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# A velocity alignment model on quotient spaces of the Euclidean space

Hansol Park
Department of Mathematical Sciences
Seoul National University, Seoul 08826, Republic of Korea
<EMAIL_ADDRESS>
###### Abstract.

The Cucker-Smale(CS) model is a velocity alignment model, and this model also
has been generalized on general manifolds. We modify the CS model on manifolds
to get rid of a-priori condition on particles’ positions and conditions on
communication functions. Since the shortest geodesic is used to define an
interaction between two particles, if there exist two or more than two
shortest geodesics, then the system is not well-defined. In this paper,
instead of using the shortest geodesic to define an interaction between two
particles, we use all geodesics to define an interaction. From this
assumption, we can relax the a-priori condition and conditions on
communication functions. We also explain the relationship between the
suggested model and previous models. Finally, we provide some emergent
behaviors on some specific manifolds(e.g. flat torus, flat Möbius strip, and
flat Klein bottle). From these results, we can discuss the effect of the
topology of the domain.

###### Key words and phrases:

Universal covering space, Cucker-Smale model, Manifold, Velocity alignment

###### 2020 Mathematics Subject Classification:

70G60, 34D06, 70F10

Acknowledgment. The work of H. Park is supported by
NRF-2020R1A2C3A01003881(National Research Foundation of Korea).

##  1\. Introduction

Analyzing the emergent behavior of dynamical systems is a one of the important
part of the applied mathematics. It can be used to analyze the following
phenomenon: flashing of fireflies [5], schooling of fish [4, 10, 20]. Also, it
can be applied to unmanned aerial vehicles [3], cooperative robot systems [14,
19, 21], variational method [7, 8, 13], shape matching problem [16], rotation
averaging problem [17], and minimization problem [18]. In this paper, we study
a velocity alignment model intensively. One of the famous velocity alignment
model is the Cucker-Smale(CS) model studied in [9], and this model is given as
follows:

(1.1) $\displaystyle\begin{cases}\dot{x}_{i}=v_{i},\quad t>0,\\\
\displaystyle\dot{v}_{i}=\frac{\kappa}{N}\sum_{j=1}^{N}\psi(x_{i},x_{j})(v_{j}-v_{i}),\quad
i=1,2,\cdots,N,\\\ x_{i}(0)=x_{i}^{0}\in\mathbb{R}^{d},\quad
v_{i}(0)=v_{i}^{0}\in\mathbb{R}^{d},\end{cases}$

where $\kappa>0$ is the coupling strength and $\psi$ is the communication
function. Under proper conditions, the velocity alignment of this system, i.e.

$\lim_{t\to\infty}\|v_{i}-v_{j}\|=0\quad\forall~{}1\leq i,j\leq N,$

occurs. System (1.1) is cefined on the Euclidean space. A canonical
generalization of this work is extension of the model on general manifolds.
This extension is studied in [15], and the CS model on general manifolds is
given as follows:

(1.2)
$\displaystyle\begin{cases}\displaystyle\frac{d}{dt}x_{i}=v_{i}\vspace{0.2cm}\\\
\displaystyle\frac{D}{dt}v_{i}=\frac{\kappa}{N}\sum_{k=1}^{N}\psi(x_{i},x_{k})(P_{ik}v_{k}-v_{i}),\\\
(x_{i}(0),v_{i}(0))=(x_{i}^{0},v_{i}^{0})\in TM\quad\forall
i\in\\{1,2,\cdots,N\\},\end{cases}$

where $P_{ik}$ is a parallel transport of a tangent vector on $T_{x_{k}}M$ to
a tangent vector on $T_{x_{i}}M$ and $\frac{D}{dt}$ is a covariant derivative
on $M$. This model was studied on the sphere [15, 2], the hyperbolic space
$\mathbb{H}^{2}$ [1], the special orthogonal group $SO(3)$ [12]. A
disadvantage of this model is that the interaction between $i^{th}$ and
$j^{th}$ particles can not be well-defined if there exist two or more than two
shortest geodesics which connect $x_{i}$ and $x_{j}$. To prevent this
situation, the authors of [15] assumed the following a-priori condition:

“A shortest geodesic between two points $x_{i}(t)$ and $x_{j}(t)$ is unique
for all $i\neq j$, $t\geq 0$.”

Actually, necessity of this a-priori condition comes from the structure of
system (1.2). The shortest geodesic is special in this system, since the
shortest geodesic between $x_{i}$ and $x_{j}$ is used to define an interaction
between $i^{th}$ and $j^{th}$ particles. In this paper, we are interested in
the following question:

$\bullet$(Q): How can we remove the speciality of the shortest geodesic
between $x_{i}$ and $x_{j}$?

To remove the speciality of the shortest geodesic, we used all geodesics which
connect $x_{i}$ and $x_{j}$ to define the interaction between $i^{th}$ and
$j^{th}$ particles. We suggest a modified system on a manifold $M$ as follows:

$\displaystyle\begin{cases}\displaystyle\frac{d}{dt}x_{i}=v_{i},\vspace{0.2cm}\\\
\displaystyle\frac{D}{dt}v_{i}=\frac{\kappa}{N}\sum_{k=1}^{N}\sum_{\gamma\in\Gamma_{x_{k}}^{x_{i}}}\varphi(|\gamma|)(P_{ik}^{\gamma}v_{k}-v_{i}),\quad
t>0,\\\ x_{i}(0)=x_{i}^{0}\in M,\quad v_{i}(0)=v_{i}^{0}\in
T_{x_{i}^{0}}M,\quad\forall i\in\mathcal{N},\end{cases}$

where $\Gamma_{x_{k}}^{x_{i}}$ is a set of all geodesics which starts from
$x_{k}$ and finish at $x_{i}$, and $P_{ik}^{\gamma}$ is a parallel transport
from $x_{k}$ to $x_{i}$ along a geodesic $\gamma$. Here, $\varphi$ is a
communication function depends on the length of a geodesic $\gamma$.
Throughout this paper, we denote the universal covering space of $M$ with the
covering metric by $\tilde{M}$ and $p:\tilde{M}\to M$ is the corresponding
covering map. We can lift this system onto the universal covering space
$\tilde{M}$ as follows:

$\displaystyle\begin{cases}\displaystyle\frac{d}{dt}\tilde{x}_{i}=\tilde{v}_{i},\vspace{0.2cm}\\\
\displaystyle\frac{D}{dt}\tilde{v}_{i}=\frac{\kappa}{N}\sum_{k=1}^{N}\sum_{\tilde{y}_{k}\in
p^{-1}(p(\tilde{x}_{k}))}\varphi(\mathrm{dist}(\tilde{x}_{i},\tilde{y}_{k}))(P_{\tilde{x}_{i}\tilde{y}_{k}}\tilde{u}^{\tilde{y}_{k}}_{k}-\tilde{v}_{i}),\quad
t>0,\\\
\tilde{x}_{i}(0)=\tilde{x}_{i}^{0}\in\tilde{M}^{d},\quad\tilde{v}_{i}(0)=\tilde{v}_{i}^{0}\in
T_{\tilde{x}_{i}^{0}}\tilde{M},\quad\forall~{}i\in\mathcal{N},\end{cases}$

where $p^{-1}_{\tilde{x}}$ is a inverse map of $p$ defined on neighborhood of
$\tilde{x}$ and
$\tilde{u}_{k}^{\tilde{y}_{k}}:=Dp^{-1}_{\tilde{y}_{k}}(v_{k})=Dp^{-1}_{\tilde{y}_{k}}\circ
Dp_{\tilde{x}_{k}}(\tilde{v}_{k})$ and $P_{\tilde{x}\tilde{y}}$ is a parallel
transport from a tangent vector at $\tilde{y}$ to a tangent vector at
$\tilde{x}$. We assume the following two conditions to $M$:

($\mathcal{M}1$): A set $p^{-1}(x)\subset\tilde{M}$ is at most countable set
for any $x\in M$.

($\mathcal{M}2$): For any points $\tilde{x},\tilde{y}\in\tilde{M}$, there
exists a unique geodesic which connects two points $\tilde{x}$ and
$\tilde{y}$.

Here, ($\mathcal{M}1$) is assumed for the well-definedness of sum
$\sum_{\tilde{y}_{k}\in p^{-1}(p(\tilde{x}_{k}))}$, and ($\mathcal{M}2$) is
assumed for the well-definedness of the parallel transport
$P_{\tilde{x}_{i}\tilde{y}_{k}}$. We provide more details about these
assumptions in Section 4.1.

Now, we define the following energy functional

$\mathcal{E}[\mathcal{V}]:=\frac{1}{2}\sum_{i=1}^{N}\|v_{i}\|^{2}$

where $\mathcal{V}:=\\{v_{i}\\}_{i=1}^{N}$. Then $\mathcal{E}$ decreases along
the time evolution(i.e. $\frac{d}{dt}\mathcal{E}\leq 0$). We combine this fact
and Barbalat’s lemma(Lemma 4.1) to obtain the long-time behavior of the system
as follows(See Theorem 4.1):

$\lim_{t\to\infty}\sum_{\gamma\in\Gamma_{x_{k}}^{x_{i}}}\psi(|\gamma|)\|P_{ik}^{\gamma}v_{k}-v_{i}\|^{2}=0\quad\forall~{}1\leq
i,k\leq N.$

If we lift this result to the universal covering space $\tilde{M}$, we have
the follows(See Corollary 4.1):

(1.3) $\displaystyle\lim_{t\to\infty}\sum_{\tilde{y}_{k}\in
p^{-1}(p(\tilde{x}_{k}))}\varphi(\mathrm{dist}(\tilde{x}_{i},\tilde{y}_{k}))\|P_{\tilde{x}_{i}\tilde{y}_{k}}\tilde{u}_{k}^{\tilde{y}_{k}}-\tilde{v}_{i}\|^{2}=0\quad\forall~{}1\leq
i,k\leq N.$

Since $P_{\tilde{x}_{i}\tilde{y}_{k}}\tilde{u}_{k}^{\tilde{y}_{k}}$ can not be
simplified on a general manifold $\tilde{M}$, we assume that $\tilde{M}$ is
the Euclidean space. Also, we simplify condition (1.3) on some specific
manifolds(e.g. flat torus, flat Möbius strip, and flat Klein bottle) in
Section 5. On the flat torus, we could obtain the ordinary velocity alignment

(1.4) $\displaystyle\lim_{t\to\infty}\|v_{i}-v_{j}\|=0\quad\forall~{}1\leq
i,k\leq N,$

under suitable conditions(see Section 5.1). Let the flat Möbius strip and the
flat Klein bottle constructed by the way introduced in Sections 5.2 and 5.3,
respectively. Then, on the flat Möbius strip and the flat Klein bottle, we
could obtain the ordinary velocity alignment (1.4) and the second component of
velocities $v_{i}$ converges for all $1\leq i\leq N$(see Sections 5.2 and
5.3).

