# Conformal trajectories in $3$-dimensional space forms

Rafael López, Marian Ioan Munteanu  Departamento de Geometría y Topología
Universidad de Granada. 18071 Granada, Spain<EMAIL_ADDRESS>University ’Al.
I. Cuza’ of Iasi, Faculty of Mathematics, Bd. Carol I, no. 11, 700506 Iasi,
Romania<EMAIL_ADDRESS>
###### Abstract.

We introduce the notion of conformal trajectories in three-dimensional
Riemannian manifolds $M^{3}$. Given a conformal vector field
$V\in\mathfrak{X}(M^{3})$, a conformal trajectory of $V$ is a regular curve
$\gamma$ in $M^{3}$ satisfying
$\nabla_{\gamma^{\prime}}\gamma^{\prime}=q\,V\times\gamma^{\prime}$, for some
fixed non-zero constant $q\in\mathbb{R}$. In this paper, we study conformal
trajectories in the space forms $\mathbb{R}^{3}$, $\mathbb{S}^{3}$ and
$\mathbb{H}^{3}$. For (non-Killing) conformal vector fields in
$\mathbb{S}^{3}$ (respectively in $\mathbb{H}^{3}$), we prove that conformal
trajectories have constant curvature and its torsion is a linear combination
of trigonometric (respectively hyperbolic) functions on the arc-length
parameter. In the case of Euclidean space $\mathbb{R}^{3}$, we obtain the same
result for the radial vector field and characterizing all conformal
trajectories.

###### Key words and phrases:

conformal vector field, space form, conformal trajectories

###### 1991 Mathematics Subject Classification:

Primary 53A10; Secondary 53C44, 53C21, 53C42.

##  1\. Introduction and motivation

In a $n$-dimensional Riemannian manifold $(M^{n},\langle\leavevmode\nobreak\
,\leavevmode\nobreak\ \rangle)$, a magnetic field is a closed $2$-form $F$. A
curve $\gamma\colon I\to M^{n}$ is called a magnetic trajectory of $F$ if
$\gamma$ satisfies the Lorentz equation
$\nabla_{\gamma^{\prime}}\gamma^{\prime}=\Phi(\gamma^{\prime})$, where $\Phi$
is the $(1,1)$-tensor field on $M$ defined by $\langle\Phi X,Y\rangle=F(X,Y)$,
$X,Y\in\mathfrak{X}(M^{n})$. The notion of magnetic trajectory extends that of
geodesic when $F=0$.

Suppose that the dimension of $M^{n}$ is $n=3$. In such a case, the study of
magnetic trajectories can be introduced with a different approach. This is due
because $2$-forms and vector fields are identified via the Hodge star operator
$*$ and the volume form $\omega$ of $M^{3}$. In particular, magnetic fields
are vector fields with zero divergence and the Lorentz force can be expressed
in terms of the cross product $\times$ of $M^{3}$. Killing vector fields are
examples of magnetic fields. Each Killing vector field defines a magnetic
field $F_{V}$ by means of $F_{V}=i_{V}\omega$. Then the Lorentz force $\Phi$
of $F_{V}$ is given by

$\Phi(X)=V\times X.$

Thus a magnetic trajectory $\gamma$ of $V$ is characterized by the identity

$\nabla_{\gamma^{\prime}}\gamma^{\prime}=V\times\gamma^{\prime}.$

Looking this identity, the property that $V$ is magnetic can be ignored and it
only uses the cross product of $M^{3}$. This observation allows us to
generalize the notion of magnetic curve by replacing a Killing vector field by
a conformal vector field. A vector field $V\in\mathfrak{X}(M^{n})$ is said to
be conformal if

$\mathcal{L}_{V}\langle\leavevmode\nobreak\ ,\leavevmode\nobreak\
\rangle=\lambda\langle\leavevmode\nobreak\ ,\leavevmode\nobreak\ \rangle,$

where $\mathcal{L}$ is the Lie derivative and $\lambda\in C^{\infty}(M^{n})$
is a function on $M^{n}$. Equivalently, we have

$\langle\nabla_{X}V,Y\rangle+\langle X,\nabla_{Y}V\rangle=\lambda\langle
X,Y\rangle,\quad X,Y\in\mathfrak{X}(M^{n}).$ (1)

In the literature, it is also used the terminology of conformal Killing vector
field. If $\lambda=0$, then $V$ is just a Killing vector field. Those ones
that are not Killing will be caller proper conformal vector fields.

######  Definition 1.1.

Let $V\in\mathfrak{X}(M^{3})$ be a conformal vector field. A conformal
trajectory of $V$ is a regular curve $\gamma\colon I\subset\mathbb{R}\to
M^{3}$ satisfying the equation

$\nabla_{\gamma^{\prime}}\gamma^{\prime}=q\,V\times\gamma^{\prime},$ (2)

where $q\in\mathbb{R}$, $q\not=0$, is fixed.

