N\in\Gamma(tr(TM))$.

Suppose $\pi$ is a non-degenerate plane of $T_{p}\overline{M}$, for
$p\in\overline{M}$. Then, the associated matrix $G_{p}$ of $\overline{g}_{p}$,
with respect to an arbitrary basis $\\{u,v\\}$, is of rank 2 given by (1.2.15)
of [5, p. 16]. Define a real number $K(\pi)=K_{p}(u,v)=\overline{R}(u,v,v,u)$,
where $\overline{R}(u,v,v,u)$ is the 4-linear mapping on $T_{p}\overline{M}$
by the curvature tensor. The smooth function $K$, which assigns to each non-
degenerate tangent plane $\pi$ the real number $K(\pi)$ is called the
sectional curvature of $\overline{M}$, which is independent of the basis
$\\{u,v\\}$. If $K$ is a constant $c$ at every point of $p\in\overline{M}$
then $\overline{M}$ is of constant sectional curvature $c$, denote by
$\overline{M}(c)$, whose curvature tensor field $\overline{R}$ is given by
$\overline{R}(X,Y)Z=c\\{\overline{g}(Y,Z)X-\overline{g}(X,Z)Y\\}$, for any
$X,Y,Z\in\Gamma(T\overline{M})$ (see [10] for details). In particular, if
$K=0$, then $\overline{M}$ is called a flat manifold for which
$\overline{R}=0$.

##  3\. Geometry of $(M,g)$ from that of a leaf of $S(TM)$

Assume that $(M,g)$ is a screen integrable null hypersurface of a Lorentzian
manifold $(\overline{M},\overline{g})$. Let $M^{*}$ be a (Riemannian)leaf of
its screen distribution $S(TM)$. Let $f:M^{*}\longrightarrow\overline{M}$ be
an isometric immersion of $M^{*}$ in $\overline{M}$, as a codimension 2
nondegenerate submanifold, then (2.1) and (2.3) gives the Gauss formula of
$M^{*}$ (in $\overline{M}$) as

$\displaystyle\overline{\nabla}_{X}Y=\nabla^{*}_{X}Y+C(X,Y)\xi+B(X,Y)N,\;\;\;\forall\,X,Y\in\Gamma(TM^{*}).$
(3.1)

It is obvious from (3.1) that the second fundamental form $h^{*}$ of $M^{*}$,
as a submanifold of $\overline{M}$, is given by
$h^{*}(X,Y)=C(X,Y)\xi+B(X,Y)N$. Next, denote by $\nabla^{*\perp}$ the normal
connection on the normal bundle $TM^{*\perp}$. Then, the Weingarten formula
for $M^{*}$ is given by

$\displaystyle\overline{\nabla}_{X}V=-A_{V}X+\nabla^{*\perp}_{X}V,\;\;\;\forall\,X\in\Gamma(TM^{*}),\;\;V\in\Gamma(TM^{*\perp}),$
(3.2)

where $A_{V}$ denotes the shape operator of $M^{*}$. Since
$TM^{*\perp}=TM^{\perp}\oplus tr(TM)$, we let $V=a\xi+bN$, such that $a,b\neq
0$. Then, it is easy to see that $W=a\xi-bN$ is another vector field of
$TM^{*\perp}$ which is orthogonal to $V$. From now on, we consider
$TM^{*\perp}$ spanned by $V$ and $W$. Putting all the above into account, we
can express the shape operator $A_{V}$ of $M^{*}$ in terms of the shape
operators $A_{\xi}^{*}$ and $A_{N}$ as follows.

######  Lemma 3.1.

Let $f:M^{*}\longrightarrow\overline{M}$ an isometric immersion such that
(3.1) and (3.2) holds. Then, the shape operator of $M^{*}$ satisfies
$A_{V}=aA_{\xi}^{*}+bA_{N}$, where $V=a\xi+bN$.

###### Proof.

Taking the $\overline{g}$-product of (3.2) with $Y\in\Gamma(TM^{*})$ and using
the fact that $\overline{\nabla}$ is a metric connection, we get
$g(A_{V}X,Y)=\overline{g}(V,\overline{\nabla}_{X}Y)$. Then, applying (3.1) to
the last relation and the fact that $M^{*}$ is nondegenerate, we get the
desired result. ∎

Let $\\{V,W\\}$ be an orthonormal basis of $T_{x}M^{*\perp}$ at $x\in M^{*}$.
Then, the mean curvature vector of a leaf $M^{*}$ in $\overline{M}$ is the
vector $H^{*}=\frac{1}{2}[(\mathrm{trace}A_{V})V+(\mathrm{trace}A_{W})W]$. We
say that $M^{*}$ is minimal in $\overline{M}$ if $H^{*}$ vanishes. It then
follows that $M^{*}$ is minimal if and only if $\mathrm{trace}A_{V}=0$ and
$\mathrm{trace}A_{W}=0$. In view of Lemma 3.1, one can easily see that
minimality of a leaf $M^{*}$ implies minimality of the underlying null
hypersurface $(M,g)$. Let us consider the curvature tensor of the normal
bundle $TM^{*\perp}$ as $R^{*\perp}:T_{x}M^{*}\times T_{x}M^{*}\times
T_{x}M^{*\perp}\longrightarrow T_{x}M^{*\perp}$, given by

$\displaystyle
R^{*\perp}(X,Y)V=\nabla^{*\perp}_{X}\nabla^{*\perp}_{Y}V-\nabla^{*\perp}_{Y}\nabla^{*\perp}_{X}V-\nabla^{*\perp}_{[X,Y]}V,$
(3.3)

for any $X,Y\in\Gamma(TM^{*})$ and $V\in\Gamma(TM^{*\perp})$. The importance
of the 1-form $\tau$ in the study of null geometry has been shown in [4] and
[5]. In fact, it has been shown that the Ricci tensor of a null submanfold is
symmetric if and only if $\tau$ is closed, that is; $d\tau=0$. In what
follows, we show that the normal curvature $R^{*\perp}$ of a leaf $M^{*}$ is
directly linked to the 1-form $\tau$ of (2.2).

######  Proposition 3.2.

Let $(M,g)$ be a screen integrable null hypersurface of a Lorentzian manifold
$(\overline{M}(c),\overline{g})$. Then, the normal curvature $R^{*\perp}$ of
any leaf $M^{*}$ of $S(TM)$ satisfies

$\displaystyle R^{*\perp}(X,Y)V$
$\displaystyle=\\{C(X,A_{V}Y)-C(Y,A_{V}X)\\}\xi$
$\displaystyle+\\{B(X,A_{V}Y)-B(Y,A_{V}X)\\}N$ (3.4)
$\displaystyle=2d\tau(X,Y)W,$ (3.5)

for any vector fields $X,Y\in\Gamma(TM^{*})$ and $V,W\in\Gamma(TM^{*\perp})$.

###### Proof.

A direct calculation using (3.1) and (3.2) leads to

$\displaystyle\overline{R}(X,Y)V$
$\displaystyle=-\nabla^{*}_{X}A_{V}Y-C(X,A_{V}Y)\xi-B(X,A_{V}Y)N-A_{\nabla^{*\perp}_{Y}V}X$
$\displaystyle+\nabla_{X}^{*\perp}\nabla_{Y}^{*\perp}V+\nabla_{Y}^{*}A_{V}X+C(Y,A_{V}X)\xi+B(Y,A_{V}X)N$
$\displaystyle+A_{\nabla^{*\perp}_{X}V}Y-\nabla_{Y}^{*\perp}\nabla_{X}^{*\perp}V+A_{V}[X,Y]-\nabla_{[X,Y]}^{*\perp}V$
$\displaystyle=-\nabla^{*}_{X}A_{V}Y+\nabla_{Y}^{*}A_{V}X-A_{\nabla^{*\perp}_{Y}V}X+A_{\nabla^{*\perp}_{X}V}Y+A_{V}[X,Y]$
$\displaystyle+R^{*\perp}(X,Y)V+\\{C(Y,A_{V}X)-C(X,A_{V}Y)\\}\xi$
$\displaystyle+\\{B(Y,A_{V}X)-B(X,A_{V}Y)\\}N,\;\;\;\forall\,X,Y\in\Gamma(TM^{*})$
(3.6)

Since $\overline{M}$ is a space of constant curvature $c$, we have
$\overline{R}(X,Y)V=0$, for any $X,Y\in\Gamma(TM^{*})$ and
$V\in\Gamma(TM^{*\perp})$. Thus, (3.6) gives

$\displaystyle R^{*\perp}(X,Y)V+$
$\displaystyle\\{C(Y,A_{V}X)-C(X,A_{V}Y)\\}\xi$
$\displaystyle+\\{B(Y,A_{V}X)-B(X,A_{V}Y)\\}N=0,$ (3.7)

which proves (3.4). Next, applying Lemma 3.1 to (3.7) and using the fact that
$B(A^{*}_{\xi}X,Y)=B(X,A^{*}_{\xi}Y)$ and $C(A_{N}X,Y)=C(X,A_{N}Y)$, for all
$X,Y\in\Gamma(S(TM))$, we get

$\displaystyle R^{*\perp}(X,Y)V+$
$\displaystyle\\{C(Y,A^{*}_{\xi}Y)-C(X,A^{*}_{\xi}X)\\}a\xi$
$\displaystyle+\\{B(Y,A_{N}X)-B(X,A_{N}Y)\\}bN=0.$ (3.8)

As $B(Y,A_{N}X)=g(A_{N}X,A^{*}_{\xi}Y)=C(X,A^{*}_{\xi}Y)$, (3.8) reduces to

$\displaystyle
R^{*\perp}(X,Y)V=\\{C(Y,A^{*}_{\xi}Y)-C(X,A^{*}_{\xi}X)\\}(a\xi-bN).$ (3.9)

Next, as $\overline{M}$ is a space of constant curvature $c$, we have
$\overline{R}(X,Y)\xi=0$, for all $X,Y\in\Gamma(TM)$. Thus, in view of (2.5)
and (3.9), we conclude that

$\displaystyle R^{*\perp}(X,Y)V=2d\tau(X,Y)W,$

where $W=a\xi-bN$, which proves (3.5) and proof is completed. ∎

The following follows directly from Proposition 3.2.