The rest of this paper is organized as follows. In Section 2, we introduce the
universal covering space and previous results of the Cucker-Smale(CS) model on
manifolds. In Section 3, we modify the CS model on manifold on the flat torus
$\mathbb{T}^{d}$. From this modification, we suggest the modified CS model on
manifold on general manifold under some assumption in Section 4. We also study
emergent behaviors of the modified CS model on some specific spaces(flat
torus, flat Möbius strip, flat Klein bottle) in Section 5. Finally, Section 6
is devoted to a brief summary of the paper.

Gallery of Notations. Now, we present some notations. Since we use the index
set frequently, we define

$\mathcal{N}:=\\{1,2,\cdots,N\\}.$

Also, for any vector $x\in\mathbb{R}^{d}$, we denote $\alpha^{th}$ component
of $x$ by $(x)_{\alpha}$.

$\text{i.e.}\quad x=((x)_{1},(x)_{2},\cdots,(x)_{d}).$

##  2\. Preliminaries

In this section, we provide some preparatory concepts. Since the goal of this
paper is to construct a velocity alignment model using universal covering
spaces, we provide a simple review of universal covering spaces and prior
studies on velocity alignment models.

###  2.1. Universal covering space of spaces of constant curvature

In this subsection, we provide a review on universal covering spaces of
constant curvature space, since we will consider quotient spaces of the
Euclidean space.

######  Theorem 2.1 (Theorem 4.1 of [11]).

Let $M$ be a $d$-dimensional complete Riemannian manifold with constant
sectional curvature zero. Then the universal covering $\tilde{M}$ of $M$, with
the covering metric, is isometric to $\mathbb{R}^{d}$.

Here, covering metric means that the covering map $p:\tilde{M}\to M$ is a
local isomorphism.

######  Corollary 2.1.

From Theorem 2.1, we have the following results:

(1) The universal covering space of the flat torus $\mathbb{T}^{d}$ is
$\mathbb{R}^{d}$.

(2) The universal covering spaces of the flat Möbius strip $\mathbb{M}$ and
the flat Klein bottle $\mathbb{K}$ are $\mathbb{R}^{2}$.

###  2.2. The Cucker-Smale model on manifolds

In this subsection, we introduce the Cucker-Smale(CS) model on manifolds and
its emergent behaviors. System (1.1) is a velocity alignment model on the
Euclidean space introduced in [9]. It is natural that generalization of this
system onto general manifolds. The authors of [15] suggested the CS model on
general manifolds as system (1.2). Now, we present the previous results of
system (1.2) on various spaces.

####  2.2.1. On the sphere $\mathbb{S}^{d}$

The CS model (1.2) on sphere was first studied in [15]. In this paper, the
authors assumed the following a-priori:

(2.5) $\displaystyle\sup_{0\leq
t<\infty}\max_{i,j\in\mathcal{N}}\mathrm{dist}(x_{i}(t),x_{k}(t))<\pi.$

This a priori assumption allows the well-posedness of the system on the sphere
$\mathbb{S}^{d}$. The authors of [2] improved these result. They assumed the
following condition on the communication function $\psi$:

(2.6)
$\displaystyle\mathrm{dist}(x,y)=\pi\quad\Longrightarrow\quad\psi(x,y)=0.$

The well-posedness of system (1.2) on the sphere can be broken when the
parallel transport $P_{ik}$ is not well defined, and this situation occurs
when $x_{i}$ and $x_{k}$ are the antipodal points of each others for some
$i,k\in\mathcal{N}$. However, if we impose condition (2.6), we do not have to
calculate $P_{ik}v_{k}-v_{i}$ since $\psi(x_{i},x_{k})$ is zero. One of the
useful lemma to show a velocity alignment is Barbalat’s lemma. We provide the
lemma without the proof.

######  Lemma 2.1 (Barbalat’s lemma [6]).

Suppose that a real-valued function $f:[0,\infty)\to\mathbb{R}$ is uniformly
continuous and it satisfies

$\lim_{t\to\infty}\int_{0}^{t}f(s)ds\quad\text{exists}.$

Then, $f$ tends to zero as $t\to\infty$. i.e. $\lim_{t\to\infty}f(t)=0$.

######  Theorem 2.2 (Emergent behavior on the sphere [2]).

Let $\\{(x_{i},v_{i})\\}_{i=1}^{N}$ be a global smooth solution to (1.2) on
the sphere $\mathbb{S}^{d}$ and assume that
$\psi:\mathbb{S}^{d}\times\mathbb{S}^{d}\to\mathbb{R}$ is a positive smooth
function satisfying (2.6). Then, we have the following dichotomy for the
asymptotic dynamics of $\\{(x_{i},v_{i})\\}_{i=1}^{N}$:

(1) either the energy converges to zero:

$\lim_{t\to\infty}\mathcal{E}(t)=0,$

where $\mathcal{E}(t)=\sum_{i=1}^{N}\|v_{i}\|^{2}$.

(2) or the energy converges to a nonzero positive value and all positions
approach to a common great circle asymptotically: for every
$i,j,k\in\mathcal{N}$ and $a,b,c\in\\{1,\cdots,d+1\\}$ we have

$\lim_{t\to\infty}\mathcal{E}(t)>0\quad\text{and}\quad\lim_{t\to\infty}\mathrm{det}\begin{pmatrix}x_{i}^{a}(t)&x_{j}^{a}(t)&x_{k}^{a}(t)\\\
x_{i}^{b}(t)&x_{j}^{b}(t)&x_{k}^{b}(t)\\\
x_{i}^{c}(t)&x_{j}^{c}(t)&x_{k}^{c}(t)\end{pmatrix}=0.$

####  2.2.2. On the hyperbolic space $\mathbb{H}^{2}$

The CS model on the hyperbolic space $\mathbb{H}^{2}$ was studied in [1]. For
any $x,y\in\mathbb{H}^{2}$, there exists the unique geodesic $\gamma$ which
connects $x$ and $y$. This means, the parallel transport $P_{ik}$ between
$x_{i}$ and $x_{k}$ is always well-defined. So, we do not need a condition
which is similar to (2.6) in this case.

######  Theorem 2.3 (Emergent behavior on the hyperbolic space [1]).

Let $\\{(x_{i},v_{i})\\}_{i=1}^{N}$ be a global smooth solution to (1.2) on
the hyperbolic space $\mathbb{H}^{2}$ and assume that
$\psi:\mathbb{H}^{2}\times\mathbb{H}^{2}\to\mathbb{R}$ is a strictly positive
smooth function. Then, we have the following dichotomy for the asymptotic
dynamics of $\\{(x_{i},v_{i})\\}_{i=1}^{N}$:

(1) either the energy tends to zero:

$\lim_{t\to\infty}\mathcal{E}(t)=0.$

(2) or the energy converges to a positive value, and position configuration
becomes coplanar asymptotically:

$\lim_{t\to\infty}\mathcal{E}(t)=\mathcal{E}^{\infty}>0\quad\text{and}\quad\lim_{t\to\infty}\mathrm{det}\Big{(}x_{i}(t)~{}\Big{|}~{}x_{j}(t)~{}\Big{|}x_{k}(t)\Big{)}=0\quad\forall~{}i,j,k\in\mathcal{N}$

where $x_{i}(t)\in\mathbb{H}^{2}\subset\mathbb{R}^{3}$ is considered as a
three-dimensional vector for all $i\in\mathcal{N}$.

####  2.2.3. On the special orthogonal group $SO(3)$

The CS model on $SO(3)$ was studied in [12]. Instead of using
$\\{(x_{i},v_{i})\\}_{i=1}^{N}$, we use the pair
$\\{(R_{i},A_{i})\\}_{i=1}^{N}$ which satisfies

$x_{i}=R_{i},\quad v_{i}=R_{i}A_{i}\quad~{}\forall~{}i\in\mathcal{N}.$

Here, $R_{i}$ is an element of $SO(3)$ and $A_{i}$ is a skew symmetric matrix
of size $3\times 3$. Then, we can express system (1.2) on $SO(3)$ as follows:

(2.7) $\displaystyle\begin{cases}\displaystyle\frac{d}{dt}R_{i}=R_{i}A_{i},\\\
\displaystyle\frac{d}{dt}\mathbf{a}_{i}=\frac{\kappa}{N}\sum_{k=1}^{N}\varphi(R_{i},R_{k})\left[\left(1-\cos\frac{\theta_{ki}}{2}\right)(\mathbf{n}_{ki}\cdot\mathbf{a}_{k})\mathbf{n}_{ki}+\sin\frac{\theta_{ki}}{2}\mathbf{a}_{k}\times\mathbf{n}_{ki}+\cos\frac{\theta_{ki}}{2}\mathbf{a}_{k}-\mathbf{a}_{i},\right]\end{cases}$

where

$\hat{\mathbf{n}}_{ki}=\frac{\hat{\mathbf{u}}_{ki}}{\theta_{ki}},\quad\mathbf{a}_{i}=\check{A}_{i}$

and

$\theta_{ki}=\arccos\left(\frac{\mathrm{tr}(R_{k}^{\top}R_{i})-1}{2}\right),\quad\hat{\mathbf{u}}_{ki}=\frac{\theta_{ki}}{2\sin\theta_{ki}}(R_{k}^{\top}R_{i}-R_{i}^{\top}R_{k}).$

Here, $\hat{\cdot}$ and $\check{\cdot}$ operators are defined as follows:

$\hat{x}:=A,\quad\check{A}:=x,$

where

$x=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3},\quad
A=\begin{pmatrix}0&-x_{3}&x_{2}\\\ x_{3}&0&-x_{1}\\\
-x_{2}&x_{1}&0\end{pmatrix}\in\mathfrak{so}(3).$

######  Theorem 2.4 (Emergent behavior on the special orthogonal group [12]).

Let $\\{(R_{i},A_{i})\\}_{i=1}^{N}$ be a solution to system (2.7). Then, we
have the following dichotomy for the asymptotic dynamics of
$\\{(R_{i},A_{i})\\}_{i=1}^{N}$:

(1) either the kinetic energy tends to zero:

$\lim_{t\to\infty}\mathcal{E}(t)=0.$

(2) or the energy converges to positive value $\mathcal{E}^{\infty}$, and

$\lim_{t\to\infty}(A_{i}(t)-A_{k}(t))=0,\quad\forall~{}i,k\in\mathcal{N}.$

Also, particles approach and rotate with constant speed along a common
geodesic.