The number $q$ in (2) is due because conformal trajectories cannot be
rescaled, such as it occurs in the case of geodesics. Indeed, if $\gamma$ is a
conformal trajectory, then

$\frac{d}{ds}\langle\gamma^{\prime}(s),\gamma^{\prime}(s)\rangle=2\langle\nabla_{\gamma^{\prime}(s)}\gamma^{\prime}(s),\gamma^{\prime}(s)\rangle=2\langle
V\times\gamma^{\prime},\gamma^{\prime}\rangle=0.$

As a consequence the speed $|\gamma^{\prime}(s)|$ of $\gamma$ is constant.

The purpose of this paper is to study conformal trajectories in the space
forms $\mathbb{R}^{3}$, $\mathbb{S}^{3}$ and $\mathbb{H}^{3}$. For magnetic
fields, the study of magnetic trajectories in space forms have been developed
in [3, 4, 5, 8, 9, 10, 12]. In each of these space forms, we will consider
particular examples of proper conformal vector fields. To be precise,

  1. (1)

In Euclidean space $\mathbb{R}^{3}$, let $V$ be the radial vector field
$V(x,y,z)=x\partial_{x}+y\partial_{y}+z\partial_{z}$.

  2. (2)

In the unit sphere $\mathbb{S}^{3}$, proper conformal vector fields are
generated by $V(p)=\vec{a}-\langle\vec{a},p\rangle p$, where
$\vec{a}\in\mathbb{R}^{4}$.

  3. (3)

In the hyperbolic space $\mathbb{H}^{3}$, consider the Lorentzian model where
$\mathbb{H}^{3}$ is viewed as a quadric of $\mathbb{R}^{4}$. The proper
conformal vector fields are generated by $V(p)=\vec{a}+\langle\vec{a},p\rangle
p$, where $\vec{a}\in\mathbb{R}^{4}_{1}$.

In all cases, when we express Eq. (2) in coordinates, the equation turns on a
second order differential equation whose explicit solutions are difficult to
find in general. However, we can summarize a property of the conformal
trajectories.

######  Theorem 1.2.

Let $M^{3}$ be a space form and let $V\in\mathfrak{X}(M^{3})$ be the conformal
vector field described above. Let $\gamma\colon I\subset\mathbb{R}\to M^{3}$
be a regular (non-geodesic) curve parametrized by arc-length
$\gamma=\gamma(s)$. If $\gamma$ is a conformal trajectory of $V$, then its
curvature $\kappa$ is a non-zero constant and its torsion $\tau$ is:

  1. (1)

in $\mathbb{R}^{3}$, $\tau(s)=a_{1}s+a_{2}$;

  2. (2)

in $\mathbb{S}^{3}$, $\tau(s)=a_{1}\sin s+a_{2}\cos s$;

  3. (3)

in $\mathbb{H}^{3}$, $\tau(s)=a_{1}\sinh s+a_{2}\cosh s$,

where $a_{1},a_{2}\in\mathbb{R}$.

The proof of this theorem will be done in Sects. 3, 4 and 5 in each one of the
space forms $\mathbb{R}^{3}$, $\mathbb{S}^{3}$ and $\mathbb{H}^{3}$,
respectively. In Sect. 2 we present all conformal vector fields in space forms
and we calculate the curvature and the torsion of conformal trajectories for
proper conformal vector fields.

##  2\. Conformal vector fields in space forms

In $3$-dimensional space forms, the space of conformal vector fields is
$10$-dimensional, from which $6$ correspond to Killing vector fields. We
describe in each space form these vector fields. General references are [2,
11].

  1. (1)

Euclidean space $\mathbb{R}^{3}$. Consider $(x,y,z)$ canonical coordinates in
$\mathbb{R}^{3}$ and let $\langle\leavevmode\nobreak\ ,\leavevmode\nobreak\
\rangle=dx^{2}+dy^{2}+dz^{2}$ be the Euclidean metric. The Killing vector
fields are generated by the translations $\partial_{x}$, $\partial_{y}$ and
$\partial_{z}$ and the rotations $-y\partial_{x}+x\partial_{y}$,
$-z\partial_{x}+x\partial_{z}$ and $-z\partial_{y}+y\partial_{z}$. The proper
conformal vector fields are generated by the radial vector field

$V=x\partial_{x}+y\partial_{y}+z\partial_{z},$ (3)

and the vector fields

$W(p)=\frac{|p|^{2}}{2}\vec{a}-\langle\vec{a},p\rangle p,\quad
p\in\mathbb{R}^{3},$ (4)

where $\vec{a}\in\mathbb{R}^{3}$. The vector fields $V$ and $W$ are conformal
because $\mathcal{L}_{V}\langle\leavevmode\nobreak\ ,\leavevmode\nobreak\
\rangle=2\langle\leavevmode\nobreak\ ,\leavevmode\nobreak\ \rangle$ and
$\mathcal{L}_{W}\langle\leavevmode\nobreak\ ,\leavevmode\nobreak\
\rangle=-2\langle\vec{a},p\rangle\langle\leavevmode\nobreak\
,\leavevmode\nobreak\ \rangle$.