######  Corollary 3.3.

In view of Proposition 3.2, the following are equivalent;

  1. (1)

$d\tau$ vanishes on $S(TM)$.

  2. (2)

$A_{\xi}^{*}\circ A_{N}=A_{N}\circ A_{\xi}^{*}$.

  3. (3)

$R^{*\perp}(x)=0$.

  4. (4)

The normal bundle of $M^{*}$ is parallel.

######  Corollary 3.4.

If $(M,g)$ is totally umbilic or screen totally umbilic in $\overline{M}$,
then $R^{*\perp}(x)=0$.

######  Remark 3.5.

Condition (2) of Corollary 3.3 implies that $A_{V}\circ A_{W}=A_{W}\circ
A_{V}$. In fact, by a simple calculation, while considering Lemma 3.1, we get
$A_{V}\circ A_{W}=a^{2}(A^{*}_{\xi}\circ A^{*}_{\xi})-ab(A_{\xi}^{*}\circ
A_{N}-A_{N}\circ A_{\xi}^{*})-b^{2}(A_{N}\circ A_{N})$. In view of (2) of
Corollary 3.3, we have $A_{V}\circ A_{W}=a^{2}(A^{*}_{\xi}\circ
A^{*}_{\xi})-b^{2}(A_{N}\circ A_{N})$. On the other hand, $A_{W}\circ
A_{V}=a^{2}(A^{*}_{\xi}\circ A^{*}_{\xi})-b^{2}(A_{N}\circ A_{N})$, which, if
compared with previous relation, proves the assertion. It then follows that
the vanishing of $d\tau$ on a leaf $M^{*}$ implies simultaneous
diagonalisation of $A_{V}$, for all $V\in\Gamma(TM^{*\perp})$.

Next, we prove the following result.

######  Theorem 3.6.

Let $(M,g)$ be a screen integrable null hypersurface of
$\mathbb{R}_{1}^{n+2}$, and $f:M^{*}\longrightarrow\mathbb{R}_{1}^{n+2}$ an
isometric immersion of a leaf $M^{*}$ of $S(TM)$ as a codimension 2
submanifold of $\mathbb{R}_{1}^{n+2}$. Suppose there exist a nonzero normal
vector field $V$ to $M^{*}$ in $\mathbb{R}_{1}^{n+2}$ such that $d\tau=0$ on
$S(TM)$ and $A_{V}^{*}=\lambda I$, $\lambda\neq 0$, then $f(M^{*})$ is
contained inside

  1. (1)

$\mathbb{S}_{1}^{n+1}\left(\frac{\sqrt{\epsilon}}{\lambda}\right)$, if
$\epsilon>0$,

  2. (2)

$\mathbb{H}_{0}^{n+1}\left(\frac{-\sqrt{-\epsilon}}{\lambda}\right)$, if
$\epsilon<0$,

where $\epsilon=\overline{g}(V,V)$. Furthermore, the null hypersurface $(M,g)$
is proper quasi-screen conformal in $\mathbb{R}_{1}^{n+2}$. Moreover, if
$A_{W}=0$, where $V,W$ spans the normal bundle $TM^{*\perp}$ then, $(M,g)$ is
a proper totally umbilic and screen totally umbilic in $\mathbb{R}_{1}^{n+2}$.

###### Proof.

Observe that the vector $f(x)+\frac{1}{\lambda}V$ is constant for all $x\in
M^{*}$. Let us denote it by $\tilde{c}$, then we have
$\overline{g}(f(x)-\tilde{c},f(x)-\tilde{c})=\frac{1}{\lambda^{2}}\overline{g}(V,V)=\frac{\epsilon}{\lambda^{2}}$.
As $d\tau=0$ on $S(TM)$, then $TM^{*\perp}$ is parallel by Corollary 3.3.
Consequently, $V$ is parallel and therefore, $f(M^{*})$ is contained in the
sphere or hyperbolic space with center $\tilde{c}$ by [9]. This proves parts
(1) and (2). Furthermore, the condition $A_{V}=\lambda I$ together with Lemma
3.1 implies that $A_{V}=aA_{\xi}^{*}+bA_{N}=\lambda I$, where $V=a\xi+bN$.
Then, in view of [11], the null hypersurface $(M,g)$ is quasi-screen conformal
in $\mathbb{R}_{1}^{n+2}$. On the other hand, if $A_{W}=0$, we have
$aA^{*}_{\xi}-bA_{N}=0$. Combining this relation with the previous one gives
$A^{*}_{\xi}=\frac{\lambda}{2a}I$ and $A_{N}=\frac{\lambda}{2b}I$, which shows
that $(M,g)$ is totally umbilic and screen totally umbilic in
$\mathbb{R}_{1}^{n+2}$, and the theorem is proved. ∎

######  Corollary 3.7.

In case the vector field $V$, in Theorem 3.6, is the mean curvature vector of
$M^{*}$ in $\mathbb{R}_{1}^{n+2}$, then $M^{*}$ is immersed minimally in
$\mathbb{S}_{1}^{n+1}$ or $\mathbb{H}_{0}^{n+1}$ (Such an immersion is called
pseudo umbilic by Chen and Yano [13]). Moreover, the underlying null
hypersurface $(M,g)$ is a proper totally umbilic, screen totally umbilic and
screen conformal in $\mathbb{R}_{1}^{n+1}$.

###### Proof.

If $V$ is the mean curvature vector of $M^{*}$ in $\mathbb{R}_{1}^{n+2}$, then
$V$ is parallel to the position vector $f(x)-\tilde{c}$ and therefore, by [9],
$M^{*}$ is minimal in either $\mathbb{S}_{1}^{n+1}$ or $\mathbb{H}_{0}^{n+1}$.
In view of (3.1) and the fact that $f(M^{*})$ is pseudo umbilic, we have
$C(X,Y)\xi+B(X,Y)N=g(X,Y)V=ag(X,Y)\xi+bg(X,Y)N$, for all
$X,Y\in\Gamma(TM^{*})$. Taking the $\overline{g}$-product of the previous
relation with $\xi$ and $N$, in turns, we get $B(X,Y)=bg(X,Y)$ and
$C(X,Y)=ag(X,Y)$, respectively. Thus, $(M,g)$ is totally umbilic and screen
totally umbilic in $\mathbb{R}_{1}^{n+1}$. As $a,b\neq 0$, we deduce that
$C(X,Y)=\psi B(X,Y)$, with $\psi=\frac{a}{b}$, showing that $M$ is screen
conformal, which completes the proof. ∎

As an example, we have the following.

######  Example 3.8 (The null cone $\Lambda_{0}^{n+1}$ of
$\mathbb{R}_{1}^{n+2}$).

Let $\mathbb{R}_{1}^{n+2}$ be the space $\mathbb{R}^{n+2}$ endowed with the
semi-Euclidean metric

$\displaystyle\overline{g}(x,y)=-x^{0}y^{0}+\sum_{a=1}^{n+1}x^{a}y^{a},\;\;\;\mbox{where}\;\;\;x=\sum_{A=0}^{n+1}x^{A}\frac{\partial}{\partial
x^{A}}.$

The null cone $\Lambda_{0}^{n+1}$ is given by the equation
$-(x^{0})^{2}+\sum_{a=1}^{n+1}(x^{a})^{2}=0$, $x^{0}\neq 0$. It is known that
$\Lambda_{0}^{n+1}$ is a null hypersurface of $\mathbb{R}_{1}^{n+2}$ and the
radical distribution is spanned by a global vector field

$\displaystyle\xi=\sum_{A=0}^{n+1}x^{A}\frac{\partial}{\partial x^{A}},$
(3.10)

on $\Lambda_{0}^{n+1}$. The unique section $N$ spanning the transversal bundle
$tr(T\Lambda_{0}^{n+1})$ is given by

$\displaystyle N=\frac{1}{2(x^{0})^{2}}\left\\{-x^{0}\frac{\partial}{\partial
x^{0}}+\sum_{a=1}^{n+1}x^{a}\frac{\partial}{\partial x^{a}}\right\\},$ (3.11)

and is also globally defined. As $\xi$ is the position vector field we get

$\displaystyle\overline{\nabla}_{X}\xi=\nabla_{X}\xi=X,\;\;\;\forall\,X\in\Gamma(T\Lambda_{0}^{n+1}).$
(3.12)

Then, $A^{*}_{\xi}X+\tau(X)\xi+X=0$. Since $A^{*}_{\xi}$ is
$\Gamma(S(T\Lambda_{0}^{n+1}))$-valued, we have

$\displaystyle
A_{\xi}^{*}X=-PX\;\;\;\mbox{and}\;\;\;\tau(X)=-\eta(X),\;\;\forall\,X\in\Gamma(T\Lambda_{0}^{n+1}).$
(3.13)

Note that any $X\in\Gamma(S(T\Lambda_{0}^{n+1}))$ is expressed as
$X=\sum_{a=1}^{n+1}X^{a}\frac{\partial}{\partial x^{a}}$, where
$(X^{1},\ldots,X^{n+1})$ satisfy $\sum_{a=1}^{n+1}x^{a}X^{a}=0$. Then,