Theorems 2.2, 2.3, and 2.4 are results on the different manifolds, however,
the common result is that the particles are aligned on to a common geodesic.

##  3\. Velocity alignment models on the flat torus $\mathbb{T}^{d}$

A velocity alignment model on general manifolds was suggested in [15] and
given as (1.2). If the following a-priori condition is guaranteed in system
(1.2):

“A shortest geodesic between two points $x_{i}(t)$ and $x_{j}(t)$ is unique
for all $i\neq j$, $t\geq 0$,”

then the solution of the system is well-defined and unique. Since the system
contains the parallel transport on the manifold, the a-priori condition should
be assumed for the well-posedness. However, if the domain $M$ has a cut
locus(i.e. there exist at least two shortest geodesic between two points
$x,y\in M$), then the system defined on $M$ is not well-defined in general.
Since we choose the shortest geodesic, we guess that singularities come from
the speciality of the shortest geodesic. Regarding this reason, in this paper,
we suggest a modified system which uses all geodesics to define the
interactions. In this section, we suggest a modified velocity alignment model
on the flat torus $\mathbb{T}^{d}$.

###  3.1. Geodesics on the flat torus

We are planning to consider all geodesics between $x_{i},x_{j}\in M$ to define
the interaction between $i^{th}$ and $j^{th}$ particles. So, in this
subsection, we study the geodesics on the flat torus $\mathbb{T}^{d}$. We
consider the flat torus $\mathbb{T}^{d}$ as a quotient space of the Euclidean
space

$\mathbb{T}^{d}\simeq\mathbb{R}^{d}/_{\sim},$

where the equivalence relation $\sim$ given as

$[x_{1},x_{2},\cdots,x_{d}]\sim[x_{1}+n_{1},x_{2}+n_{2},\cdots,x_{d}+n_{d}]\quad\forall~{}(n_{1},\cdots,n_{d})\in\mathbb{Z}^{d}.$

Note that the metric of the torus was induced from $\mathbb{R}^{d}$.

####  3.1.1. The case when $d=1$

Before we study general dimension, we consider the simplest case. Since the
universal covering space of $\mathbb{T}^{1}$ is $\mathbb{R}$, we can define a
covering map $p$ as follows:

$p:\mathbb{R}\to\mathbb{T}^{1},\quad x\mapsto x+\mathbb{Z}.$

Let $x,y\in\mathbb{T}^{1}$ be two points on the one dimensional flat torus and
$\gamma$ be a one of geodesic which connects two points $x$ and $y$. Then
there exists two points $\tilde{x},\tilde{y}\in\mathbb{R}$ such that
$\gamma=p([\tilde{x},\tilde{y}])$. Now we will express the following set:

$\Gamma_{x}^{y}:=\\{\text{geodesics connecting $x$ and $y$}\\}.$

From the previous argument, we can express the set of geodesics which connects
$x$ and $y$ as follows:

$\tilde{\gamma}:[0,1]\to\mathbb{R}^{1},\quad\tilde{\gamma}(t)=\tilde{x}+t(\tilde{y}-\tilde{x}),\quad\gamma=p\circ\tilde{\gamma}.$

Since $\tilde{x}$ and $\tilde{y}$ are not unique, we can express all geodesics
$\gamma_{n}$ with $n\in\mathbb{Z}$ as follows:

$\displaystyle\tilde{\gamma}_{n}:[0,1]\to\mathbb{R}^{1},\quad\tilde{\gamma}_{n}(t)=\tilde{x}+t(\tilde{y}-\tilde{x}+n),$
$\displaystyle\gamma_{n}:[0,1]\to\mathbb{T}^{1},\quad\gamma_{n}=p\circ\tilde{\gamma}_{n}.$

Then $\Gamma_{x}^{y}=\\{\gamma_{n}\\}_{n\in\mathbb{Z}}$ is the set of geodesic
from $x$ to $y$.

######  Remark 3.1.

From the above definition, we have the following properties.

  1. (1)

The length of $\gamma_{n}$ is given as

$|\gamma_{n}|=|\tilde{y}-\tilde{x}+n|.$

  2. (2)

Since $\mathbb{R}^{1}/\mathbb{T}^{1}\simeq\mathbb{Z}$, we can find a
correspondence between $\Gamma_{x}^{y}$ and $\mathbb{Z}$.

####  3.1.2. The general case $d\geq 2$

Now we make some arguments on $\mathbb{T}^{d}$ case. Recall that the universal
covering of $\mathbb{T}^{d}$ is $\mathbb{R}^{d}$ with the following covering
map $p$:

$p:\mathbb{R}^{d}\to\mathbb{T}^{d},\quad x\mapsto x+\mathbb{Z}^{d}.$

From a similar argument that we made in the case of $d=1$, we can express
$\Gamma_{x}^{y}$ as follows:

$\Gamma_{x}^{y}=\\{\gamma_{\vec{n}}\\}_{\vec{n}\in\mathbb{Z}^{d}},$

where $\vec{n}\in\mathbb{Z}^{d}$ and

(3.8) $\displaystyle\begin{aligned}
&\tilde{\gamma}_{\vec{n}}:[0,1]\to\mathbb{R}^{d},\quad\tilde{\gamma}_{\vec{n}}(t)=\tilde{x}+t(\tilde{y}-\tilde{x}+\vec{n}),\\\
&\gamma_{\vec{n}}:[0,1]\to\mathbb{T}^{d},\quad\gamma_{\vec{n}}=p\circ\tilde{\gamma}_{\vec{n}}.\end{aligned}$

$x$$y$$p$$p^{-1}$ Figure 1. The universal covering of the two dimensional flat
torus $\tilde{\mathbb{T}}^{2}$ on $\mathbb{R}^{2}$. White points are
$p^{-1}(x)$ and a black point is an element of $p^{-1}(y)$. Drawn line
segments are geodesic connecting the black point and white points.

######  Remark 3.2.

From the above definition, we have the following properties.

  1. (1)

The length of $\gamma_{\vec{n}}$ with $\vec{n}\in\mathbb{Z}^{d}$ is given as

$|\gamma_{\vec{n}}|=\|\tilde{y}-\tilde{x}+\vec{n}\|.$

  2. (2)

Since $\mathbb{R}^{d}/\mathbb{T}^{d}\simeq\mathbb{Z}^{d}$, we can find a
correspondence between $\Gamma_{x}^{y}$ and $\mathbb{Z}^{d}$.

See Figure 1 for the case of $d=2$, we express the geodesics of the flat torus
on its universal covering space. We expressed all of the geodesics between
$x,y\in M$ as follows:

(3.9)
$\displaystyle\Gamma_{x}^{y}=\big{\\{}p([\tilde{x}_{0},\tilde{y}]):\text{ for
a fixed }\tilde{x}_{0}\in p^{-1}(x)\text{ and for all }\tilde{y}\in
p^{-1}(y)\big{\\}},$

where $[\tilde{x}_{0},\tilde{y}]$ is a line segment on $\mathbb{R}^{d}$ which
connects $\tilde{x}_{0}$ and $\tilde{y}$. Now, we are ready to construct a
modified model on the flat torus.

###  3.2. Construction of the modified model on the flat torus

Before we define a system on the flat torus, we define a system on its
covering space $\mathbb{R}^{d}$. Since we expressed the set of geodesics
between two points $x,y$ as (3.9), we can construct a system which considers
all geodesics between two particles as follows:

(3.10)
$\displaystyle\begin{cases}\displaystyle\frac{d}{dt}\tilde{x}_{i}=\tilde{v}_{i},\vspace{0.2cm}\\\
\displaystyle\frac{D}{dt}\tilde{v}_{i}=\frac{\kappa}{N}\sum_{k=1}^{N}\sum_{\tilde{y}_{k}\in
p^{-1}(p(\tilde{x}_{k}))}\psi(\tilde{x}_{i},\tilde{y}_{k})(P_{\tilde{x}_{i}\tilde{y}_{k}}\tilde{v}_{k}-\tilde{v}_{i}),\quad
t>0,\\\
\tilde{x}_{i}(0)=\tilde{x}_{i}^{0}\in\mathbb{R}^{d},\quad\tilde{v}_{i}(0)=\tilde{v}_{i}^{0}\in
T_{\tilde{x}_{i}^{0}}\mathbb{R}^{d},\quad\forall~{}i\in\mathcal{N},\end{cases}$

where $P_{\tilde{x}_{i}\tilde{y}_{k}}$ is a parallel transport from a tangent
vector at $\tilde{y}_{k}$ to a tangent vector at $\tilde{x}_{k}$. Here, we
assume that the sum $\sum_{\tilde{y}_{k}\in p^{-1}(p(\tilde{x}_{k}))}$ is
well-defined. We will discuss conditions for $\psi$ in the last of this
subsection. Since the flat torus has zero curvatures, the parallel transport
$P_{\tilde{x}_{i}\tilde{y}_{k}}$ is not necessary in this case. However, for a
future generalization on general manifolds, we leave
$P_{\tilde{x}_{i}\tilde{y}_{k}}$. Now, we assume that $\psi(x,y)$ is a
function of $\mathrm{dist}_{\mathbb{R}^{d}}(x,y)$ and define a function
$\varphi:\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ as follows:

$\varphi(\mathrm{dist}_{\mathbb{R}^{d}}(x,y))=\psi(x,y)\quad\forall~{}x,y\in\mathbb{T}^{d}.$

If we put $x_{i}=p(\tilde{x}_{i})$ and $v_{i}=Dp(\tilde{v}_{i})$ for all
$i\in\mathcal{N}$, we can reduce system (3.10) to $M$ as follows:

(3.11)
$\displaystyle\begin{cases}\displaystyle\frac{d}{dt}x_{i}=v_{i},\vspace{0.2cm}\\\
\displaystyle\frac{D}{dt}v_{i}=\frac{\kappa}{N}\sum_{k=1}^{N}\sum_{\gamma\in\Gamma_{x_{k}}^{x_{i}}}\varphi(|\gamma|)(P^{\gamma}_{ik}v_{k}-v_{i}),\quad
t>0,\\\ x_{i}(0)=x_{i}^{0}\in\mathbb{T}^{d},\quad v_{i}(0)=v_{i}^{0}\in
T_{x_{i}^{0}}\mathbb{T}^{d},\quad\forall~{}i\in\mathcal{N},\end{cases}$

where $P^{\gamma}_{ik}$ is a parallel transport from a tangent vector at
$x_{k}$ to a tangent vector at $x_{i}$ along a curve $\gamma$ and $|\gamma|$
is the length of $\gamma$. Recall that
$P_{ik}^{\gamma_{1}}=P_{ik}^{\gamma_{2}}$ for any
$\gamma_{1},\gamma_{2}\in\Gamma_{x_{k}}^{x_{i}}$, since the domain of this
system is the flat torus. So, we denote that $P_{ik}:=P_{ik}^{\gamma}$ for all
$\gamma\in\Gamma_{x_{k}}^{x_{i}}$. If we define