  2. (2)

Sphere $\mathbb{S}^{3}$. Let $\langle\leavevmode\nobreak\
,\leavevmode\nobreak\ \rangle$ be the Euclidean metric on $\mathbb{S}^{3}$.
Let $(x,y,z,t)$ be canonical coordinates in $\mathbb{R}^{4}$ and let identify
a vector field with its coordinates $(x,y,z,t)$ with respect to the canonical
basis $\\{\partial_{x},\partial_{y},\partial_{z},\partial_{t}\\}$ of
$\mathbb{R}^{4}$. Then the Killing vector fields are generated by

$\displaystyle K_{1}=(-y,x,-t,z),\quad$ $\displaystyle K_{2}=(-y,x,t,-z),$
$\displaystyle K_{3}=(-z,-t,x,y,)\quad$ $\displaystyle K_{4}=(-z,t,x,-y),$
$\displaystyle K_{5}=(-t,z,-y,x),\quad$ $\displaystyle K_{6}=(-t,-z,y,x).$

Proper conformal vector fields are the projections on $\mathbb{S}^{3}$ of the
analogous vector fields $W$ given in (4) but defined in $\mathbb{R}^{4}$.
Thus, there is a description of these vector fields free of coordinates,
namely,

$V(p)=\vec{a}-\langle\vec{a},p\rangle p,\quad p\in\mathbb{S}^{3},$ (5)

where $\vec{a}\in\mathbb{R}^{4}$. Here
$\mathcal{L}_{V}\langle\leavevmode\nobreak\ ,\leavevmode\nobreak\
\rangle=-2\langle\vec{a},p\rangle\langle\leavevmode\nobreak\
,\leavevmode\nobreak\ \rangle$.

  3. (3)

Hyperbolic space $\mathbb{H}^{3}$. Consider the Lorentzian model, that is, the
Euclidean space $\mathbb{R}^{4}$ with canonical coordinates $(x,y,z,t)$,
endowed with the Lorentzian metric $\langle\leavevmode\nobreak\
,\leavevmode\nobreak\ \rangle=dx^{2}+dy^{2}+dz^{2}-dt^{2}$. Then
$\mathbb{H}^{3}$ is the quadric $\mathbb{H}^{3}=\\{p\in\mathbb{R}^{4}:\langle
p,p\rangle=-1,\leavevmode\nobreak\ t>0\\}$. As in $\mathbb{S}^{3}$, we
identify a vector field with its coordinates with respect to the canonical
basis of tangent vectors. The Killing vector fields are generated by

$\displaystyle K_{1}=(-y,x,t,z)\quad$ $\displaystyle K_{2}=(y,-x,t.z)$
$\displaystyle K_{3}=(-z,t,x,y)\quad$ $\displaystyle K_{4}=(z,t,-x,y)$
$\displaystyle K_{5}=(t,-z,y,x)\quad$ $\displaystyle K_{6}=(t,z,-y,x).$

Similarly as in the case of the sphere $\mathbb{S}^{3}$, the space of proper
conformal vector fields are generated by

$V(p)=\vec{a}+\langle\vec{a},p\rangle p,\quad p\in\mathbb{H}^{3},$ (6)

where $\vec{a}\in\mathbb{R}^{4}$. In this case
$\mathcal{L}_{W}\langle\leavevmode\nobreak\ ,\leavevmode\nobreak\
\rangle=2\langle\vec{a},p\rangle\langle\leavevmode\nobreak\
,\leavevmode\nobreak\ \rangle$.

Let $V\in\mathfrak{X}(M^{3})$ be a conformal vector field with
$\mathcal{L}_{V}\langle\leavevmode\nobreak\ ,\leavevmode\nobreak\
\rangle=\lambda\langle\leavevmode\nobreak\ ,\leavevmode\nobreak\ \rangle$. Let
$\gamma\colon I\to M^{3}$ be a regular curve parametrized by arc-length. If
$\gamma$ is a geodesic, then it is immediate the following result.

######  Proposition 2.1.

A geodesic $\gamma\colon I\to M^{3}$ is a conformal trajectory of $V$ if and
only if $V(\gamma(s))$ and $\gamma^{\prime}(s)$ are collinear for all $s\in
I$.