$\displaystyle\nabla_{\xi}X=\overline{\nabla}_{\xi}X=\sum_{A=0,a=1}^{n+1}x^{A}\frac{\partial
X^{a}}{\partial x^{A}}\frac{\partial}{\partial x^{a}},$ (3.14)

from which we obtain

$\displaystyle
g(\nabla_{\xi}X,\xi)=\sum_{A=0,a=1}^{n+1}x^{a}X^{A}\frac{\partial
X^{a}}{\partial x^{A}}=-\sum_{a=1}^{n+1}x^{a}X^{a}=0.$ (3.15)

From (3.14) and (3.15), we have
$\nabla_{\xi}X\in\Gamma(S(T\Lambda_{0}^{n+1}))$, that is, $A_{N}\xi=0$.
Moreover, by simple calculations, we have

$\displaystyle
C(X,Y)=\overline{g}(\nabla_{X}Y,N)=\overline{g}(\overline{\nabla}_{X}Y,N)=-\frac{1}{2(x^{0})^{2}}g(X,Y).$
(3.16)

Clearly, $S(T\Lambda_{0}^{n+1})$ is integrable. Denote by $M^{*}$ its leaf,
then

$\displaystyle\overline{\nabla}_{X}Y=\nabla^{*}_{X}Y+\frac{g(X,Y)}{x^{0}}(-\frac{1}{2x^{0}}\xi-x^{0}N),\;\;\;\forall\,X,Y\in\Gamma(TM^{*}).$
(3.17)

It is obvious that $M^{*}$ is a totally umbilic Riemannian submanifold of
codimension 2 of $\mathbb{R}_{1}^{n+2}$. Moreover, using (3.13) and (3.16), we
have

$\displaystyle
d\tau(X,Y)=\frac{1}{2}\\{C(X,PY)-C(Y,PX)\\}=0,\;\;\;\forall\,X,Y\in\Gamma(T\Lambda_{0}^{n+1}).$
(3.18)

As $x^{0}\neq 0$, we may suppose $x^{0}>0$ (for $x^{0}<0$ we proceed
analogously), and consider in the normal bundle $TM^{*\perp}$, the vector
fields

$\displaystyle
V_{1}=-\frac{1}{2x^{0}}\xi-x^{0}N\;\;\;\mbox{and}\;\;\;V_{2}=-\frac{1}{2x^{0}}\xi+x^{0}N.$
(3.19)

Note that $\\{V_{1},V_{2}\\}$ is an orthonormal basis, where $V_{1}$ and
$V_{2}$ are spacelike and timelike, respectively. Using the expressions of
$A_{\xi}^{*}$ and $A_{N}$, we get $A_{V_{1}}=\frac{1}{x^{0}}I$, from which
$\lambda=\frac{1}{x^{0}}$. From the expressions of $\xi$ and $N$, we have
$\overline{\nabla}_{X}V_{1}=-\frac{1}{x^{{0}}}X$ and
$\overline{\nabla}_{X}V_{2}=0$, for all $X\in\Gamma(TM^{*})$. Therefore, from
the Weingarten formula (3.2) for $M^{*}$, we get $\nabla^{*\perp}_{X}V_{1}=0$
and $\nabla^{*\perp}_{X}V_{2}=0$. Clearly, $\\{V_{1},V_{2}\\}$ is a parallel
basis with respect to the normal connection $\nabla^{*\perp}$ of $M^{*}$. As
the vector field $V_{1}$ is spacelike, that is, $\overline{g}(V_{1},V_{1})=1$
and also parallel to the mean curvature of $M^{*}$ in $\mathbb{R}_{1}^{n+2}$
(see (3.17)), we conclude by Corollary 3.7 that $M^{*}$ is minimally immersed
in the sphere $\mathbb{S}_{1}^{n+1}(x^{0})$. Note also that $(M,g)$ is totally
umbilic, screen totally umbilic and screen conformal in
$\mathbb{R}_{1}^{n+2}$.

A subbundle $\mathcal{D}$ of the normal bundle $TM^{*\perp}$ is said to be
parallel in the normal bundle if it is invariant by parallel translation with
respect to the normal connection $\nabla^{*\perp}$; that is, if
$V\in\Gamma(\mathcal{D})$ then $\nabla_{X}^{*\perp}V\in\Gamma(\mathcal{D})$,
for any $X\in\Gamma(TM^{*})$. We also say that the curvature tensor
$R^{*\perp}$ of the normal connection $\nabla^{*\perp}$ is parallel in the
normal bundle if $\nabla^{*\perp}R^{*\perp}=0$; that is, for any
$X,Y,Z\in\Gamma(TM^{*})$ and $V\in\Gamma(TM^{*\perp})$ , we have

$\displaystyle(\nabla^{*\perp}_{Z}R^{*\perp}(X,Y))V=\nabla^{*\perp}_{Z}R^{*\perp}(X,Y)V-R^{*\perp}(X,Y)\nabla^{*\perp}_{Z}V=0.$
(3.20)

As an example, the curvature tensor $R^{*\perp}$ of the normal bundle
$TM^{*\perp}$ of Example 3.8 is parallel in the normal bundle. This is due to
the fact that the normal bundle is parallel; that is $\nabla^{*\perp}V=0$, for
any $V\in\Gamma(TM^{*\perp})$.

Next, we define the first normal space $\mathcal{Q}(x)$ at $x\in M^{*}$ as the
orthogonal complement in $T_{x}M^{*\perp}$ of $\\{V(x)\in
T_{x}M^{*\perp}:A_{V(x)}=0\\}$.

######  Lemma 3.9.

Let $(M,g)$ be a screen integrable null hypersurface of $\overline{M}(c)$.
Assume that $f:M^{*}\longrightarrow\overline{M}$ is an immersion of a leaf
$M^{*}$ of $S(TM)$ as a codimension 2 submanifold of $\overline{M}$. Suppose
that the curvature tensor $R^{*\perp}$ of the normal bundle to $M^{*}$ is
parallel in the normal bundle. For each $x\in M^{*}$ let
$\mathcal{D}(x)=\\{V(x)\in
T_{x}M^{*\perp}:R^{*\perp}(X,Y)V=0,\forall\,X,Y\\}$. Then, $\mathcal{D}$ is
parallel in the normal bundle $TM^{*\perp}$. Moreover, $d\tau$ vanishes on
$TM^{*}$.

###### Proof.

A proof of the first assertion follows similar steps as in [2]. The vanishing
of $d\tau$ on $TM^{*}$ follows from Proposition 3.2 and the definition of
$\mathcal{D}(x)$. ∎

Next, we proof the following result.

######  Theorem 3.10.

Let $(M,g)$ be an $(n+1)$-dimensional screen integrable null hypersurface of a
Lorentzian manifold $\overline{M}(c)$. Assume that
$f:M^{*}\longrightarrow\overline{M}$ is a minimal, nontotally geodesic,
immersion of a leaf $M^{*}$ of $S(TM)$ as a codimension 2 submanifold of
$\overline{M}$. Suppose that the curvature tensor $R^{*\perp}$ of the normal
bundle to $M^{*}$ is parallel in the normal bundle. Then there exists an
$(n+1)$-dimensional totally geodesic submanifold $M^{\prime}$ of
$\overline{M}$ such that $f$ is a minimal immersion of $M^{*}$ in
$M^{\prime}$. Furthermore, the underlying null hypersurface $(M,g)$ is minimal
in $\overline{M}$ and its shape operator $A_{N}$ satisfies
$\mathrm{trace}_{|_{S(TM)}}A_{N}=0$.

###### Proof.

First note that; as $f$ is nontotally geodesic, then the first normal space of
$f$ has constant dimension $1$. We first prove the case when the normal bundle
$TM^{*\perp}$ is parallel. To that end, let $\mathcal{Q}(x)$ be the first
normal space at $x$. As $\dim\mathcal{Q}$ is constant,
$\mathcal{P}=\mathcal{Q}^{\perp}$, where $\perp$ is the orthogonal complement
in the normal bundle $TM^{*\perp}$, is a subbundle of the normal bundle. We
want to show that $\mathcal{P}$ is parallel in the normal bundle and then, use
a result of [9] to draw conclusions. Given $x\in M^{*}$, choose a unit vector
field $V_{1}$, spanning $\mathcal{Q}$ at each point in a neighborhood
$\mathscr{U}$ of $x\in M^{*}$. Let us extent the above field to
$\\{V_{1},V_{2}\\}$ so that the latter span the normal space at a point of
$\mathscr{U}$. Consider the vector field $V_{2}$ which generate the subbundle
$\mathcal{P}$ over $\mathscr{U}$. It then suffices to show that $\mathcal{P}$
is parallel in the normal bundle $TM^{*\perp}$.