(3.12)
$\displaystyle\Phi(x_{i},x_{k}):=\sum_{\gamma\in\Gamma_{x_{k}}^{x_{i}}}\varphi(|\gamma|)$

for all $i,k\in\mathcal{N}$, then system (3.11) can be expressed as follows:

(3.13)
$\displaystyle\begin{cases}\displaystyle\frac{d}{dt}x_{i}=v_{i},\vspace{0.2cm}\\\
\displaystyle\frac{D}{dt}v_{i}=\frac{\kappa}{N}\sum_{k=1}^{N}\Phi(x_{i},x_{k})(P_{ik}v_{k}-v_{i}),\quad
t>0,\\\ x_{i}(0)=x_{i}^{0}\in\mathbb{T}^{d},\quad v_{i}(0)=v_{i}^{0}\in
T_{x_{i}^{0}}\mathbb{T}^{d},\quad\forall~{}i\in\mathcal{N}.\end{cases}$

Now, we discuss conditions for the communication function $\psi$ to guarantee
the well-posedness of system (3.10). Actually, it is equivalent to find a
condition for the convergence of $\Phi$ in (3.12). From the definition of
$\Phi$ and Remark 3.2 (1), we have the following calculation:

$\displaystyle\Phi(x,y)=\sum_{\gamma\in\Gamma_{y}^{x}}\varphi(|\gamma|)=\sum_{\vec{n}\in\mathbb{Z}^{d}}\varphi(\|\tilde{y}-\tilde{x}+\vec{n}\|).$

Now we assume the following two conditions for $\varphi$:

($\mathcal{A}1$) A function $\varphi:\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq
0}$ is a continuous decreasing function.

($\mathcal{A}2$) A sum
$\displaystyle\sum_{\vec{n}\in\mathbb{Z}^{d}}\varphi(\|\tilde{y}-\tilde{x}+\vec{n}\|)$
converges.

Here, we assumed $(\mathcal{A}1)$, since $\psi$ is a communication function.
$(\mathcal{A}2)$ is assumed for the well-definedness of system (3.10) as we
mentioned before. For a further argument, we provide the following lemma.

######  Lemma 3.1.

Let $\tilde{x},\tilde{y}\in\mathbb{R}^{d}$, then we have the following
inequality:

$\min_{\vec{n}\in\mathbb{Z}^{d}}\|\tilde{y}-\tilde{x}+\vec{n}\|\leq\frac{\sqrt{d}}{2}.$

###### Proof.

From a simple calculation, we get

$\min_{\vec{n}\in\mathbb{Z}^{d}}\|\tilde{y}-\tilde{x}+\vec{n}\|^{2}=((\tilde{y})_{1}-(\tilde{x})_{1}+(\vec{n})_{1})^{2}+\cdots+((\tilde{y})_{d}-(\tilde{x})_{d}+(\vec{n})_{d})^{2}.$

Now, we pick $\vec{m}\in\mathbb{Z}^{d}$ as follows:

(3.14)
$\displaystyle(\vec{m})_{i}=\begin{cases}[(\tilde{x})_{i}-(\tilde{y})_{i}]\qquad&\text{if
}\quad
0\leq((\tilde{x})_{i}-(\tilde{y})_{i})-[(\tilde{x})_{i}-(\tilde{y})_{i}]<\frac{1}{2},\\\
[(\tilde{x})_{i}-(\tilde{y})_{i}]+1\qquad&\text{if
}\quad\frac{1}{2}\leq((\tilde{x})_{i}-(\tilde{y})_{i})-[(\tilde{x})_{i}-(\tilde{y})_{i}]<1\end{cases}$

where $[x]$ is the greatest integer which is not greater than $x$. Then we
know that
$-\frac{1}{2}\leq(\tilde{y})_{i}-(\tilde{x})_{i}+(\vec{m})_{i}\leq\frac{1}{2}$
for all $1\leq i\leq N$ and this yields

$\min_{\vec{n}\in\mathbb{Z}^{d}}\|\tilde{y}-\tilde{x}+\vec{n}\|^{2}\leq\|\tilde{y}-\tilde{x}+\vec{m}\|^{2}\leq
d\times\left(\frac{1}{2}\right)^{2}.$

Finally we can conclude that

$\min_{\vec{n}\in\mathbb{Z}^{d}}\|\tilde{y}-\tilde{x}+\vec{n}\|\leq\frac{\sqrt{d}}{2}.$

This is the desired result. ∎

###  3.3. Condition of the communication functions $\varphi$

In this subsection, we study the condition for $\varphi$ which satisfies

$\Phi(x,y)=\sum_{\vec{n}\in\mathbb{Z}^{d}}\varphi(\|\tilde{y}-\tilde{x}+\vec{n}\|)<\infty\quad\forall~{}x,y\in
M.$

Since $\varphi$ is continuous, non-negative and decreasing, we can apply the
integral test on $\mathbb{R}^{d}$ to get

$\sum_{\vec{n}\in\mathbb{Z}^{d}}\varphi(\|\tilde{y}-\tilde{x}+\vec{n}\|)\quad\text{converges}\quad\Leftrightarrow\quad\int_{\mathbb{R}^{d}}\varphi(|x|)dx<\infty.$

From the radial symmetry, we have

$\int_{\mathbb{R}^{d}}\varphi(|x|)dx=|\mathbb{S}^{d-1}|\int_{0}^{\infty}r^{d-1}\varphi(r)dr,$

where $|\mathbb{S}^{d-1}|$ is a $(d-1)$-dimensional Hausdorff measure of the
unit $(d-1)$-dimensional sphere. Actually, $|\mathbb{S}^{d-1}|$ can be
expressed as

(3.15)
$\displaystyle|\mathbb{S}^{d-1}|=\frac{d\pi^{d/2}}{\Gamma\left(\frac{d}{2}+1\right)},$

where $\Gamma$ is the gamma function. Since (3.15) only depends on $d$, we
have

$\sum_{\vec{n}\in\mathbb{Z}^{d}}\varphi(\|\tilde{y}-\tilde{x}+\vec{n}\|)\quad\text{converges}\quad\Leftrightarrow\quad\int_{0}^{\infty}r^{d-1}\varphi(r)dr<\infty.$

So we can conclude that the condition ($\mathcal{A}1$) and ($\mathcal{A}2$) is
equivalent to the following condition.

($\mathcal{A}$): A function $\varphi:\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq
0}$ is a continuous decreasing function and satisfies
$\int_{0}^{\infty}r^{d-1}\varphi(r)dr$ exists.

Now, we study some examples of communication functions $\varphi$ which satisfy
($\mathcal{A}$).

######  Example 3.1.

(1) When $\varphi$ has a compact support: Let
$\mathrm{supp}\varphi\subset[0,A]$. Then of course,
$\int_{\mathbb{R}^{d}}r^{d-1}\varphi(r)dr<\infty$.

(2) When $\varphi(r)=e^{-r}$: From a simple calculation, we have

$\displaystyle\int_{0}^{\infty}r^{d-1}e^{-r}dr$
$\displaystyle=[-r^{d-1}e^{-r}]_{0}^{\infty}+(d-1)\int_{0}^{\infty}r^{d-2}e^{-r}dr$
$\displaystyle=\cdots$
$\displaystyle=(d-1)!\int_{0}^{\infty}e^{-r}dr=(d-1)!<\infty.$

If $d=1$, we can calculate the explicit form of $\Phi(x,y)$. From a similar
argument that we used in (3.14), we can choose $\tilde{x}\in p^{-1}(x)$ and
$\tilde{y}\in p^{-1}(y)$ which satisfy $0\leq\delta:=\tilde{x}-\tilde{y}<1$.
Then we have

$\displaystyle\Psi(x,y)$
$\displaystyle=\sum_{n\in\mathbb{Z}}e^{-|\tilde{x}-\tilde{y}+n|}=\sum_{n\in\mathbb{Z}}e^{-|\delta+n|}=\sum_{n=0}^{\infty}e^{-(\delta+n)}+\sum_{n=1}^{\infty}e^{(\delta-n)}$
$\displaystyle=e^{-\delta}\frac{1}{1-e^{-1}}+e^{\delta}\frac{e^{-1}}{1-e^{-1}}=\frac{e^{1-\delta}+e^{\delta}}{e-1}=\frac{\sinh(\delta-1/2)}{\sinh(1/2)}.$

(3) When $\varphi(r)=\frac{1}{(1+r^{2})^{\alpha}}$ with $\alpha>\frac{d}{2}$:
From a simple calculation, we have

$\displaystyle\int_{1}^{\infty}\frac{r^{d-1}}{(1+r^{2})^{\alpha}}dr<\int_{1}^{\infty}r^{d-1-2\alpha}dr<\infty.$

##  4\. Velocity alignment models on general manifold

In this section, we generalize the result of Section 3 to a general manifold
$M$.

###  4.1. Construction of the modified model on general manifold

We can simply generalize system (3.11) on general manifold $M$ as follows:

(4.16)
$\displaystyle\begin{cases}\displaystyle\frac{d}{dt}x_{i}=v_{i},\vspace{0.2cm}\\\
\displaystyle\frac{D}{dt}v_{i}=\frac{\kappa}{N}\sum_{k=1}^{N}\sum_{\gamma\in\Gamma_{x_{k}}^{x_{i}}}\varphi(|\gamma|)(P_{ik}^{\gamma}v_{k}-v_{i}),\quad
t>0,\\\ x_{i}(0)=x_{i}^{0}\in M,\quad v_{i}(0)=v_{i}^{0}\in
T_{x_{i}^{0}}M,\quad\forall i\in\mathcal{N}.\end{cases}$

We only have to concern about the well-definedness and the convergence of the
sum $\sum_{\gamma\in\Gamma_{x_{k}}^{x_{i}}}$. Since arguing these issues on
the universal covering space is easier, we lift system (4.16) defined on $M$
to a system on its universal covering space. We bring system (3.10) on general
manifold as follows:

(4.17)
$\displaystyle\begin{cases}\displaystyle\frac{d}{dt}\tilde{x}_{i}=\tilde{v}_{i},\vspace{0.2cm}\\\
\displaystyle\frac{D}{dt}\tilde{v}_{i}=\frac{\kappa}{N}\sum_{k=1}^{N}\sum_{\tilde{y}_{k}\in
p^{-1}(p(\tilde{x}_{k}))}\varphi(\mathrm{dist}(\tilde{x}_{i},\tilde{y}_{k}))(P_{\tilde{x}_{i}\tilde{y}_{k}}\tilde{u}^{\tilde{y}_{k}}_{k}-\tilde{v}_{i}),\quad
t>0,\\\
\tilde{x}_{i}(0)=\tilde{x}_{i}^{0}\in\tilde{M}^{d},\quad\tilde{v}_{i}(0)=\tilde{v}_{i}^{0}\in
T_{\tilde{x}_{i}^{0}}\tilde{M},\quad\forall~{}i\in\mathcal{N},\end{cases}$

where $p^{-1}_{\tilde{x}}$ is a inverse map of $p$ defined on neighborhood of
$\tilde{x}$ and
$\tilde{u}_{k}^{\tilde{y}_{k}}=Dp^{-1}_{\tilde{y}_{k}}(v_{k})=Dp^{-1}_{\tilde{y}_{k}}\circ
Dp_{\tilde{x}_{k}}(\tilde{v}_{k})$ and $P_{\tilde{x}\tilde{y}}$ is a parallel
transport from a tangent vector at $\tilde{y}$ to a tangent vector at
$\tilde{x}$. Since a sum $\sum_{\tilde{y}_{k}\in p^{-1}(p(\tilde{x}_{k}))}$
should be well-defined, we assume the follows:

($\mathcal{M}1$): A set $p^{-1}(x)\subset\tilde{M}$ is at most countable set
for any $x\in M$.