From now on, suppose that $\gamma$ is not a geodesic. Then we can assign to
$\gamma$ its Frenet frame $\\{T,N,B\\}$, where $T(s)=\gamma^{\prime}(s)$ and
$T\times N=B$. The Frenet equations are given by

$\begin{split}\nabla_{T}T&=\kappa N,\\\ \nabla_{T}N&=-\kappa T+\tau B,\\\
\nabla_{T}B&=-\tau B,\end{split}$

where $\kappa$ and $\tau$ are the curvature and the torsion of $\gamma$,
respectively. Let denote $V(s)=V(\gamma(s))$

In the following result, we study the tangent part of $V$ along $\gamma$ for
those conformal vector fields $V$ satisfying that $\nabla_{T}V$ is
proportional to $T$ along $\gamma$.

######  Proposition 2.2.

Let $\gamma\colon I\to M^{3}$ be a curve parametrized by arc-length. If
$\gamma$ is a conformal trajectory of $V$ then

$\langle
V(s),\gamma^{\prime}(s)\rangle=\frac{1}{2}\int^{s}\lambda(\gamma(t))dt.$ (7)

If, in addition, $\nabla_{T}V$ is proportional to $T$ for all $s\in I$, then
the curvature $\kappa$ is constant, $\kappa(s)=\kappa_{0}$ and the torsion is

$\tau(s)=q\langle V(s),\gamma^{\prime}(s)\rangle.$ (8)

###### Proof.

We know $\nabla_{\gamma^{\prime}}\gamma^{\prime}=\nabla_{T}T=\kappa N$. We now
compute the right hand-side of (2). For this, let us write

$V(s)=a(s)T(s)+b(s)N(s)+c(s)B(s),$ (9)

where $a,b,c\in C^{\infty}(I)$. Then $V\times\gamma^{\prime}=V\times T=cN-bB$.
Thus Eq. (2) becomes

$\kappa N=q(cN-bB).$

This implies $b=0$ and $\kappa=qc$. In particular, we have $c\not=0$. Using
this, we obtain

$\nabla_{\gamma^{\prime}}V=a^{\prime}T+\kappa(a-\frac{\tau}{q})N+\frac{\kappa^{\prime}}{q}B.$
(10)

Multiplying by $T$ and using (1) because $V$ is a conformal vector field, we
have

$\lambda=\lambda\langle\gamma^{\prime},\gamma^{\prime}\rangle=2\langle\nabla_{\gamma^{\prime}}V,\gamma^{\prime}\rangle=2a^{\prime}.$

This proves (7). If now we assume that $\nabla_{T}V$ is proportional to $T$,
then (10) yields $\kappa^{\prime}(s)=0$ for all $s\in I$ and $\tau(s)=qa(s)$.
Thus $\kappa$ is a nonzero constant and the torsion is given by (8) because of
(7). ∎

We can use this proposition for the proper conformal vector field in space
forms described previously.

######  Theorem 2.3.

Let $\gamma\colon I\to M^{3}$ be a curve parametrized by arc-length, where
$M^{3}$ is a space form $\mathbb{R}^{3}$, $\mathbb{S}^{3}$ or
$\mathbb{H}^{3}$. Suppose that $\gamma$ is not a geodesic.

  1. (1)

Case $\mathbb{R}^{3}$. Let $V$ be the radial vector field in $\mathbb{R}^{3}$
defined in (3). If $\gamma$ is a conformal trajectory of $V$, then the
curvature of $\gamma$ is constant and its torsion is

$\tau(s)=q\langle\gamma(s),\gamma^{\prime}(s)\rangle.$

  2. (2)

Case $\mathbb{S}^{3}$. Let $V$ be a proper conformal vector field defined in
(5). If $\gamma$ is a conformal trajectory of $V$, then the curvature of
$\gamma$ is constant and its torsion is

$\tau(s)=q\langle\vec{a},\gamma^{\prime}(s)\rangle.$

  3. (3)

Case $\mathbb{H}^{3}$. Let $V$ be a proper conformal vector field defined in
(6). If $\gamma$ is a conformal trajectory of $V$, then the curvature of
$\gamma$ is constant and its torsion is

$\tau(s)=q\langle\vec{a},\gamma^{\prime}(s)\rangle.$

###### Proof.

In all cases, we have $\nabla_{T}V$ is proportional to $T$, hence we can apply
Prop. 2.2. ∎

##  3\. The Euclidean space $\mathbb{R}^{3}$: the radial vector field

In this section we study the conformal trajectories of the radial vector field
$V$ defined in (3). Notice that the conformal transformations of $V$ are
dilations from the origin. By Prop. 2.1, a straight-line
$\gamma(s)=p+s\vec{v}$ is a conformal trajectory of $V$ if and only if $p$ and
$\vec{v}$ are collinear, that $\gamma$ is the straight-line acrosses the
origin of $\mathbb{R}^{3}$.