Given $y\in\mathscr{U}$, let $X_{1},\ldots,X_{n}$ be coordinate vector fields
in a neighborhood $\mathscr{U}^{\prime}\subset\mathscr{U}$ of $y$ which
diagonalize, at $y$, all the shape operators of $M^{*}$. This is possible
because the normal bundle is flat, and it is a result of Cartan that the
normal bundle is flat if and only if at each point all the second fundament
forms are simultaneously diagonalizable. As $\overline{M}$ is a space of
constant curvature $c$, we have $\overline{R}(X_{j},X_{i})V_{\alpha}=0$, for
$1\leq i,j\leq n$ and $1\leq\alpha\leq 2$. Then, by (3.6), we have

$\displaystyle-\nabla^{*}_{X_{j}}A_{V_{\alpha}}X_{i}+\nabla_{X_{i}}^{*}A_{V_{\alpha}}X_{j}-A_{\nabla^{*\perp}_{X_{i}}V_{\alpha}}X_{j}+A_{\nabla^{*\perp}_{X_{j}}V_{\alpha}}X_{i}+A_{V_{\alpha}}[X_{j},X_{i}]$
$\displaystyle+R^{*\perp}(X_{j},X_{i})V_{\alpha}+\\{C(X_{i},A_{V_{\alpha}}X_{j})-C(X_{j},A_{V_{\alpha}}X_{i})\\}\xi$
$\displaystyle+\\{B(X_{i},A_{V_{\alpha}}X_{j})-B(X_{j},A_{V_{\alpha}}X_{i})\\}N=0,\;\;\;\forall\,X_{i},X_{j}\in
T_{x}M^{*}.$ (3.21)

For $V_{\alpha}\in\mathcal{P}$, that is $\alpha=2$, (3.21) gives

$\displaystyle
R^{*\perp}(X_{j},X_{i})V_{2}-A_{\nabla^{*\perp}_{X_{i}}V_{2}}X_{j}+A_{\nabla^{*\perp}_{X_{j}}V_{2}}X_{i}=0,$

from which we get

$\displaystyle-
A_{\nabla^{*\perp}_{X_{i}}V_{2}}X_{j}+A_{\nabla^{*\perp}_{X_{j}}V_{2}}X_{i}=0,$
(3.22)

after considering the assumption $R^{*\perp}=0$. Since $X_{i},\ldots,X_{n}$
diagonalize all the shape operators at $y$, we have, at $y$,
$A_{\nabla^{*\perp}_{X_{j}}V_{2}}X_{i}=a_{j}^{i}X_{i}$ and
$A_{\nabla^{*\perp}_{X_{i}}V_{2}}X_{j}=b_{i}^{j}X_{j}$, for some numbers
$a_{j}^{i}$, $b_{i}^{j}$. Therefore, (3.22) implies that
$A_{\nabla^{*\perp}_{X_{j}(y)}V_{2}}X_{i}(y)=0$, for all $i\neq j$. As the
immersion is minimal, we have $A_{\nabla^{*\perp}_{X_{j}(y)}V_{2}}X_{i}(y)=0$,
for all $i,j$, and therefore, $A_{\nabla^{*\perp}_{X_{j}(y)}V_{2}}=0$ for all
$j$. This in turn implies that
$\nabla^{*\perp}_{X_{j}(y)}V_{2}\in\mathcal{P}(y)$, following the definition
of $\mathcal{P}$. That is, $\mathcal{P}$ over $\mathscr{U}$ is parallel in the
normal bundle. This implies that $\mathcal{Q}$ is also parallel in the normal
bundle. In fact, take $V_{2}\in\mathcal{P}$ and $V_{1}\in\mathcal{Q}$. As
$\nabla^{*\perp}$ is a metric connection, we have
$X\overline{g}(V_{2},V_{1})=\overline{g}(\nabla_{X}^{*\perp}V_{2},V_{1})+\overline{g}(V_{2},\nabla_{X}^{*\perp}V_{1})=0$,
from which we see that $\mathcal{Q}$ is also parallel in the normal bundle. By
Theorem 0.2 of [9, p. 33], there exist a totally geodesic submanifold
$M^{\prime}$ of $\overline{M}$ such that $\dim M^{\prime}=n+1$ and
$f(M^{*})\subset M^{\prime}$.

Next, since $\dim\mathcal{Q}=1$ we have $\dim\mathcal{P}=1$. Let
$V_{2}\in\mathcal{P}$, then $A_{V_{2}}=0$ by the definition of $\mathcal{P}$.
Thus, Lemma 3.1 gives $aA^{*}_{\xi}+bA_{N}=0$. Taking the trace of this
relation along $TM^{*}$, we get
$a\,\mathrm{trace}A^{*}_{\xi}+b\,\mathrm{trace}A_{N}=0$. On the other hand,
since $M^{*}$ is minimal in $\overline{M}$, we have
$(\mathrm{trace}A_{V_{1}})V_{1}+(\mathrm{trace}A_{V_{2}})V_{2}=0$, where
$V_{1}\in\mathcal{Q}$. In view of Lemma 3.1, we have
$a\,\mathrm{trace}A^{*}_{\xi}-b\,\mathrm{trace}A_{N}=0$. Therefore, solving
gives $\mathrm{trace}A^{*}_{\xi}=0$ and $\mathrm{trace}A_{N}=0$, showing that
$(M,g)$ is minimal in $\overline{M}$. This completes the first case of the
proof.

Turning to the second case, that is; the normal bundle is not flat. Set
$\mathcal{D}(x)=\\{V(x)\in
T_{x}M^{*\perp}:R^{*\perp}(X,Y)V=0,\forall\,X,Y\\}$. By Lemma 3.9,
$\mathcal{D}$ is parallel in the normal bundle. Let $\mathcal{P}$ be the
orthogonal complement of $\mathcal{Q}$ in the normal bundle $TM^{*\perp}$. It
is obvious that $\mathcal{P}\subset\mathcal{D}$. Observe that, by Lemma 3.9,
$\mathcal{D}$ is parallel and , by (3.7), all the shape operators $A_{V}$,
$V\in\mathcal{D}$, can be simultaneously diagonalized. Therefore, we can apply
the same arguments as in the first case above, with $\mathcal{D}$ in place of
$TM^{*\perp}$, to conclude that $\mathcal{P}$, hence $\mathcal{Q}$, is
parallel in the normal bundle. Again, as in the first case, by Megid’s theorem
0.2 in [9], $f(M^{*})\subset M^{\prime}$, where $M^{\prime}$ is a totally
geodesic submanifold of $\overline{M}$ with dimension $n+1$. The minimality of
$(M,g)$, as a null hypersurface of $\overline{M}$, also follows as in the
previous case. This completes the second case and so the theorem is proved. ∎

## References

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# The energy landscape of the Kuramoto model in one-dimensional random
geometric graphs with a hole

Cecilia De Vita  Departamento de Matemática Facultad de Ciencias Exactas y
Naturales Universidad de Buenos Aires IMAS-UBA-CONICET Buenos Aires, Argentina
<EMAIL_ADDRESS>,  Julián Fernández Bonder  Departamento de Matemática
Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires Instituto
de Cálculo-UBA-CONICET Buenos Aires, Argentina<EMAIL_ADDRESS>and  Pablo
Groisman  Departamento de Matemática Facultad de Ciencias Exactas y Naturales
Universidad de Buenos Aires IMAS-UBA-CONICET Buenos Aires, Argentina
<EMAIL_ADDRESS>
###### Abstract.

We study the energy function of the Kuramoto model in random geometric graphs
defined in the unit circle as the number of nodes diverges. We prove the
existence of at least one local minimum for each winding number
$q\in\mathbb{Z}$ with high probability. Hence providing a large family of
graphs that support patterns that are generic. These states are in
correspondence with the explicit twisted states found in WSG and other highly
symmetric networks, but in our situation there is no explicit formula due to
the lack of symmetry. The method of proof is simple and robust. It allows
other types of graphs like $k-$nn graphs or the boolean model and holds also
for graphs defined in any simple closed curve or even a small neighborhood of
the curve and for weighted graphs. It seems plausible that the method can be
extended also to higher dimensions, but a more careful analysis is required.

###### Key words and phrases:

interacting dynamical systems; Kuramoto model; geometric random graphs;
twisted states; energy landscape; non-convex optimization

##  1\. Introduction

The study of local minima and the whole geometry of high-dimensional random
non-convex functions is highly relevant in areas as diverse as deep-learning,
statistical mechanics, complex networks and synchronicity.

Phase synchronization of systems of coupled oscillators is a phenomenon that
has attracted the mathematical and scientific community because of its
intrinsic mathematical interest [8, 15, 6, 10] and its ubiquity in
technological, physical and biological models [18, 24, 3, 7, 4, 11, 21, 22].

One of the most popular models for describing synchronization of a system of
coupled oscillators is the Kuramoto model. The model has been studied both by
means of rigorous mathematical proofs and heuristics arguments and simulations
in different families of graphs. Here we focus in the first class of evidence.

We consider graphs $G_{n}=G=(V,E)$ where the set of nodes
$V=\\{x_{0},x_{1},\dots,x_{n-1}\\}\subset\mathbb{S}^{1}$ is a sample of $n$
i.i.d uniform random variables. The distance between two nodes is given by the
Euclidean distance in $\mathbb{C}$, which is equivalent to the geodesic
distance in $\mathbb{S}^{1}$. For convenience, we assume that the nodes $V$
are labeled counterclockwise with $\arg(x_{0})=0$ and we denote
$x_{n}:=x_{0}$.

The random geometric graph in the circle $\mathbb{S}^{1}$ with parameters
$n,\epsilon_{n}$ is the graph that has $V$ as the set of nodes in which we
declare $\\{x_{i},x_{j}\\}\in E$ (we denote this by $i\sim j$) if and only if
$|x_{i}-x_{j}|<\epsilon_{n}$.

We can think of $x_{0},\dots,x_{n-1}$ as points in $[0,2\pi]$ and $|\cdot|$ as
the absolute value function with the convention that everything is understood
mod $2\pi$.

For the sequence of random geometric graphs defined above, we are going to
work in the regime

(1.1) $n\epsilon_{n}^{2}\to 0,\qquad\frac{n\epsilon_{n}}{\log
n}\to\infty,\qquad\text{ as }n\to\infty.$

The first condition implies $\epsilon_{n}\to 0$, which is important to obtain
Proposition 3.1 below (this proposition does not hold if
$\epsilon_{n}\nrightarrow 0$). It will also be used for the conclusion of the
main theorem. However, we expect the conclusion of our main theorem to hold
even without Proposition 3.1 (but some bound from above is needed on
$\epsilon_{n}$ to avoid high connectivity that leads to global synchronization
[23]). The second condition is required to guarantee connectivity of the graph
with high probability and so, it can’t be removed without altering the
behavior of the system. Observe that both conditions are verified for any
sequence of the form $\epsilon_{n}\approx n^{-a}$ with $1/2<a<1$.