When $M=\mathbb{T}^{d}$, we have already showed that there exists a one-to-one
correspondence between $p^{-1}(x)$ and $\mathbb{Z}^{d}$ in Section 3.1. So, in
this case, obviously $p^{-1}(x)$ is countable for any $x\in\mathbb{T}^{d}$. On
the other hand, we know that $P_{\tilde{x}_{i}\tilde{y}_{k}}$ only depends on
two points $\tilde{x}_{i}$ and $\tilde{y}_{k}$. We should assume the following
property of $\tilde{M}$:

($\mathcal{M}2$): For any points $\tilde{x},\tilde{y}\in\tilde{M}$, there
exists a unique geodesic which connects two points $\tilde{x}$ and
$\tilde{y}$.

If $\tilde{M}$ is Euclidean space or has constant negative
curvature(Hyperbolic space), then it satisfies ($\mathcal{M}2$). System (4.17)
given on a manifold $M$ is well defined for any $M$ which satisfies
($\mathcal{M}1$) and ($\mathcal{M}2$). Especially, if
$\tilde{M}=\mathbb{R}^{d}$, then the parallel transport
$P_{\tilde{x}_{i}\tilde{y}_{k}}$ in (4.17) can be omitted. In this case,
analyzing system (4.17) is easier than analyzing system (4.16), since we do
not have to consider the parallel transport on $M$. In Section 5, we study
some examples of $M$(e.g. flat torus, flat Möbius strip, and flat Klein
bottle) which of the universal covering space is the Euclidean spaces.

###  4.2. Reduction to the original CS model on manifold

In this section, we introduce the relationship between system (4.16) and
previous velocity alignment models. If we consider that system (4.16) is
defined on the Euclidean space(i.e. $M=\mathbb{R}^{d}$), then there is the
unique geodesic $\gamma$ which connects two points $x,y\in M$ and its length
is $\mathrm{dist}(x,y)$. So the set of geodesics $\Gamma_{x}^{y}$ only
contains one element. This implies that system (4.16) can be reduced to the
original CS system (1.1).

Now, we compare two systems (1.2) and (4.16) given on $\mathbb{T}^{1}$. We
define $\varphi$ which satisfies $\varphi\left(\frac{1}{2}\right)=0$. If
$x,y\in\mathbb{T}^{1}$, then the length of geodesics are
$|\tilde{y}-\tilde{x}+n|$ for any $n\in\mathbb{Z}$. We know that there are at
most one $m\in\mathbb{Z}$ such that $|\tilde{y}-\tilde{x}+m|<\frac{1}{2}$.
This implies that only the shortest geodesic between two points $x_{i}$ and
$x_{k}$ determine the interaction between $i^{th}$ and $k^{th}$ particles.
i.e. system (4.16) can be reduced as follows:

$\displaystyle\begin{cases}\displaystyle\frac{d}{dt}x_{i}=v_{i},\vspace{0.2cm}\\\
\displaystyle\frac{D}{dt}v_{i}=\frac{\kappa}{N}\sum_{k=1}^{N}\varphi(\mathrm{dist}(x_{i},x_{k}))(P_{ik}v_{k}-v_{i}),\quad
t>0,\\\ x_{i}(0)=x_{i}^{0}\in\mathbb{T}^{1},\quad v_{i}(0)=v_{i}^{0}\in
T_{x_{i}^{0}}\mathbb{T}^{1},\quad\forall i\in\mathcal{N},\end{cases}$

where $P_{ik}$ is a parallel transport from the tangent space at $x_{i}$ to
the tangent space at $x_{k}$ along shortest geodesic between them. This system
is exactly same with system (1.2). So, we can conclude that a modified
velocity alignment system (4.16) can be reduced to a previous system (1.2).

###  4.3. Emergent behaviors of the modified model

In this subsection, we study the emergent behaviors of system (4.16). We
define the following energy functional:

$\displaystyle\mathcal{E}(\mathcal{V})=\frac{1}{2}\sum_{i=1}^{N}\|v_{i}\|^{2},$

where $\mathcal{V}=\\{v_{i}\\}_{i=1}^{N}$.

######  Lemma 4.1.

Let $(\mathcal{X},\mathcal{V})$ be a solution of system (4.16). Then we have

$\displaystyle\frac{d}{dt}\mathcal{E}(\mathcal{V})=-\frac{\kappa}{2N}\sum_{i,k=1}^{N}\sum_{\gamma\in\Gamma_{x_{k}}^{x_{i}}}\psi(|\gamma|)\|P_{ik}^{\gamma}v_{k}-v_{i}\|^{2}.$

i.e. the energy of the system is a non-increasing function.

###### Proof.

From the simple calculation, we have

$\displaystyle\frac{d}{dt}\mathcal{E}(\mathcal{V})$
$\displaystyle=\sum_{i=1}^{N}\left\langle\frac{D}{dt}v_{i},v_{i}\right\rangle$
$\displaystyle=\frac{\kappa}{N}\sum_{i,k=1}^{N}\sum_{\gamma\in\Gamma_{x_{k}}^{x_{i}}}\psi(|\gamma|)\langle
P_{ik}^{\gamma}v_{k}-v_{i},v_{i}\rangle$
$\displaystyle=\frac{\kappa}{2N}\sum_{i,k=1}^{N}\sum_{\gamma\in\Gamma_{x_{k}}^{x_{i}}}\psi(|\gamma|)\langle
P_{ik}^{\gamma}v_{k}-v_{i},v_{i}\rangle+\frac{\kappa}{2N}\sum_{i,k=1}^{N}\sum_{\gamma\in\Gamma_{x_{i}}^{x_{k}}}\psi(|\gamma|)\langle
P_{ki}^{\gamma}v_{i}-v_{k},v_{k}\rangle$
$\displaystyle=\frac{\kappa}{2N}\sum_{i,k=1}^{N}\sum_{\gamma\in\Gamma_{x_{k}}^{x_{i}}}\psi(|\gamma|)\langle
P_{ik}^{\gamma}v_{k}-v_{i},v_{i}\rangle+\frac{\kappa}{2N}\sum_{i,k=1}^{N}\sum_{\gamma\in\Gamma_{x_{k}}^{x_{i}}}\psi(|\gamma|)\langle
P_{ki}^{-\gamma}v_{i}-v_{k},v_{k}\rangle$
$\displaystyle=\frac{\kappa}{2N}\sum_{i,k=1}^{N}\sum_{\gamma\in\Gamma_{x_{k}}^{x_{i}}}\psi(|\gamma|)\Big{(}\langle
P_{ik}^{\gamma}v_{k}-v_{i},v_{i}\rangle+\langle
P_{ki}^{-\gamma}v_{i}-v_{k},v_{k}\rangle\Big{)}$
$\displaystyle=-\frac{\kappa}{2N}\sum_{i,k=1}^{N}\sum_{\gamma\in\Gamma_{x_{k}}^{x_{i}}}\psi(|\gamma|)\|P_{ik}^{\gamma}v_{k}-v_{i}\|^{2},$

where $-\gamma$ stands for the inverse path of $\gamma$. We have the desired
result. ∎

######  Remark 4.1.

It is easy to show that the second temporal derivative of
$\mathcal{E}(\mathcal{V})$ is bounded.

We combine Lemma 4.1 and Barbalat’s lemma(Lemma ) to obtain the following
theorem.

######  Theorem 4.1.

Let $(\mathcal{X},\mathcal{V})$ be a solution of system (4.16). Then we have

$\displaystyle\lim_{t\to\infty}\sum_{\gamma\in\Gamma_{x_{k}}^{x_{i}}}\psi(|\gamma|)\|P_{ik}^{\gamma}v_{k}-v_{i}\|^{2}=0$

for any $i,k\in\mathcal{N}$.

From this theorem, we have the following corollary.

######  Corollary 4.1.

Let $(\tilde{\mathcal{X}},\tilde{\mathcal{V}})$ be a solution of system
(4.17). Then we have

$\displaystyle\lim_{t\to\infty}\sum_{\tilde{y}_{k}\in
p^{-1}(p(\tilde{x}_{k}))}\varphi(\mathrm{dist}(\tilde{x}_{i},\tilde{y}_{k}))\|P_{\tilde{x}_{i}\tilde{y}_{k}}\tilde{u}_{k}^{\tilde{y}_{k}}-\tilde{v}_{i}\|^{2}=0,$

where $P_{\tilde{x}_{i}\tilde{y}_{k}}\tilde{u}_{k}^{\tilde{y}_{k}}$ is defined
in (4.17).

Since given models (4.16) and (4.17) are defined on too general manifold $M$,
we can not provide more specific emergent behaviors. In Section 5, we provide
more specific emergent behaviors of system (4.16) and (4.17) on the specific
manifold $M$.