From now, we will assume that $\gamma$ is not a straight-line. Suppose that
$\gamma$ is parametrized by arc-length. The left hand-side of (2) is
$\nabla_{\gamma^{\prime}}\gamma^{\prime}=\gamma^{\prime\prime}$. Thus $\gamma$
is a conformal trajectory if and only if

$\gamma^{\prime\prime}=q\,\gamma\times\gamma^{\prime}.$ (11)

######  Theorem 3.1.

If $\gamma$ is a conformal trajectory of the radial vector field (3), then
$\gamma$ is a curve with constant curvature $\kappa(s)=\kappa_{0}$, and its
torsion is an affine function in the parameter of the curve. Moreover,

$\gamma(s)=(s+a_{0})\gamma^{\prime}(s)+\frac{1}{q}\gamma^{\prime}(s)\times\gamma^{\prime\prime}(s).$
(12)

###### Proof.

Notice that $\nabla_{T}V=\nabla_{\gamma^{\prime}}V=\gamma^{\prime}=T$. Thus we
are under the assumptions of Prop. 2.2, in particular, the curvature $\kappa$
is constant and (7) gives

$\langle\gamma(s),\gamma^{\prime}(s)\rangle=\int^{s}\ ds=s+a_{0},$

where $a_{0}\in\mathbb{R}$ is an integration constant. Moreover, from (8), we
obtain $\tau(s)=q(s+a_{0})$. Using the same notation as in the proof of Prop.
2.2, equation (10) implies $c(s)=c_{0}\not=0$ is constant and
$\kappa(s)=qc_{0}$. Notice that $cB=\frac{\kappa}{q}T\times
N=\frac{1}{q}\gamma^{\prime}\times\gamma^{\prime\prime}$. This yields (12). ∎

Equation (12) implies that the radial vector of $\gamma$ lies in the
rectifying plane at each point. This class of curves were studied by B-Y. Chen
in [1]. However identity (12) does not characterize conformal trajectories of
the radial vector field. This is because the curvature and torsion determine a
spatial curve of $\mathbb{R}^{3}$, up to rigid motions of $\mathbb{R}^{3}$.
However, identity (2) depends on the position vector of the curve.

A consequence of (12) is

$|\gamma(s)|^{2}=(s+a_{0})^{2}+c_{0}^{2}.$

This allows to give the converse of Thm. 3.1, obtaining a characterization of
conformal trajectories of the radial vector field.

######  Theorem 3.2.

Let $\gamma\colon I\subset\mathbb{R}\to\mathbb{R}^{3}$ be a curve parametrized
by arc-length. Suppose that the curvature of $\gamma$ is constant,
$\kappa(s)=\kappa_{0}$ and that there are $a_{0},a_{1}\in\mathbb{R}$,
$a_{1}>0$ such that

$|\gamma(s)|^{2}=(s+a_{0})^{2}+a_{1}^{2}.$ (13)

Then $\gamma$ is a conformal trajectory of $V$ for some $q\in\mathbb{R}$.

###### Proof.

Differentiating twice (13), we have $\langle\gamma,T\rangle=s+a_{0}$ and
$\langle\gamma,N\rangle=0$. By expressing $\gamma$ in terms of the Frenet
frame, and using (13), we obtain

$\gamma(s)=(s+a_{0})T+a_{1}B.$

Then $\gamma\times\gamma^{\prime}=a_{1}N$. On the other hand, the first Frenet
equation gives $\gamma^{\prime\prime}=k_{0}N$. Then taking
$q=\frac{\kappa_{0}}{a_{1}}$, we have identity (11). ∎

######  Remark 3.3.

For a curve $\gamma$ in Euclidean space $\mathbb{R}^{3}$ parametrized by arc-
length $\gamma=\gamma(s)$, the identity (13) is a characterization to be a
rectifying curve: see [1, Theorem 1].

In Fig. 1 we have different numerical graphics of conformal trajectories of
$V$. In all them, we have solved the system (11), which in coordinates writes
as

$\left\\{\begin{split}x^{\prime\prime}(s)&=q(y(s)z^{\prime}(s)-y^{\prime}(s)z(s))\\\
y^{\prime\prime}(s)&=q(x^{\prime}(s)z(s)-x(s)z^{\prime}(s))\\\
z^{\prime\prime}(s)&=q(x(s)y^{\prime}(s)-x^{\prime}(s)y(s)),\end{split}\right.$
(14)

where $\gamma(s)=(x(s),y(s),z(s))$. We have considered different values of $q$
and initial conditions. In Fig. 1, left, the initial conditions are

$\gamma(0)=(1,0,0),\quad\gamma^{\prime}(0)=(0,1,0).$ (15)

The second condition on $\gamma^{\prime}(0)$ implies that $\gamma$ is
parametrized by arc-length. Since
$a_{0}=\langle\gamma(0),\gamma^{\prime}(0)\rangle$ by (12), we obtain
$a_{0}=0$. By (13), we know $a_{1}=1$. On the other hand,
$|\gamma(0)\times\gamma^{\prime}(0)|=1$. Thus
$\kappa(s)=\kappa(0)=q|V(0)\times\gamma^{\prime}(0)|=q$. Now $\kappa(s)=q$ and
$\tau(s)=qs$. In Fig 1, right, the initial conditions are

$\gamma(0)=(1,0,0),\quad\gamma^{\prime}(0)=(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2},0).$
(16)

Then $a_{0}=\langle\gamma(0),\gamma^{\prime}(0)\rangle=\frac{\sqrt{2}}{2}$.
Thus $\kappa(s)=\sqrt{2}q/2$ and $\tau(s)=q(s+\frac{\sqrt{2}}{2})$.