Let $u^{n}\colon[0,\infty)\times V\to\mathbb{R}$ be the unique solution to a
system of $n$ homogeneous Kuramoto equations

(1.2)
$\displaystyle\begin{cases}\displaystyle{\frac{d}{dt}}u^{n}(t,x_{i})=\frac{1}{n}\displaystyle{\frac{1}{\epsilon_{n}^{2}N_{i}}}\sum_{j\sim
i}\sin\left(u^{n}(t,x_{j})-u^{n}(t,x_{i})\right),\\\
u^{n}(0,x_{i})=\bar{u}^{n}(x_{i}),\quad\quad i=0,1,...,n-1.\end{cases}$

The random integer $N_{i}$ denotes the number of neighbors of $x_{i}$. To
lighten notation we call $u^{n}_{i}=u^{n}(t,x_{i})$ and we also omit the
dependence on $n$ if it is not nessesary. Equations (1.2) define a gradient
system. It is a direct computation to see that

$\dot{u}=-\nabla E_{n}(u),$

for

$E_{n}(u)=E_{n}(u_{0},\dots,u_{n-1})=\frac{1}{n}\sum_{i=1}^{n}\displaystyle{\frac{1}{2\epsilon_{n}^{2}N_{i}}}\sum_{j\sim
i}(1-\cos(u_{j}-u_{i})).\\\ $

The sine function in (1.2) can be replaced by an odd $2\pi-$periodic symmetric
smooth function $J$ with Taylor expansion $J(\theta)=\theta+o(\theta^{2})$
without altering the conclusion of our main result.

Our interest in the Kuramoto model in graphs with this structure is threefold:
on the one hand this kind of graphs is relevant to model several natural
situations in which spatial considerations are important to determine the
strength of the links between oscillators. On the other hand they form a large
family of model networks with persistent behavior (robust to small
perturbations) for which we expect to observe patterns.

Last but not least, there has been a recent interest to understand the
behavior of the Kuramoto model on diverse models of random and non-random
graphs [13, 1, 2, 12]. The main goal is to decide if the networks foster
synchronization or not. In [2] the authors have recently shown that in
expander graphs and in particular in Erdős-Rény graphs above the connectivity
threshold, synchronization occurs with high probability as $n\to\infty$. Our
results can be seen as a complement of those in the sense that we are
exhibiting a class of random graphs that are not expanders for which global
synchronization fails. Up to our knowledge this is the first rigorous proof of
non-synchronization in random graphs.

Twisted states have been defined for particular classes of graphs as explicit
equilibria of (1.2). They have been shown to be stable equilibria in rings in
which each node is connected to its $k$ nearest neighbors on each side [23],
in Cayley graphs and in random graphs with a particular structure [16].They
have also been studied in small-world networks [14] and in the continuum limit
[17] among others.

Our notion of twisted state is a bit different since we don’t expect to find
explicit equilibria in our context besides complete synchronization. We think
of them as stable equilibria that can be identified in some way to with the
functions $u_{q}(x_{i})=qx_{i}$. Precise definitions in the next section. We
remark that we are considering functions that take values in $\mathbb{S}^{1}$
rather than $\mathbb{R}$. Alternatively, we can think of them as functions
$u\colon[0,2\pi]\to\mathbb{R}$ with $u(2\pi)=u(0)+2q\pi$ for some
$q\in\mathbb{Z}$.

Situations in which the twisted states are explicit are not expected to be
robust and persistent. Our interest is to find twisted states that are generic
in some sense and for this same reason we do not expect us to be able to
compute them explicitly.

Remark that as far as we know, in most of the literature that give rigorous
proofs about existence of twisted states they are computed explicitly by
exploding graph symmetries and the issue is to prove their stability. Here
(and in most typical real situations with spatial structure and local
interactions) the issue is to prove their existence. We are going to get the
stability for free.

The Kuramoto model in random geometric graphs has been studied in [1]. In that
work the authors are interested in the optimization landscape of the energy
function determined by (1.2) as well as we are here, but they work on a
different regime: in their setting the graphs are constructed on the sphere
$\mathbb{S}^{d-1}$ rather than in the circle and $d\to\infty$ as $n\to\infty$.
In that context, they obtain guarantees for global spontaneous synchronization
(i.e. the global minimum $\theta_{1}=\theta_{2}=\cdots=\theta_{n}$ is the
unique local minima of the energy). This is pretty different from our
situation as we will see.

Besides the Kuramoto model, our work enters in the framework of random non-
convex optimization, which is relevant not just in the study of dynamics of
complex networks but also in deep-learning and statistical mechanics. In the
first case due to the fact that most of modern learning algorithms (i.e.,
artificial neural networks) rely on the adequate optimization of a loss
functions which is typically highly non-convex and random [20, 9, 5, 19].

In particular, our results show that for this kind of random energies, while
the energy at a typical point diverges to infinity with the size of the graph,
at local minima is of order one.

We are going to state our results for random geometric graphs but they can be
exported straightforwardly to different kinds of graphs defined in any closed
and simple curve like $k-$nn graphs or even deterministic graphs. We discuss
this in Section 4.

For a continuous function $u\colon[0,2\pi]\to\mathbb{R}$ with
$u(2\pi)=u(0)+2q\pi$ for some $q\in\mathbb{Z}$ we define its index by
$I(u)=q$. If $u$ is defined only in a discrete set $\\{x_{0},\dots,x_{n-1}\\}$
we define its index as the index of its linear interpolation (a more precise
definition is given below).

Our main result reads as follows.

######  Theorem 1.1.

For each $q\in\mathbb{Z}$ we have,

$\lim_{n\to\infty}\mathbb{P}\left(\eqref{eq:km}\text{ has an asymptotically
stable equilibrium with index $q$}\right)=1.$

To prove this theorem we first consider in Section 2 a partition of the space
$\mathcal{S}^{n}:=(\mathbb{S}^{1})^{n}$ where $E_{n}$ is defined. Next, in
Section 3 we prove Proposition 3.1 which is one of the main ingredients and
then Theorem 1.1.

##  2\. Geometry of the space $\mathcal{S}^{n}$

For a rectifiable closed curve $\gamma:[a,b]\to\mathbb{C}$ that does not
contain the origin we define the index (or winding number) of $\gamma$ (around
the origin) as the total number of times that the curve travels
counterclockwise around $\omega$. More precisely,

$\displaystyle I(\gamma)=\frac{1}{2\pi\mathrm{i}}\int_{\gamma}\frac{dz}{z}.$

For each
$\mathbf{z}=(z_{0},\ldots,z_{n-1})\in\mathcal{S}^{n}\subset\mathbb{C}^{n}$,
consider $\gamma_{\mathbf{z}}=\sum_{j=1}^{n}\gamma_{j}$ where $\gamma_{j}$ is
the geodesic from $z_{j-1}$ to $z_{j}$ in $\mathbb{S}^{1}$, $z_{n}=z_{0}$ and
$\gamma_{i}+\gamma_{j}$ is the curve that results from concatenating
$\gamma_{i}$ and $\gamma_{j}$ in the given order. Observe that
$\gamma_{\mathbf{z}}$ is a well-defined picewise differentiable closed curve
in $\mathbb{S}^{1}$ as far as $z_{j}\neq-z_{j-1}$ for every $j$ (otherwise the
geodesic from $z_{j-1}$ to $z_{j}$ is not unique). We abbreviate the notation
by writing $I({\mathbf{z}}):=I(\gamma_{\mathbf{z}})$.

If we write $z_{j}=e^{{\rm i}\theta_{j}}$ for some $\theta_{j}\in[0,2\pi)$,
then

$\displaystyle I({\mathbf{z}})$ $\displaystyle=\frac{1}{2\pi{\rm
i}}\int_{\gamma_{\mathbf{z}}}\frac{dz}{z}$ $\displaystyle=\frac{1}{2\pi{\rm
i}}\sum_{j=1}^{n}\int_{\gamma_{j}}\frac{dz}{z}$
$\displaystyle=\frac{1}{2\pi}\sum_{j=1}^{n}\theta_{j}\circleddash\theta_{j-1}.$

Here $\theta_{j}\circleddash\theta_{j-1}$ is the signed length of the geodesic
from $z_{j-1}$ to $z_{j}$. If for each $\theta,\theta^{\prime}$ we choose
$\bar{\theta}$ and $\bar{\theta}^{\prime}$ such that $e^{{\rm
i}\bar{\theta}}=e^{{\rm i}\theta}$, $e^{{\rm i}\bar{\theta}^{\prime}}=e^{{\rm
i}\theta^{\prime}}$ with $-\pi<\bar{\theta}-\bar{\theta}^{\prime}<\pi$, it can
be computed as

$\theta\circleddash\theta^{\prime}=\bar{\theta}-\bar{\theta}^{\prime}.$

Sometimes we will slightly abuse notation by writing
$I(\theta_{0},\dots,\theta_{n-1})$ instead of $I(z_{0},\dots,z_{n-1})$. This
is not a problem since the value of $I$ is independent of the choice of
$\theta_{0},\dots,\theta_{n-1}$.