###  4.4. Self-interaction effect

In a previous velocity alignment model on manifold (1.2), $i^{th}$ particle
does not interact with itself, since $P_{ii}$ is the identity map and this
yields $P_{ii}v_{i}-v_{i}=0$. However, system (4.16) is different. Since we
considered all geodesics which is connecting $x_{i}$ and $x_{j}$ to define the
interaction between $i^{th}$ and $j^{th}$ particles, if there exists non-
trivial geodesic starts from $x_{i}$ and finish at $x_{i}$ then $i^{th}$
particle interacts with itself. In this subsection, we focus on this effect.
We put $N=1$ and $(x,v):=(x_{1},v_{1})$ to system (4.16). Then we get

(4.18)
$\displaystyle\begin{cases}\displaystyle\frac{d}{dt}x=v,\vspace{0.2cm}\\\
\displaystyle\frac{D}{dt}v=\kappa\sum_{\gamma\in\Gamma_{x}^{x}}\varphi(|\gamma|)(P^{\gamma}v-v),\quad
t>0,\\\ x(0)=x^{0}\in M,\quad v(0)=v^{0}\in T_{x^{0}}M,\end{cases}$

where $P^{\gamma}$ is a parallel transport along $\gamma$. From Lemma 4.1, we
know that

(4.19)
$\displaystyle\frac{d}{dt}\left(\frac{1}{2}\|v\|^{2}\right)=-\frac{\kappa}{2}\sum_{\gamma\in\Gamma_{x}^{x}}\varphi(|\gamma|)\|P^{\gamma}v-v\|^{2}.$

This yields

$\frac{d}{dt}\|v\|^{2}=-\kappa\sum_{\gamma\in\Gamma_{x}^{x}}\varphi(|\gamma|)\|P^{\gamma}v-v\|^{2}\geq-4\kappa\sum_{\gamma\in\Gamma_{x}^{x}}\varphi(|\gamma|)\|v\|^{2},$

and from a simple fact of ODE, we know that if $v^{0}\neq 0$, then $v(t)\neq
0$ for all $t\geq 0$. So we can define $u(t)$ for all $t\geq 0$ as follows:

(4.20) $\displaystyle u(t):=\frac{v(t)}{\|v(t)\|}.$

Now we substitute (4.20) into (4.19) to get

$\frac{d}{dt}\|v\|^{2}=-\kappa\|v\|^{2}\sum_{\gamma\in\Gamma_{x}^{x}}\varphi(|\gamma|)\|P^{\gamma}u-u\|^{2}$

or equivalently,

$\frac{d}{dt}\ln\|v\|=-\frac{\kappa}{2}\sum_{\gamma\in\Gamma_{x}^{x}}\varphi(|\gamma|)\|P^{\gamma}u-u\|^{2}.$

From this, we can conclude that the self-interaction effect reduces the speed
of particle, and the ratio is at most exponential. If we apply Theorem 4.1 to
system (4.18), then we get

(4.21)
$\displaystyle\lim_{t\to\infty}\|v\|^{2}\sum_{\gamma\in\Gamma_{x}^{x}}\varphi(|\gamma|)\|P^{\gamma}u-u\|^{2}=0.$

Finally, we can obtain a dichotomy for the long-time behaviors of system
(4.18):

(1) The speed converges to zero. i.e.

$\lim_{t\to\infty}\|v\|=0.$

(2) The direction of the velocity $u$ satisfies

$\lim_{t\to\infty}\sum_{\gamma\in\Gamma_{x}^{x}}\varphi(|\gamma|)\|P^{\gamma}u-u\|^{2}=0.$

This effect comes from the topology of the domain. This argument has not been
argued before.

##  5\. Systems on quotient spaces of the Euclidean space

In this section, we study system (4.16) on $M$ which is quotient spaces of the
Euclidean space. We also assume that the universal covering space of $M$ is
the Euclidean space $\mathbb{R}^{d}$. If we consider system (4.16) on its
universal covering space $\tilde{M}=\mathbb{R}^{d}$, then we have system
(4.17). Since $\tilde{M}=\mathbb{R}^{d}$, we can omit the parallel transport
$P_{\tilde{x}_{i}\tilde{y}_{k}}$ in system (4.17). So we have

(5.22)
$\displaystyle\begin{cases}\displaystyle\frac{d}{dt}\tilde{x}_{i}=\tilde{v}_{i},\vspace{0.2cm}\\\
\displaystyle\frac{d}{dt}\tilde{v}_{i}=\frac{\kappa}{N}\sum_{k=1}^{N}\sum_{\tilde{y}_{k}\in
p^{-1}(p(\tilde{x}_{k}))}\varphi(\mathrm{dist}(\tilde{x}_{i},\tilde{y}_{k}))(Dp^{-1}_{\tilde{y}_{k}}\circ
Dp_{\tilde{x}_{k}}(\tilde{v}_{k})-\tilde{v}_{i}),\quad t>0,\vspace{0.2cm}\\\
\tilde{x}_{i}(0)=\tilde{x}_{i}^{0}\in\tilde{M}^{d},\quad\tilde{v}_{i}(0)=\tilde{v}_{i}^{0}\in
T_{\tilde{x}_{i}^{0}}\tilde{M},\quad\forall~{}i\in\mathcal{N}.\end{cases}$

Here we use the original temporal derivative of $\tilde{v}_{i}$, since the
domain is the Euclidean space. The only complicate thing in system (5.22) is
calculating $Dp_{\tilde{y}}^{-1}\circ Dp_{\tilde{x}}$ for
$\tilde{x},\tilde{y}\in p^{-1}(x)$. In this section, we provide some specific
spaces which are quotient spaces of the Euclidean space(e.g. The flat torus,
Möbius strip, and the flat Klein bottle), and calculate the explicit form of
$Dp_{\tilde{y}}^{-1}\circ Dp_{\tilde{x}}$ to obtain more specific emergent
behaviors of the modified velocity alignment system.

###  5.1. Example 1: The flat torus $\mathbb{T}^{d}$

We consider the flat torus $\mathbb{T}^{d}$ in this subsection. We have
already studied the covering map $p$ between $\mathbb{R}^{d}$ and
$\mathbb{T}^{d}$ in Section 3.1 as follows:

$p(\tilde{x})=((\tilde{x})_{1}-[(\tilde{x})_{1}],(\tilde{x})_{2}-[(\tilde{x})_{2}],\cdots,(\tilde{x})_{d}-[(\tilde{x})_{d}])\quad\forall~{}\tilde{x}\in\mathbb{R}^{d}.$

This yields that $Dp^{-1}_{\tilde{y}}\circ Dp_{\tilde{x}}$ is the identity
map. Also, we have already express the set of geodesics between
$x,y\in\mathbb{T}^{d}$ as (LABEL:C-0-1). From the above results, we can reduce
system (5.22) as follows:

(5.23)
$\displaystyle\begin{cases}\displaystyle\frac{d}{dt}\tilde{x}_{i}=\tilde{v}_{i},\vspace{0.2cm}\\\
\displaystyle\frac{d}{dt}\tilde{v}_{i}=\frac{\kappa}{N}\sum_{k=1}^{N}\left(\sum_{\vec{n}\in\mathbb{Z}^{d}}\varphi(\mathrm{dist}(\tilde{x}_{i},\tilde{x}_{k}+\vec{n}))\right)(\tilde{v}_{k}-\tilde{v}_{i}),\quad
t>0,\vspace{0.2cm}\\\
\tilde{x}_{i}(0)=\tilde{x}_{i}^{0}\in\mathbb{R}^{d},\quad\tilde{v}_{i}(0)=\tilde{v}_{i}^{0}\in\mathbb{R}^{d},\quad\forall~{}i\in\mathcal{N}.\end{cases}$

Now, we apply Corollary 4.1, we can obtain the following theorem.

######  Proposition 5.1.

Let $(\mathcal{X},\mathcal{V})$ be a solution of system (4.16) defined on the
flat torus $\mathbb{T}^{d}$. Let $\varphi$ satisfies $(\mathcal{A})$ and

$\varphi\left(\frac{\sqrt{d}}{2}\right)>0.$

Then, we have

$\displaystyle\lim_{t\to\infty}(\tilde{v}_{k}-\tilde{v}_{i})=0\quad\forall~{}i,k\in\mathcal{N}.$

###### Proof.

We apply Corollary 4.1 to system (5.23) to get

(5.24)
$\displaystyle\lim_{t\to\infty}\sum_{\vec{n}\in\mathbb{Z}^{d}}\varphi(\mathrm{dist}(\tilde{x}_{i},\tilde{x}_{k}+\vec{n})\|\tilde{v}_{k}-\tilde{v}_{i}\|^{2}=0.$

Since $\varphi\geq 0$, we also have

$\displaystyle\sum_{\vec{n}\in\mathbb{Z}^{d}}\varphi(\mathrm{dist}(\tilde{x}_{i},\tilde{x}_{k}+\vec{n})\geq\inf_{\vec{n}\in\mathbb{Z}^{d}}\varphi(\mathrm{dist}(\tilde{x}_{i},\tilde{x}_{k}+\vec{n})).$

Now we apply Lemma 3.1 to get

$\sum_{\vec{n}\in\mathbb{Z}^{d}}\varphi(\mathrm{dist}(\tilde{x}_{i},\tilde{x}_{k}+\vec{n})\geq\varphi\left(\frac{\sqrt{d}}{2}\right).$

Now, we substitute the above relation into (5.24) to get

$\varphi\left(\frac{\sqrt{d}}{2}\right)\lim_{t\to\infty}\|\tilde{v}_{k}-\tilde{v}_{i}\|^{2}=0.$

Since $\varphi\left(\frac{\sqrt{d}}{2}\right)>0$, we get

$\lim_{t\to\infty}\|\tilde{v}_{k}-\tilde{v}_{i}\|=0.$

∎

######  Remark 5.1 (Self-interaction effect of the flat torus).

Since $P^{\gamma}$ is identity map for any closed curve $\gamma$, we have
$P^{\gamma}v-v=0$ for any tangent vector $v$. So, relation (4.21) implies
nothing in this case. So we can conclude that there is no self-interaction
effect on the flat torus.