Figure 1. Conformal trajectories of the radial vector field (3) with different
values for $q$: $q=1$ (black) and $q=2$ (grey). Left: initial conditions (15).
Right: initial conditions (16).

##  4\. The unit sphere $\mathbb{S}^{3}$

In this section we study the conformal trajectories on the unit sphere
$\mathbb{S}^{3}$ for all proper conformal vector fields. We know from Sect. 2
that the proper conformal vector fields on $\mathbb{S}^{3}$ are generated by
the vector fields $V$ defined in (5). Recall that
$\lambda(p)=-2\langle\vec{a},p\rangle$.

######  Theorem 4.1.

Let $V\in\mathfrak{X}(\mathbb{S}^{3})$ be a proper conformal vector field
given by (5). Let $\gamma\colon I\to\mathbb{S}^{3}$ be a curve parametrized by
arc-length. If $\gamma$ is a conformal trajectory of $V$, then

$\langle\gamma(s),\vec{a}\rangle=a_{1}\cos(s)+a_{2}\sin(s),$ (17)

where $a_{1},a_{2}\in\mathbb{R}$. If $\gamma$ is a geodesic, then $\gamma$ is
a great circle of $\mathbb{S}^{3}$ obtained by intersecting the sphere with a
2-plane containing $\vec{a}$. Otherwise, the curvature of $\gamma$ is
constant, $\kappa(s)=|q|\sqrt{|\vec{a}|^{2}-(a_{1}^{2}+a_{2}^{2})}$, and the
torsion is

$\tau(s)=-q(a_{1}\sin(s)-a_{2}\cos(s)).$

###### Proof.

We know the relation between the Levi-Civita connection $\nabla$ in
$\mathbb{S}^{3}$ and that of $\mathbb{R}^{4}$, $\nabla^{e}$, namely,

$\nabla_{X}Y=\nabla_{X}^{e}Y+\langle X,Y\rangle p.$

As $\gamma$ is parametrized by arc-length, we deduce that
$\nabla_{\gamma^{\prime}}\gamma^{\prime}=\gamma^{\prime\prime}+\gamma$.

Let $\omega$ be the volume element on $\mathbb{S}^{3}$ defined by
$\omega(u,v,w)=\mbox{det}(u,v,w,p)$, where $u,v,w\in T_{p}\mathbb{S}^{3}$. If
$u,v\in T_{p}\mathbb{S}^{3}$, the cross product $u\times v$ on
$\mathbb{S}^{3}$ is defined as the unique vector in $T_{p}\mathbb{S}^{3}$ such
that $\langle u\times v,w\rangle=\omega(u,v,w)=\mbox{det}(u,v,w,p)$ for any
$w\in T_{p}\mathbb{S}^{3}$. Therefore, the cross product
$V\times\gamma^{\prime}$ may be expressed as

$V\times\gamma^{\prime}=\left|\begin{matrix}a&x&x^{\prime}&\partial_{x}\\\
b&y&y^{\prime}&\partial_{y}\\\ c&z&z^{\prime}&\partial_{z}\\\
d&t&t^{\prime}&\partial_{t}\end{matrix}\right|$ (18)

where $\vec{a}=(a,b,c,d)$. Obviously, $V\times\gamma^{\prime}$ is
perpendicular to $\vec{a}$. Thus, taking the scalar product with $\vec{a}$ in
(2) we obtain

$\frac{d^{2}}{ds^{2}}\langle\gamma,\vec{a}\rangle+\langle\gamma,\vec{a}\rangle=0.$

It follows that there exist two constants $a_{1}$ and $a_{2}$ such that

$\langle\gamma(s),\vec{a}\rangle=a_{1}\cos(s)+a_{2}\sin(s).$

By definition of curvature, we have
$\kappa^{2}=\langle\nabla_{\gamma^{\prime}}\gamma^{\prime},\nabla_{\gamma^{\prime}}\gamma^{\prime}\rangle$.

The special case when $\gamma$ is a geodesic implies on one hand that
$\gamma^{\prime\prime}(s)+\gamma(s)=0$ and on the other hand
$V(s)\times\gamma^{\prime}(s)=0$. The first equation yields
$\gamma(s)=\cos(s)\vec{v}+\sin(s)\vec{w}$, where $\vec{v}$ and $\vec{w}$ are
unitary and perpendicular. The second condition implies that the first three
columns in the determinant (18) are linearly dependent. As a consequence,
$\vec{a}$ lies in the 2-plane spanned by $\vec{v}$ and $\vec{w}$. This proves
the first statement of theorem.