Observe that the set of points $\mathbf{z}\in\mathcal{S}^{n}$ for which the
index $I(\mathbf{z})$ is well defined is open and that the function $I$ is
continuous in its domain and integer valued. Hence it is constant in each
connected component and in fact the sets

$\displaystyle{K}_{q}:=\\{\mathbf{z}\in\mathcal{S}^{n}:I(\mathbf{z})=q\\},$

define the connected components of the domain of $I$. Note that each $K_{q}$
is open and
$\partial{K}_{q}=\\{\mathbf{z}\in\overline{{K}_{q}}:I(\mathbf{z})\text{ is not
defined}\\}$. So, we have the decomposition

$\mathcal{S}^{n}=\bigcup_{q\in\mathbb{Z}}K_{q}\cup\left(\bigcup_{q\in\mathbb{Z}}\partial
K_{q}\right).$

Remark that for a given $n$, the sets $K_{q}=\emptyset$ for
$|q|>\lfloor\frac{n-1}{2}\rfloor$. Also remark that for
$|q|\leq\lfloor\frac{n-1}{2}\rfloor$ we have $\partial K_{q}\cap\partial
K_{q^{\prime}}\neq\emptyset$. In fact the point
$(0,\pi,0,\pi,\dots)\in\partial K_{q}$ for every
$q\leq\lfloor\frac{n-1}{2}\rfloor$.

We will prove that for each $q\in\mathbb{Z}$, the energy $E_{n}$ restricted to
$K_{q}$ attains a minimum with high probability as $n\to\infty$. Since the
sets $K_{q}$ are open, this minima are forced to be local minima of $E_{n}$.

##  3\. Proof of the main theorem

To prove Theorem 1.1 we will need the following proposition applied to the
functions $u_{q}(x)=qx$ but we state it for general smooth functions $u$ due
to its independent interest.

######  Proposition 3.1.

Assume $u\in C^{2}([0,2\pi],\mathbb{R})$, then

$\lim_{n\to\infty}E_{n}(u)=\frac{1}{12\pi}\int_{0}^{2\pi}|u^{\prime}(x)|^{2}\,dx,\qquad\text{in
probability.}$

###### Proof.

We need to Poissonize. Consider an infinite sequence of independent uniform
random variables in $\mathbb{S}^{1}$, $x_{0},x_{1},\dots$. Let $\mathbf{n}$ be
an independent Poisson random variable with parameter $n$. Define for every
$k\in\mathbb{N}$, $V_{k}=\\{x_{0},\dots,x_{k-1}\\}$. Then $V=V_{n}$ and we
denote $\mathbf{V}:=V_{\mathbf{n}}$. The point process $\mathbf{V}$ is a
Poisson Point Process in $\mathbb{S}^{1}$. Let us consider the Poissonized
version of the energy,

$\mathbf{E}_{n}(u)=\mathbf{E}_{n}(u_{0},\dots,u_{\mathbf{n}-1})=\frac{1}{n}\sum_{i=1}^{\mathbf{n}}\displaystyle{\frac{1}{2\epsilon_{n}^{2}\mathbf{N}_{i}}}\sum_{\begin{subarray}{c}j\sim
i\\\ x_{j}\in\mathbf{V}\end{subarray}}(1-\cos(u_{j}-u_{i})).\\\ $

Here $\mathbf{N}_{i}$ is the number of neighbors of $i$ in the graph
constructed with $\mathbf{V}$ instead of $V$. We also need to consider
versions of the energy for different sets of nodes. So, define for any
$V_{k}$,

$E^{V_{k}}_{n}(u)=E^{V_{k}}_{n}(u_{0},\dots,u_{k-1})=\frac{1}{n}\sum_{i=1}^{k}\displaystyle{\frac{1}{2\epsilon_{n}^{2}N^{V_{k}}_{i}}}\sum_{\begin{subarray}{c}j\sim
i\\\ x_{j}\in{V_{k}}\end{subarray}}(1-\cos(u_{j}-u_{i})).\\\ $

Similarly, $N^{V_{k}}_{i}$ denotes the number of neighbors of $i$ in the graph
constructed with $V_{k}$. With this notation we have $E_{n}=E^{V_{n}}_{n}$ and
$\mathbf{E}_{n}=E^{\mathbf{V}}_{n}$. There is a random variable $C$
independent of $n$ and $u$ such that for every integer $k\geq-n+2$,

$|E^{V_{n+k}}_{n}(u)-E^{V_{n}}_{n}(u)|\leq
C\frac{|k|}{n}\|(u^{\prime})^{2}\|_{\infty}.$

In particular,

$|\mathbf{E}_{n}(u)-E_{n}(u)|\leq
C\frac{|\mathbf{n}-n|}{n}\|(u^{\prime})^{2}\|_{\infty}.$

Since $\frac{|\mathbf{n}-n|}{n}\to 0$ a.s., it is enough to prove

$\lim_{n\to\infty}\mathbf{E}_{n}(u)=\frac{1}{12\pi}\int_{0}^{2\pi}|u^{\prime}(x)|^{2}\,dx,\qquad\text{in
probability.}$

We proceed to do that. First, observe that

(3.1) $\lim_{\epsilon\to
0}\frac{1}{\epsilon}\int_{0}^{2\pi}\int_{0}^{2\pi}\left(\frac{u(y)-u(x)}{\epsilon}\right)^{2}\mathbf{1}\\{|y-x|<\epsilon\\}\,dy\,dx={\frac{2}{3}}\int_{0}^{2\pi}|u^{\prime}(x)|^{2}\,dx,$

and moreover

(3.2) $\lim_{\epsilon\to
0}\frac{1}{\epsilon}\int_{0}^{2\pi}\int_{0}^{2\pi}\left(\frac{1-\cos(u(y)-u(x))}{\epsilon^{2}}\right)\mathbf{1}\\{|y-x|<\epsilon\\}\,dy\,dx=\frac{1}{3}\int_{0}^{2\pi}|u^{\prime}(x)|^{2}\,dx.$

Let ${\mathcal{N}}(i)$ be the set of neighbors of $i$ in the graph determined
by $\mathbf{V}$. That is,

${\mathcal{N}}(i):=\\{j\colon 0\leq
j\leq\mathbf{n},|x_{j}-x_{i}|<\epsilon_{n}\\}.$

Conditional on $i\leq\mathbf{n}$, $x_{i}=x$ and $\mathbf{\mathcal{N}}(i)$, the
variables

$\displaystyle\zeta_{j}^{i}$
$\displaystyle:=[1-\cos(u(x_{j})-u(x_{i}))]\mathbf{1}\\{x_{j}\sim x_{i}\\}$
$\displaystyle=[1-\cos(u(x+\epsilon
z^{i}_{j})-u(x))]\mathbf{1}\\{|z^{i}_{j}|<1\\}$

are i.i.d. and their absolute values are bounded by
$\|(u^{\prime})^{2}\|_{\infty}\epsilon_{n}^{2}$. Here
$z^{i}_{j}:=(x_{j}-x_{i})/\epsilon$. Moreover, since the conditional
distribution of $z^{i}_{j}$, $j\in\mathcal{N}(i)$ is uniform in $[-1,1]$, we
have that

$\displaystyle\mathbb{E}\left(\zeta_{j}^{i}\,|\,x_{i}=x,\mathcal{N}(i)\right)$
$\displaystyle=\frac{1}{2}\int_{-1}^{1}[1-\cos(u(x+\epsilon z)-u(x))]\,dz$
$\displaystyle=\frac{1}{2\epsilon_{n}}\int_{0}^{2\pi}[1-\cos(u(y)-u(x))]\mathbf{1}\\{|y-x|<\epsilon_{n}\\}\,dy.$

Then

$\displaystyle\mathbb{E}\left[\frac{1}{\epsilon_{n}^{2}\mathbf{N}_{i}}\sum_{j\in\mathcal{N}(i)}\zeta_{j}^{i}\right]$
$\displaystyle=\frac{1}{4\pi\epsilon_{n}^{3}}\int_{0}^{2\pi}\int_{0}^{2\pi}[1-\cos(u(y)-u(x))]\mathbf{1}\\{|y-x|<\epsilon_{n}\\}\,dydx.$

and

$\displaystyle\mathbb{E}\left[\frac{1}{\epsilon_{n}^{2}\mathbf{N}_{i}}\sum_{j\in\mathcal{N}(i)}\zeta_{j}^{i}\right]^{2}$
$\displaystyle\leq\|(u^{\prime})^{2}\|_{\infty}.$

We conclude by applying the following two lemmas to the variables

$Z^{n}_{i}:=\frac{1}{\epsilon_{n}^{2}\mathbf{N}_{i}}\sum_{j\in\mathcal{N}(i)}\zeta_{j}^{i},\qquad
0\leq i\leq\mathbf{n}-1.$

For the sake of well-definiteness we construct $Z^{n}_{i}$ for
$i\geq\mathbf{n}$ using independent copies of the process.