###  5.2. Example 2: The flat Möbius strip $\mathbb{M}$

We consider the flat Möbius strip in this subsection and denote it as
$\mathbb{M}$. We consider that $\mathbb{M}$ is given on
$[0,1]\times\mathbb{R}\subset\mathbb{R}^{2}$ with the identification
$(0,t)\sim(1,-t)$ where $t\in\mathbb{R}$. We can express the universal
covering $\tilde{\mathbb{M}}=\mathbb{R}^{2}$ with the following covering map
$p:\tilde{\mathbb{M}}\to[0,1)\times\mathbb{R}\simeq\mathbb{M}$:

$\displaystyle
p(\tilde{x})=\begin{cases}((\tilde{x})_{1}-[(\tilde{x})_{1}],(\tilde{x})_{2}),\quad&\text{if
$[(\tilde{x})_{1}]$ is even},\\\
((\tilde{x})_{1}-[(\tilde{x})_{1}],-(\tilde{x})_{2}),\quad&\text{if
$[(\tilde{x})_{1}]$ is
odd},\end{cases}\quad\forall~{}\tilde{x}\in\mathbb{R}^{2},$

where $[x]$ is the largest integer not greater than $x\in\mathbb{R}$. From the
above covering map $p$, we get

$Dp^{-1}_{\tilde{y}}\circ
Dp_{\tilde{x}}(\tilde{v})=\begin{cases}((\tilde{v})_{1},(\tilde{v})_{2}),\quad&\text{if
}[(\tilde{y})_{1}]-[(\tilde{x})_{1}]\text{ is even},\\\
((\tilde{v})_{1},-(\tilde{v})_{2}),\quad&\text{if
}[(\tilde{y})_{1}]-[(\tilde{x})_{1}]\text{ is odd}.\end{cases}$

For the notation simplicity, we define the operator $J$ as follows:

$J(\tilde{x})=((\tilde{x})_{1},-(\tilde{x})_{2})\quad\forall~{}\tilde{x}\in\mathbb{R}^{2}.$

We also denote $J^{n}$ be the $n$-th power of the operator $J$. Then, we can
express $p^{-1}(p(\tilde{x}))$ as follows:

$p^{-1}(p(\tilde{x}))=\\{J^{n}(\tilde{x})+(n,0):n\in\mathbb{Z}\\}$

and

$Dp^{-1}_{\tilde{y}}\circ Dp_{\tilde{x}}(\tilde{v})=J^{n}(\tilde{v})$

if $\tilde{y}=J^{n}(\tilde{x})+(n,0)$ for some $n\in\mathbb{Z}$. Using these
notations, we can reduce system (5.22) into

(5.25)
$\displaystyle\begin{cases}\displaystyle\frac{d}{dt}\tilde{x}_{i}=\tilde{v}_{i},\vspace{0.2cm}\\\
\displaystyle\frac{d}{dt}\tilde{v}_{i}=\frac{\kappa}{N}\sum_{k=1}^{N}\sum_{n\in\mathbb{Z}}\varphi(\mathrm{dist}(\tilde{x}_{i},J^{n}(\tilde{x}_{k})+(n,0))(J^{n}(\tilde{v}_{k})-\tilde{v}_{i}),\quad
t>0,\\\
\tilde{x}_{i}(0)=\tilde{x}_{i}^{0}\in\mathbb{R}^{2},\quad\tilde{v}_{i}(0)=\tilde{v}_{i}^{0}\in\mathbb{R}^{2},\quad\forall~{}i\in\mathcal{N}.\end{cases}$

From a similar argument that we made in Section 3.3, we can prove

(5.26)
$\displaystyle\sum_{n\in\mathbb{Z}}\varphi\big{(}\mathrm{dist}(\tilde{x},J^{n}(\tilde{y})+(n,0))\big{)}<\infty\quad\forall~{}\tilde{x},\tilde{y}\in\mathbb{R}^{2}\quad\Longleftrightarrow\quad\int_{0}^{\infty}\varphi(r)dr<\infty$

from the integral test. If we apply Corollary 4.1 to system (5.25), we get

(5.27)
$\displaystyle\lim_{t\to\infty}\sum_{n\in\mathbb{Z}}\varphi(\mathrm{dist}(\tilde{x}_{i},J^{n}(\tilde{x}_{k})+(n,0))\|J^{n}(\tilde{v}_{k})-\tilde{v}_{i}\|=0.$

Since $J^{2}=\mathrm{Id}$, we have $J^{2m+1}=J$ and $J^{2m}=\mathrm{Id}$ for
all integer $m$. From this fact, we get

$\displaystyle\sum_{n\in\mathbb{Z}}\varphi(\mathrm{dist}(\tilde{x}_{i},J^{n}(\tilde{x}_{k})+(n,0))\|J^{n}(\tilde{v}_{k})-\tilde{v}_{i}\|^{2}$
$\displaystyle=\left(\sum_{m\in\mathbb{Z}}\varphi(\mathrm{dist}(\tilde{x}_{i},\tilde{x}_{k}+(2m,0))\right)\|\tilde{v}_{k}-\tilde{v}_{i}\|^{2}$
$\displaystyle\hskip
142.26378pt+\left(\sum_{m\in\mathbb{Z}}\varphi(\mathrm{dist}(\tilde{x}_{i},J(\tilde{x}_{k})+(2m+1,0))\right)\|J(\tilde{v}_{k})-\tilde{v}_{i}\|^{2}.$

We combine the above result and (5.27) to obtain

(5.28)
$\displaystyle\begin{cases}\displaystyle\lim_{t\to\infty}\left(\sum_{m\in\mathbb{Z}}\varphi(\mathrm{dist}(\tilde{x}_{i},\tilde{x}_{k}+(2m,0))\right)\|\tilde{v}_{k}-\tilde{v}_{i}\|=0,\\\
\displaystyle\lim_{t\to\infty}\left(\sum_{m\in\mathbb{Z}}\varphi(\mathrm{dist}(\tilde{x}_{i},J(\tilde{x}_{k})+(2m+1,0))\right)\|J(\tilde{v}_{k})-\tilde{v}_{i}\|=0,\end{cases}$

for all $i,k\in\mathcal{N}$. Now we have the following lemma.

######  Lemma 5.1.

Let $p$ be a covering map between two dimensional the Euclidean space and the
flat Möbius strip. Let $x,y\in[0,1]\times\mathbb{R}/_{\sim}\simeq\mathbb{M}$
and $|x_{2}|,|y_{2}|<L$ for some positive number $L$. Then, we have

$\displaystyle\inf_{m\in\mathbb{Z}}\mathrm{dist}(\tilde{x},\tilde{y}+(2m,0))<\sqrt{1+4L^{2}},\quad\inf_{m\in\mathbb{Z}}\mathrm{dist}(\tilde{x},J(\tilde{y})+(2m+1,0))<\sqrt{1+4L^{2}},$

where $\tilde{x}\in p^{-1}(x),\tilde{y}\in p^{-1}(y)$.

###### Proof.

First, we fix two points $x,y\in\mathbb{M}$ and we choose
$\tilde{x},\tilde{y}\in\mathbb{R}^{2}$ such that $\tilde{x}\in p^{-1}(x)$ and
$\tilde{y}\in p^{-1}(y)$:

$\tilde{x}=(\tilde{x}_{1},\tilde{x}_{2}),\quad\tilde{y}=(\tilde{y}_{1},\tilde{y}_{2}).$

Without loss of generality, we can assume that
$-1\leq\tilde{x}_{1}-\tilde{y}_{1}<1$. Then we have

$\mathrm{dist}(\tilde{x},\tilde{y}+(2m,0))^{2}=(\tilde{x}_{1}-\tilde{y}_{1}-2m)^{2}+(\tilde{x}_{2}-\tilde{y}_{2})^{2}\leq(\tilde{x}_{1}-\tilde{y}_{1})^{2}+(\tilde{x}_{2}-\tilde{y}_{2})^{2}.$

Here, the last equality only holds for $m=0$. Since
$-1\leq\tilde{x}_{1}-\tilde{y}_{1}<1$ and
$-2L<\tilde{x}_{2}-\tilde{y}_{2}<2L$, we have

$\inf_{m\in\mathbb{Z}}\mathrm{dist}(\tilde{x},\tilde{y}+(2m,0))^{2}<1+4L^{2}.$

Similarly, we have

$\displaystyle\inf_{m\in\mathbb{Z}}\mathrm{dist}(\tilde{x},J(\tilde{y})+(2m+1,0))^{2}$
$\displaystyle=\inf_{m\in\mathbb{Z}}\left((\tilde{x}_{1}-\tilde{y}_{1}-2m-1)^{2}+(\tilde{x}_{2}+\tilde{y}_{2})^{2}\right)<1+4L^{2}.$

∎

From this lemma and (5.28), we get the following proposition.

######  Proposition 5.2.

Let $(\mathcal{X},\mathcal{V})$ be a solution of system (4.16) defined on
$M=\mathbb{M}$. If we assume the following a-priori assumption: “the second
component of $x_{i}\in[0,1)\times\mathbb{R}$ lies in $(-L,L)$ for all $t\geq
0$ and $i\in\mathcal{N}$.” We also assume that $\varphi$ satisfies (5.26) and
$\varphi(\sqrt{1+4L^{2}})>0$. Then, we have

$\lim_{t\to\infty}(\tilde{v}_{i}-\tilde{v}_{k})=0\quad\forall~{}i,k\in\mathcal{N},$

and the second components of all velocities converges to zero. i.e.

$\lim_{t\to\infty}(\tilde{v}_{i})_{2}=0\quad\forall i\in\mathcal{N}.$

###### Proof.

From Lemma 5.1, we have

(5.29) $\displaystyle\begin{aligned}
\sum_{m\in\mathbb{Z}}\varphi(\mathrm{dist}(\tilde{x}_{i},\tilde{x}_{k}+(2m,0))&\geq\inf_{m\in\mathbb{Z}}\varphi(\mathrm{dist}(\tilde{x}_{i},\tilde{x}_{k}+(2m,0))\geq\varphi(\sqrt{1+4L^{2}}),\\\
\sum_{m\in\mathbb{Z}}\varphi(\mathrm{dist}(\tilde{x}_{i},J(\tilde{x}_{k})+(2m+1,0))&\geq\inf_{m\in\mathbb{Z}}\varphi(\mathrm{dist}(\tilde{x}_{i},J(\tilde{x}_{k})+(2m+1,0))\geq\varphi(\sqrt{1+4L^{2}}).\end{aligned}$

Here, we used the given a-priori assumption. We substitute (5.29) into (5.28)
to obtain

$\varphi(\sqrt{1+4L^{2}})\lim_{t\to\infty}\|\tilde{v}_{k}-\tilde{v}_{i}\|=0,\quad\varphi(\sqrt{1+4L^{2}})\lim_{t\to\infty}\|J(\tilde{v}_{k})-\tilde{v}_{i}\|=0.$

Since $\varphi(\sqrt{1+4L^{2}})>0$, we have

(5.30)
$\displaystyle\lim_{t\to\infty}\|\tilde{v}_{k}-\tilde{v}_{i}\|=0,\quad\lim_{t\to\infty}\|J(\tilde{v}_{k})-\tilde{v}_{i}\|=0.$

We consider the case when $k=i$ of second equality for (5.30) to get

$\lim_{t\to\infty}\|J(\tilde{v}_{i})-\tilde{v}_{i}\|=0\quad\forall
i\in\mathcal{N}.$

From the definition of the operator $J$, we could obtain that the second
component of $\tilde{v}_{i}$ converges to zero. ∎

######  Remark 5.2 (Self-interaction effect of the flat Möbius strip).

If we put $x_{i}=x_{k}=x$ into (5.27), we get (4.21) on the Möbius strip as
follows:

$\lim_{t\to\infty}\sum_{n\in\mathbb{Z}}\varphi(\mathrm{dist}(\tilde{x},J^{n}(\tilde{x})+(n,0)))\|J^{n}(\tilde{v})-\tilde{v}\|=0.$

From the definition of $J$, we can simplify the above relation as follows:

$\lim_{t\to\infty}\left(\sum_{m\in\mathbb{Z}}\varphi(\mathrm{dist}(\tilde{x},J(\tilde{x})+(2m+1,0)))\right)\|J(\tilde{v})-\tilde{v}\|=0.$

The simple calculation yields

$1\leq\inf_{m\in\mathbb{Z}}\mathrm{dist}(\tilde{x},J(\tilde{x})+(2m+1,0))=\inf_{m\in\mathbb{Z}}\sqrt{(2m+1)^{2}+4(\tilde{x})_{2}^{2}}\leq\sqrt{1+4L^{2}}.$

If $\varphi(1)=0$, then
$\sum_{m\in\mathbb{Z}}\varphi(\mathrm{dist}(\tilde{x},J(\tilde{x})+(2m+1,0)))=0$
and there is no self-interaction effect. However, if we assume
$\varphi(\sqrt{1+4L^{2}})>0$ and a-priori condition: $|x^{2}|<L$, then we have

$\lim_{t\to\infty}\|J(\tilde{v})-\tilde{v}\|=0.$

This also implies

$\lim_{t\to\infty}(\tilde{v})_{2}=0.$

So, in this case, the self-interaction effect of the flat Möbius strip yields
that the second coordinate of the velocity converges to zero.