If $\gamma$ is not a geodesic, the unit normal vector is defined by
$\nabla_{\gamma^{\prime}}\gamma^{\prime}=\kappa N$. To obtain the curvature,
we use (17) to compute

$\begin{split}\langle
V\times\gamma^{\prime},V\times\gamma^{\prime}\rangle&=\langle
V,V\rangle\langle\gamma^{\prime},\gamma^{\prime}\rangle-\langle
V,\gamma^{\prime}\rangle^{2}=\langle\vec{a},\vec{a}\rangle-\langle\gamma,\vec{a}\rangle^{2}-\langle\gamma^{\prime},\vec{a}\rangle^{2}\\\
&=|\vec{a}|^{2}-(a_{1}^{2}+a_{2}^{2}).\end{split}$

Hence $\gamma$ has constant curvature
$\kappa=|q|\sqrt{|\vec{a}|^{2}-(a_{1}^{2}+a_{2}^{2})}$. The expression of the
torsion $\tau$ is obtained by Thm. 2.3 and identity (17).

∎

We show an explicit example of non-geodesic conformal trajectory.

######  Example 4.2.

We prove that non-great circles of $\mathbb{S}^{3}$ are conformal
trajectories. To be precise, let $\vec{a}=(a,b,c,d)\in\mathbb{R}^{4}$ and the
corresponding conformal vector field $V(p)=\vec{a}-\langle\vec{a},p\rangle p$.
In the 3-dimensional space $\vec{a}^{\perp}$ consider the orthonormal basis
$\\{\epsilon_{1},\epsilon_{2},\epsilon_{3}\\}$ given by

$\epsilon_{1}=\frac{(-d,c,-b,a)}{|\vec{a}|}\leavevmode\nobreak\
,\epsilon_{2}=\frac{(-c,-d,a,b)}{|\vec{a}|}\leavevmode\nobreak\
,\epsilon_{3}=\frac{(-b,a,d,-c)}{|\vec{a}|}\leavevmode\nobreak\ .$

Let $R\in(0,1)$ and consider the curve $\gamma$ on $\mathbb{S}^{3}$ defined by

$\gamma(s)=R\Big{(}\cos\frac{s}{R}\leavevmode\nobreak\
\epsilon_{1}+\sin\frac{s}{R}\leavevmode\nobreak\
\epsilon_{2}\Big{)}+\sqrt{1-R^{2}}\leavevmode\nobreak\ \epsilon_{3}.$

Obviously $\gamma$ is parametrized by the arc-length and
$V(\gamma(s))=\vec{a}$ for any $s$. Straightforward computations yield

$\begin{split}\nabla_{\gamma^{\prime}}\gamma^{\prime}&=\gamma^{\prime\prime}(s)+\gamma(s)=\frac{R^{2}-1}{R}\left(\cos\frac{s}{R}\leavevmode\nobreak\
\epsilon_{1}+\sin\frac{s}{R}\leavevmode\nobreak\
\epsilon_{2}\right)+\sqrt{1-R^{2}}\leavevmode\nobreak\ \epsilon_{3},\\\
V(\gamma(s))\times\gamma^{\prime}(s)&=|\vec{a}|\left(\sqrt{1-R^{2}}\cos\frac{s}{R}\leavevmode\nobreak\
\epsilon_{1}+\sqrt{1-R^{2}}\sin\frac{s}{R}\leavevmode\nobreak\
\epsilon_{2}-R\leavevmode\nobreak\ \epsilon_{3}\right).\end{split}$

By choosing $q=-\frac{\sqrt{1-R^{2}}}{|\vec{a}|R}$, then $\gamma$ satisfies
Eq. (2). Notice that $\langle\gamma(s),\vec{a}\rangle=0$ and thus
$a_{1}=a_{2}=0$ in (17). This shows that the torsion of $\gamma$ is $0$, as
expected for a circle in $\mathbb{S}^{3}$.

We particularize for a conformal vector field. Let $\vec{a}=(0,0,0,1)$ in (5).
Then

$V=-tx\partial_{x}-ty\partial_{y}-tz\partial_{z}+(1-t^{2})\partial_{t}.$

Here $\lambda(x,y,z,t)=-2t$. If $\gamma(s)=(x(s),y(s),z(s),t(s))$ is
parametrized by arc-length, then $\langle\gamma(s),\vec{a}\rangle=t(s)$ and
$t(s)$ satisfies the differential equation

$t^{\prime\prime}(s)+t(s)=0.$

In Fig. 2 we show some graphics of conformal trajectories of $V$ once we take
stereographic projection from $\mathbb{S}^{3}$ in $\mathbb{R}^{3}$.