######  Lemma 3.2.

For $Z_{i}^{n}$ defined as above, we have for $i\neq j$,

$|{\rm
Cov}(Z_{i}^{n},Z_{j}^{n})|\leq\frac{\|(u^{\prime})^{2}\|^{2}_{\infty}\epsilon_{n}}{\pi}.$

###### Proof.

Observe that for every $i,n$, $|Z_{i}^{n}|\leq\|(u^{\prime})^{2}\|_{\infty}$.
We compute for $i\neq j$,

$\displaystyle\mathbb{E}[Z_{i}^{n}Z_{j}^{n}|$ $\displaystyle
x_{i},x_{j},\mathcal{N}(i),\mathcal{N}(j)]=$
$\displaystyle=\mathbb{E}\left[Z_{i}^{n}Z_{j}^{n}\mathbf{1}\\{|x_{i}-x_{j}|>2\epsilon_{n}\\}|x_{i},x_{j},\mathcal{N}(i),\mathcal{N}(j)\right]$
$\displaystyle+\mathbb{E}\left[Z_{i}^{n}Z_{j}^{n}\mathbf{1}\\{|x_{i}-x_{j}|\leq
2\epsilon_{n}\\}|x_{i},x_{j},\mathcal{N}(i),\mathcal{N}(j)\right].$

But,

$\displaystyle\mathbb{E}\left[Z_{i}^{n}Z_{j}^{n}\mathbf{1}\\{|x_{i}-x_{j}|>2\epsilon_{n}\\}|x_{i},x_{j},\mathcal{N}(i),\mathcal{N}(j)\right]=\mathbb{E}\left[Z_{i}^{n}|x_{i},\mathcal{N}(i)\right]\mathbb{E}\left[Z_{j}^{n}|x_{j},\mathcal{N}(j)\right]\mathbf{1}\\{|x_{i}-x_{j}|>2\epsilon_{n}\\}$

and

$\displaystyle\mathbb{E}\left[Z_{i}^{n}Z_{j}^{n}\mathbf{1}\\{|x_{i}-x_{j}|\leq
2\epsilon_{n}\\}\right]\leq\|(u^{\prime})^{2}\|^{2}_{\infty}\mathbb{P}(|x_{i}-x_{j}|\leq
2\epsilon_{n})\leq\|(u^{\prime})^{2}\|^{2}_{\infty}\frac{2\epsilon_{n}}{\pi}.$

Hence,

$\displaystyle\big{|}\mathbb{E}(Z_{i}^{n}Z_{j}^{n})$
$\displaystyle-\mathbb{E}(Z_{i}^{n})\mathbb{E}(Z_{j}^{n})\big{|}=$
$\displaystyle=\left|\mathbb{E}(Z_{i}^{n})\mathbb{E}(Z_{j}^{n})\mathbb{P}(|x_{i}-x_{j}|>2\epsilon_{n})+\mathbb{E}\left[Z_{i}^{n}Z_{j}^{n}\mathbf{1}\\{|x_{i}-x_{j}|\leq
2\epsilon_{n}\\}\right]-\mathbb{E}(Z_{i}^{n})\mathbb{E}(Z_{j}^{n})\right|$
$\displaystyle=\left|-\mathbb{E}(Z_{i}^{n})\mathbb{E}(Z_{j}^{n})\mathbb{P}(|x_{i}-x_{j}|\leq
2\epsilon_{n})+\mathbb{E}\left[Z_{i}^{n}Z_{j}^{n}\mathbf{1}\\{|x_{i}-x_{j}|\leq
2\epsilon_{n}\\}\right]\right|$
$\displaystyle\leq\frac{4\|(u^{\prime})^{2}\|^{2}_{\infty}\epsilon_{n}}{\pi}.$

∎

######  Lemma 3.3.

For $Z^{n}_{1},Z_{2}^{n},\dots,Z_{\mathbf{n}}^{n}$ defined as above we have,

$\frac{1}{n}\sum_{i=1}^{\mathbf{n}}Z_{i}^{n}\to\mu:=\frac{1}{12\pi}\int_{0}^{2\pi}|u^{\prime}(x)|^{2}\,dx,\qquad\text{
in probability.}$

###### Proof.

Call $\bar{Z}_{n}=\frac{1}{n}\sum_{i=1}^{\mathbf{n}}Z_{i}^{n}$. We compute,

$\displaystyle\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^{\mathbf{n}}Z_{i}^{n}\right]^{2}$
$\displaystyle=\frac{\mathbb{E}(\mathbf{n})}{n^{2}}\mathbb{E}(Z_{1}^{n})^{2}+\frac{1}{n^{2}}\mathbb{E}\left[\sum_{i\neq
j}^{\mathbf{n}}Z_{i}^{n}Z_{j}^{n}\right]$
$\displaystyle=\frac{1}{n}\mathbb{E}(Z_{1}^{n})^{2}+\frac{\mathbb{E}(\mathbf{n}(\mathbf{n}-1))}{n^{2}}{\rm
Cov}(Z_{1}^{n},Z_{2}^{n})+\frac{\mathbb{E}(\mathbf{n}(\mathbf{n}-1))}{n^{2}}\mathbb{E}^{2}(Z_{1}^{n}).$

We have that $\mathbb{E}(Z_{1}^{n})^{2}\leq\|(u^{\prime})^{2}\|_{\infty}$ and
${\rm Cov}(Z_{1}^{n},Z_{2}^{n})\to 0$. Then
$\mathbb{E}(\bar{Z}_{n}^{2})\to\mu^{2}$. Since
$\mathbb{E}(\bar{Z}_{n})\to\mu$, the variance ${\rm Var}(\bar{Z}_{n})\to 0$.
By means of Tchebychev inequality, $\bar{Z}_{n}\to\mu$ in probability. ∎

Lemma 3.3 proves that $\mathbf{E}_{n}(u)\to\mu$ and hence the same holds for
$E_{n}(u)$.

∎

We are ready to prove our main theorem.

###### Proof of Theorem 1.1.

First, by Bernstein’s inequality and union bound, we have that

(3.3)
$\displaystyle\mathbb{P}\left(\sup_{i=1}^{n}|N_{i}-\frac{\epsilon_{n}n}{\pi}|>\lambda\right)\leq
2ne^{-\frac{\frac{1}{2}\lambda^{2}}{{\epsilon_{n}n}/{\pi}+\lambda/3}},\quad\lambda>0.$

For $\lambda=\frac{\epsilon_{n}n}{\pi}$ we obtain

(3.4)
$\displaystyle\mathbb{P}\left(\sup_{i=1}^{n}N_{i}\geq\frac{2\epsilon_{n}n}{\pi}\right)\leq
2ne^{-c\epsilon_{n}n}.$

Similarly, if we call
$N_{ij}=|\\{k\colon|x_{i}-x_{k}|<\epsilon_{n},|x_{j}-x_{k}|<\epsilon_{n}\\}|$
the number of common neighbors of $i$ and $j$, we have
$\mathbb{E}(N_{ij}|{i}\sim{j})\geq\frac{n\epsilon_{n}}{2\pi}$ and

$\displaystyle\mathbb{P}\left(N_{ij}\leq\frac{\epsilon_{n}n}{4\pi}\right)\leq
e^{-c\epsilon_{n}n}.$

Hence,

(3.5) $\displaystyle\mathbb{P}\left(\inf_{i\sim
j}N_{ij}<\frac{n\epsilon_{n}}{4\pi}\right)$
$\displaystyle\leq\sum_{i,j}\mathbb{P}\left(N_{ij}<\frac{n\epsilon_{n}}{4\pi}\Big{|}{i}\sim{j}\right)\mathbb{P}({i}\sim{j})\leq
n^{2}e^{-c\epsilon_{n}n}(\epsilon_{n}/\pi).$

Let $\mathbf{z}\in\mathcal{S}^{n}$, $\mathbf{z}=(z_{0},\dots,z_{n-1})=(e^{{\rm
i}\theta_{0}},\dots,e^{{\rm i}\theta_{n-1}})$ such that $I(\mathbf{z})$ is not
defined. Then there is $k$ with $z_{k}=-z_{k-1}$ and hence we have
$\cos(\theta_{k-1}-\theta_{k})=-1$. For any $\theta\in[0,2\pi)$ we have
$\cos(\theta_{k-1}-\theta)\wedge\cos(\theta_{k}-\theta)\leq 0$. If $G$ is
connected and $r$ is a neighbor of both $k$ and $k-1$ we have

$\sum_{j\sim r}(1-\cos(\theta_{j}-\theta_{r}))\geq 1.$

Hence

(3.6)
$E_{n}(\theta_{0},\dots,\theta_{n-1})=\frac{1}{n}\sum_{i=1}^{n}\displaystyle{\frac{1}{2\epsilon_{n}^{2}N_{i}}}\sum_{j\sim
i}(1-\cos(\theta_{j}-\theta_{i}))\geq\displaystyle{\frac{N_{k,k-1}}{2n\epsilon_{n}^{2}(N_{k}\vee
N_{k-1})}}.$

Due to (3.4) and (3.5) we have for every $q\in\mathbb{Z}$ ,

$\displaystyle\mathbb{P}\left(\displaystyle\inf_{\bm{\theta}\in\partial
K_{q}}E(\bm{\theta})\leq\displaystyle{\frac{1}{16n\epsilon_{n}^{2}}}\right)$
$\displaystyle\leq\mathbb{P}\left(\sup_{i}N_{i}\geq\frac{2\epsilon_{n}n}{\pi}\right)+\mathbb{P}\left(\inf_{i,j}N_{ij}\leq\frac{\epsilon_{n}n}{4\pi}\right)+\mathbb{P}\left(\bigcup_{i=1}^{n}\\{{i}\nsim{i-1}\\}\right)$
$\displaystyle\leq
2ne^{-c\epsilon_{n}n}+n^{2}e^{-c\epsilon_{n}n}+n\left(\frac{\pi-\epsilon_{n}}{\pi}\right)^{n-1}.$

Condition (1.1) guarantees $n\epsilon_{n}^{2}\to 0$ and that
$\sum_{n}n^{2}e^{-c\epsilon_{n}n}<\infty$. Thus,

(3.7) $\lim_{n\to\infty}\displaystyle\inf_{(e^{{\rm
i}\theta_{0}},\dots,e^{{\rm i}\theta_{n-1}})\in\partial
K_{q}}E(\theta_{0},\dots,\theta_{n-1})=+\infty,\qquad\text{a.s.}$

For $q\in\mathbb{Z}$ we consider the function $u_{q}(x)=qx$. Observe that
$(u_{q}(x_{0}),\dots,u_{q}(x_{n-1}))\in K_{q}$ and compute