###  5.3. Example 3: The flat Klein bottle $\mathbb{K}$

We consider the flat Klein bottle in this subsection and denote it as
$\mathbb{K}$. We consider that the $\mathbb{K}$ is given on
$[0,1]^{2}\subseteq\mathbb{R}^{2}$ as Figure 2.

$O=(0,0)$$A=(1,0)$$B=(1,1)$$C=(0,1)$ Figure 2. The flat Klein bottle
$\mathbb{K}$ on $\mathbb{R}^{2}$

Here, two directed line segments $OC$ and $BA$ are identified as
$(0,t)\sim(1,1-t)$ and other two directed line segments $OA$ and $CB$ are
identified as $(t,0)\sim(t,1)$ where $0\leq t\leq 1$. We can express the
universal covering $\tilde{\mathbb{K}}$ of $\mathbb{K}$ on $\mathbb{R}^{2}$.
Also, we can express the covering map
$p:\tilde{\mathbb{K}}=\mathbb{R}^{2}\to[0,1)^{2}\subset\mathbb{K}$ as follows:

$\displaystyle
p(\tilde{x})=\begin{cases}((\tilde{x})_{1}-[(\tilde{x})_{1}],(\tilde{x})_{2}-[(\tilde{x})_{2}]),\quad&\text{if
$[(\tilde{x})_{1}]$ is even},\\\
((\tilde{x})_{1}-[(\tilde{x})_{1}],-(\tilde{x})_{2}-[-(\tilde{x})_{2}]),\quad&\text{if
$[(\tilde{x})_{1}]$ is odd},\end{cases}\quad\tilde{x}\in\mathbb{R}^{2}$

From this, we get

$Dp^{-1}_{\tilde{y}}\circ
Dp_{\tilde{x}}(\tilde{v})=\begin{cases}((\tilde{v})_{1},(\tilde{v})_{2}),\quad&\text{if
}[(\tilde{y})_{1}]-[(\tilde{x})_{1}]\text{ is even},\\\
((\tilde{v})_{1},-(\tilde{v})_{2}),\quad&\text{if
}[(\tilde{y})_{1}]-[(\tilde{x})_{1}]\text{ is odd}.\end{cases}$

Now, we use $J$ defined in Section 5.2 to get

$p^{-1}(p(\tilde{x}))=\\{J^{n}\tilde{x}+(n,m):(n,m)\in\mathbb{Z}^{2}\\}.$

We also have that if $\tilde{y}=J^{n}(\tilde{x})+(n,m)$ for some
$(n,m)\in\mathbb{Z}^{2}$, then

$Dp_{\tilde{y}}^{-1}\circ Dp_{\tilde{x}}(\tilde{v})=J^{n}(\tilde{v}).$

This notations allow us to simplify system (5.22) defined on $\mathbb{M}$ as
follows:

(5.31)
$\displaystyle\begin{cases}\displaystyle\frac{d}{dt}\tilde{x}_{i}=\tilde{v}_{i},\vspace{0.2cm}\\\
\displaystyle\frac{D}{dt}\tilde{v}_{i}=\frac{\kappa}{N}\sum_{k=1}^{N}\sum_{(n,m)\in\mathbb{Z}^{2}}\varphi\big{(}\mathrm{dist}(\tilde{x}_{i},J^{n}(\tilde{x}_{k})+(n,m))\big{)}(J^{n}(\tilde{v}_{k})-\tilde{v}_{i}),\quad
t>0,\\\
\tilde{x}_{i}(0)=\tilde{x}_{i}^{0}\in\mathbb{R}^{2},\quad\tilde{v}_{i}(0)=\tilde{v}_{i}^{0}\in\mathbb{R}^{2},\quad\forall~{}i\in\mathcal{N}.\end{cases}$

Again $J^{2}=\mathrm{Id}$, we have

$\displaystyle\sum_{(n,m)\in\mathbb{Z}^{2}}\varphi(\mathrm{dist}(\tilde{x}_{i},J^{n}(\tilde{x}_{k})+(n,m)))\|J^{n}(\tilde{v}_{k})-\tilde{v}_{i}\|^{2}$
$\displaystyle=\sum_{(l,m)\in\mathbb{Z}^{2}}\varphi(\mathrm{dist}(\tilde{x}_{i},\tilde{x}_{k}+(2l,m)))\|\tilde{v}_{k}-\tilde{v}_{i}\|^{2}+\sum_{(l,m)\in\mathbb{Z}^{2}}\varphi(\mathrm{dist}(\tilde{x}_{i},J(\tilde{x}_{k})+(2l+1,m)))\|J(\tilde{v}_{k})-\tilde{v}_{i}\|^{2}$

Now, we apply Corollary 4.1 to get

(5.32)
$\displaystyle\begin{cases}\displaystyle\lim_{t\to\infty}\sum_{(l,m)\in\mathbb{Z}^{2}}\varphi(\mathrm{dist}(\tilde{x}_{i},\tilde{x}_{k}+(2l,m)))\|\tilde{v}_{k}-\tilde{v}_{i}\|^{2}=0,\\\
\displaystyle\lim_{t\to\infty}\sum_{(l,m)\in\mathbb{Z}^{2}}\varphi(\mathrm{dist}(\tilde{x}_{i},J(\tilde{x}_{k})+(2l+1,m)))\|J(\tilde{v}_{k})-\tilde{v}_{i}\|^{2}=0,\end{cases}$

for all $i,k\in\mathcal{N}$. Now, we have the following lemma.

######  Lemma 5.2.

Let $\tilde{x},\tilde{y}\in\mathbb{R}^{2}$. Then we have

$\displaystyle\inf_{(l,m)\in\mathbb{Z}^{2}}\mathrm{dist}(\tilde{x},\tilde{y}+(2l,m))\leq\frac{\sqrt{5}}{2},\quad\inf_{(l,m)\in\mathbb{Z}^{2}}\mathrm{dist}(\tilde{x},J(\tilde{y})+(2l+1,m))\leq\frac{\sqrt{5}}{2}.$

Since the proof of the above lemma is similar to the proof of Lemmas 3.1 and
Lemma 5.1, we omitted it. From this lemma, we have the following proposition.

######  Proposition 5.3.

Let $(\mathcal{X},\mathcal{V})$ be a solution of system (4.16) defined on
$M=\mathbb{K}$. If $\varphi$ satisfies $(\mathcal{A})$ and
$\varphi\left(\frac{\sqrt{5}}{2}\right)>0$, then we have

$\lim_{t\to\infty}(\tilde{v}_{i}-\tilde{v}_{k})=0\quad\forall~{}i,k\in\mathcal{N}$

and the second component of $\tilde{v}_{i}$ converges to zero. i.e.

$\lim_{t\to\infty}(\tilde{v}_{i})_{2}=0\quad\forall~{}i\in\mathcal{N}.$

###### Proof.

From a similar argument that we used in the proof of Proposition 5.2, we can
prove this proposition with (5.28) and Lemma 5.2 ∎

######  Remark 5.3 (Self-interaction effect of the flat Klein bottle).

If we put $x_{i}=x_{k}=x$ into (5.32), we get (4.21) on the flat Klein bottle
as follows:

$\lim_{t\to\infty}\sum_{(l,m)\in\mathbb{Z}^{2}}\varphi(\mathrm{dist}(\tilde{x},J(\tilde{x})+(2l+1,m)))\|J(\tilde{v})-\tilde{v}\|^{2}=0$

The simple calculation yields

$1\leq\inf_{(l,m)\in\mathbb{Z}^{2}}\mathrm{dist}(\tilde{x},J(\tilde{x})+(2l+1,m))=\inf_{(l,m)\in\mathbb{Z}^{2}}\sqrt{(2l+1)^{2}+(2\tilde{x}^{2}-m)^{2}}\leq\frac{\sqrt{5}}{2}.$

From a similar argument that we made in Remark 5.2, if $\varphi(1)=0$, then
there is no self-interaction effect. However, if
$\varphi\left(\frac{\sqrt{5}}{2}\right)>0$, then the flat Klein bottle yields
the self-interaction effect which makes that the second coordinate of the
velocity converges to zero.

##  6\. Conclusion

Throughout this paper, we constructed a velocity alignment model on manifolds
and studied the emergent dynamics in various spaces. Previously in [15], the
CS model on manifold uses the shortest geodesic which connects two particles
to define an interaction between them. One disadvantage of this model is that
if there exist two or more than two shortest geodesics then the system can not
be well-defined. We tried to remove the speciality of the shortest geodesic.
So, we consider all of geodesics which is connecting $i^{th}$ particle and
$j^{th}$ particle to define an interaction between $i^{th}$ and $j^{th}$
particles. In this way, we could obtain a modified velocity alignment system.
One of the interesting properties of a modified system is the self-interaction
effect. This effect depends on the structure of the domain of the system. In
the last section, we provided the emergent behaviors of the modified system
defined on some specific spaces(flat torus, flat Möbius strip, flat Klein
bottle). However, our model can not be applied on the sphere $\mathbb{S}^{d}$
with $d\geq 2$. Since there are uncountably many geodesics that connecting
antipodal pair points, we can not define the sum of interactions in this case.
We need another setup to cover this case, and it can be nice future work.

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