$\lim_{n\to\infty}E_{n}(u_{q})=\frac{1}{12\pi}\int_{0}^{2\pi}|u^{\prime}_{q}(x)|^{2}\,dx=\frac{q^{2}}{6},\qquad\text{
in probability}.$

Define the event

$A_{n,q}:=\left\\{\inf_{\mathbf{z}\in\partial
K_{q}}E_{n}(\mathbf{z})>\frac{q^{2}}{4}\text{ and
}E_{n}(u_{q})<\frac{q^{2}}{5}\right\\}.$

Due to Proposition 3.1 and (3.7), $\mathbb{P}(A_{n,q})\to 1$ as $n\to\infty$.
Finally, observe that since $\overline{K_{q}}$ is compact and $E_{n}$ is
continuous, it attains a minimum at $\overline{K_{q}}$. If $A_{n,q}$ occurs
this minimum can not be attained in $\partial K_{q}$ and hence there is a
point $u_{q}^{*}\in K_{q}$ with

$E_{n}(u_{q}^{*})\leq E_{n}(u),\qquad\text{ for every }u\in K_{q}.$

Since $K_{q}$ is open, $u_{q}^{*}$ is a local minimum of $E_{n}$ and hence a
stable equilibrium for (1.2). We have proved that for every $q\in\mathbb{Z}$

$\mathbb{P}\left(\eqref{eq:km}\text{ has a stable equilibrium with index
$q$}\right)\geq\mathbb{P}(A_{n,q})\to 1.$

To ensure that $u_{q}^{*}$ is a strict local minima and hence asymptotically
stable, we verify a well-known condition that implies that the Hessian
$D^{2}E_{n}(u_{q}^{*})$ is positive definite, namely

(3.8) $|u_{q}^{*}(x_{i})-u_{q}^{*}(x_{j})|<\frac{\pi}{2},\qquad\text{ for
every }i\sim j,$

(see [13]). Let $B_{n,q}:=\\{\omega\colon\omega\in A_{n,q}\text{ and
\eqref{regularity.uq} does not hold}\\}$. If there is $k\sim\ell$ with
$|u_{q}^{*}(x_{k})-u_{q}^{*}(x_{\ell})|\geq\frac{\pi}{2}$, proceeding as in
(3.6) we bound from below

$E_{n}(u_{q}^{*})=\frac{1}{n}\sum_{i=1}^{n}\displaystyle{\frac{1}{2\epsilon_{n}^{2}N_{i}}}\sum_{j\sim
i}(1-\cos(u_{q}^{*}(x_{i})-u_{q}^{*}(x_{j})))\geq\displaystyle{\frac{N_{k,\ell}}{n\epsilon_{n}^{2}(N_{k}\vee
N_{\ell})}}.$

Since in $A_{n,q}$ we have $E_{n}(u_{q}^{*})\leq E_{n}(u_{q})\leq q^{2}/5$,
using Bernstein’s inequality again we obtain for $n$ large enough,

$\displaystyle\mathbb{P}(B_{n,q})$ $\displaystyle\leq$
$\displaystyle\leq\mathbb{P}\left(A_{n,q}\cap\left\\{E_{n}(u_{q}^{*})\geq\inf_{i\sim
j}\frac{N_{i,j}}{n\epsilon_{n}^{2}(N_{i}\vee N_{j})}\right\\}\right)$
$\displaystyle\leq\mathbb{P}\left(\sup_{i}N_{i}\geq\frac{2\epsilon_{n}n}{\pi}\right)+\mathbb{P}\left(\inf_{i\sim
j}N_{ij}\leq\frac{\epsilon_{n}n}{4\pi}\right)$
$\displaystyle\leq(2n+n^{2})e^{-c\epsilon_{n}n}.$

Since $\mathbb{P}(A_{n,q})\to 1$, we get that

$\mathbb{P}(A_{n,q}\text{ and \eqref{regularity.uq} holds})\to 1.$

In particular,

$\lim_{n\to\infty}\mathbb{P}\left(\eqref{eq:km}\text{ has an asymptotically
stable equilibrium with index $q$}\right)=1.$

∎

##  4\. Discussion

In this section we discuss other models for which our results still holds,
possible extensions and other considerations.

###  4.1. Other graph models.

Using the same arguments, Theorem 1.1 also holds for the following families of
graphs. All are based on nodes $V=\\{x_{0},\dots,x_{n-1}\\}$ i.i.d uniformly
distributed on the unit circle. Different models correspond to different sets
of edges.

a. $k-$nn graphs Two vertices $x_{i}$ and $x_{j}$ are connected by an edge if
the distance between $x_{i}$ and $x_{j}$ is among the $k_{n}-$th smallest
distances from $x_{i}$ to other nodes from $x_{i}$ or vice versa. Condition
(1.1) becomes

$\frac{k_{n}^{2}}{n}\to 0,\qquad\frac{k_{n}}{\log n}\to\infty.$

b. Boolean model. For each node $x_{i}$ we consider a random radius $r_{i}$.
We assume the radii are i.i.d. We declare two nodes $x_{i}$, $x_{j}$ neighbors
if

$(x_{i}-r_{i},x_{i}+r_{i})\cap(x_{j}-r_{j},x_{j}+r_{j})\neq\emptyset.$

The role of $r_{i}$ is similar to the one of $\epsilon_{n}/2$ in the original
model but now they are random. Condition (1.1) becomes

$n\mathbb{E}(r_{i}^{2})\to 0,\qquad\frac{n\mathbb{E}(r_{i})}{\log
n}\to\infty,\qquad\text{ as }n\to\infty.$

c. Random $N-$nn. This is similar to the $k-$nn graph but instead of
considering a deterministic $k$ we choose a random number $N_{i}$ for each
$x_{i}$. The variables $(N_{i})_{0\leq i\leq n-1}$ are i.i.d.

d. Weighted graphs. In any of the previous models or even in WSG networks
(with small $k$) we can consider (random or deterministic) weights as far as
they don’t degenerate as $n\to\infty$. To get a tractable model it is better
to consider a kernel $k:\mathbb{R}\to\mathbb{R}_{\geq 0}$ to be a symmetric,
smooth function with compact support in $(-1,1)$ and $\int k(z)dz=1.$ Then we
consider the weighted graph $G=(V,E)$, where the weights are given by
$w_{ij}=k\left(\epsilon_{n}^{-1}(x_{j}-x_{i})\right)$. For these graphs,
condition (1.1) remains unchanged.

e. Random geometric graphs in an $\epsilon_{n}-$neighborhood of a simple
closed curve. Consider a simple closed curve $\gamma$ and its $\epsilon_{n}$
neighborhood

$\gamma^{\epsilon_{n}}:=\\{x\in\mathbb{R}^{d}\colon
d(x,\gamma)<\epsilon_{n}\\}.$

Here $d(x,\gamma)=\inf_{y\in\gamma}|x-y|$. For $\epsilon_{n}$ small enough
$\gamma^{\epsilon_{n}}$ is homeomorphic to an $\epsilon_{n}-$neighborhood of
the unit circle $C^{\epsilon_{n}}$ and we can work on that setting without
loss of generality. So, consider in $\mathbb{R}^{d}$ the set
$C^{\epsilon_{n}}$ with

$C=\\{(x,y,0,\dots,0)\in\mathbb{R}^{d}\colon x^{2}+y^{2}=1\\}.$

We consider as in the whole manuscript a sample $V=\\{x_{0},\dots,x_{n-1}\\}$
of $n$ i.i.d. uniform points in $C^{\epsilon_{n}}$ and we declare $x_{i}\sim
x_{j}$ if and only if their projections in the unit circle are at distance
less than $\epsilon_{n}$. Observe that this implies that the distance between
them is less than $3\epsilon_{n}$. By working with the projections, we obtain
a random geometric graph in the circle and hence we can apply Theorem 1.1.

###  4.2. Bounds for the existence of $u_{q}$

In the course of the proof of Theorem 1.1 we saw that with high probability
the infimum of the energy on the boundary of any $K_{q}$ is bounded below by
$(16n\epsilon_{n}^{2})^{-1}$. This bound is sharp. Then we expect the event
$A_{n,q}$ to have small probability for $(16n\epsilon_{n}^{2})^{-1}<q^{2}/4$
and large probability when $(16n\epsilon_{n}^{2})^{-1}>q^{2}/4$, which is
equivalent to

$|q|<\frac{1}{2\sqrt{n}\epsilon_{n}}\to\infty.$

Hence, the larger the $|q|$, the larger the $n$ we need to get the existence
of a $q-$twisted state with high probability.

In fact, following the same arguments it can be proved that if
$q_{n}<\frac{1}{2\sqrt{n}\epsilon_{n}}$ for $n$ large enough, then

$\lim_{n\to\infty}\mathbb{P}\left(\eqref{eq:km}\text{ has an asymptotically
stable equilibrium with index $q_{n}$}\right)=1.$

###  4.3. The role of the scaling factor.

Equation (1.2) is scaled according to the factor ${1}/{n\epsilon_{n}^{2}}$.
The goal of this factor is to obtain Proposition 3.1, but once we obtain the
existence of $q-$twisted states for a specific value of $n$, the scaling
factor plays no role and the same conclusion can be obtained for any other
constant used to normalize the energy $E_{n}$.

Acknowledgments. We thank Steven Strogatz for illuminating discussions.

Pablo Groisman and Cecilia De Vita are partially supported by CONICET Grant
PIP 2021 11220200102825CO, UBACyT Grant 20020190100293BA and PICT 2021-00113
from Agencia I+D.

Julián Fernández Bonder is partially supported by CONICET under grant PIP
11220150100032CO and PIP 11220210100238CO and by ANPCyT under grants PICT
2019-3837 and PICT 2019-3530.

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