Figure 2. Conformal trajectories of the vector field (5) when $q=1$ and
$\gamma(0)=(0,1,0,0)$. Left: $\gamma^{\prime}(0)=(-1,0,0,0)$. Middle and
right: $\gamma^{\prime}(0)=(0,0,1,1)$.

##  5\. The hyperbolic space $\mathbb{H}^{3}$

Consider the Lorentzian model for $\mathbb{H}^{3}$. Here we follow the
notation of Sect. 2. We know that the proper conformal vector fields on
$\mathbb{H}^{3}$ are generated by the vector field defined in (6).

######  Theorem 5.1.

Let $V\in\mathfrak{X}(\mathbb{H}^{3})$ be a proper conformal vector field
given by (6). Let $\gamma\colon I\to\mathbb{H}^{3}$ be a curve parametrized by
arc-length. If $\gamma$ is a conformal trajectory of $V$, then

$\langle\gamma(s),\vec{a}\rangle=a_{1}\cosh(s)+a_{2}\sinh(s),$ (19)

where $a_{1},a_{2}\in\mathbb{R}$. If $\gamma$ is a geodesic, then $\gamma$ is
obtained by intersecting $\mathbb{H}^{3}$ with a Minkowski $2$-plane
containing $\vec{a}$. If $\gamma$ is not a geodesic, then the curvature of
$\gamma$ is constant,
$\kappa(s)=|q|\sqrt{||\vec{a}|^{2}+a_{1}^{2}-a_{2}^{2}|}$, and the torsion is

$\tau(s)=q(a_{1}\sinh(s)+a_{2}\cosh(s)).$

###### Proof.

The proof is similar as in Thm. 4.1. We know the relation between the Levi-
Civita connection $\nabla$ in $\mathbb{H}^{3}$ and that of $\mathbb{R}^{4}$,
$\nabla^{e}$, namely,

$\nabla_{X}Y=\nabla_{X}^{e}Y-\langle X,Y\rangle p.$

As $\gamma$ is parametrized by arc-length, we deduce that
$\nabla_{\gamma^{\prime}}\gamma^{\prime}=\gamma^{\prime\prime}-\gamma$. Since
$\langle V\times\gamma^{\prime},\vec{a}\rangle=0$, from (2) we have

$\frac{d^{2}}{ds^{2}}\langle\gamma,\vec{a}\rangle-\langle\gamma,\vec{a}\rangle=0.$

This gives (19). If $\gamma$ is a geodesic on $\mathbb{H}^{3}$, then
$\gamma^{\prime\prime}(s)-\gamma(s)=0$ and this implies
$\gamma(s)=\cosh(s)\vec{v}+\sinh(s)\vec{w}$, where
$\langle\vec{v},\vec{v}\rangle=-1$, $\langle\vec{w},\vec{w}\rangle=1$ and
$\langle\vec{v},\vec{w}\rangle=0$. Since $V(s)\times\gamma^{\prime}(s)=0$,
then $\vec{a}$ lies in the 2-plane spanned by $\vec{v}$ and $\vec{w}$.

Suppose now that $\gamma$ is not a geodesic. Using (19) and $\kappa
N=V\times\gamma^{\prime}$, we obtain

$\begin{split}\kappa^{2}&=\langle
V\times\gamma^{\prime},V\times\gamma^{\prime}\rangle=\langle
V,V\rangle\langle\gamma^{\prime},\gamma^{\prime}\rangle-\langle
V,\gamma^{\prime}\rangle^{2}\\\
&=\langle\vec{a},\vec{a}\rangle+\langle\gamma,\vec{a}\rangle^{2}-\langle\gamma^{\prime},\vec{a}\rangle^{2}=|\vec{a}|^{2}+a_{1}^{2}-a_{2}^{2}.\end{split}$

Thus $\gamma$ has constant curvature. For the last part of the theorem, we use
Thm. 2.3 in combination with (19).

∎

######  Example 5.2.

Let $\vec{a}=(1,0,0,0)$ and $\alpha>0$. Consider the curve
$\gamma:\mathbb{R}\rightarrow\mathbb{H}^{3}$ given by

$\gamma(s)=\left(0,s,\alpha s^{2}+\alpha-\frac{1}{4\alpha},\alpha
s^{2}+\alpha+\frac{1}{4\alpha}\right).$

Obviously $\gamma$ is parametrized by the arc-length and
$V(\gamma(s))=\vec{a}$ for any $s$. Direct calculations lead to
$\gamma^{\prime\prime}-\gamma=V\times\gamma^{\prime}$ meaning that $\gamma$ is
a (non-geodesic) conformal trajectory on $\mathbb{H}^{3}$ with $q=1$.
Moreover,
$\langle\gamma^{\prime\prime}-\gamma,\gamma^{\prime\prime}-\gamma\rangle=1$
which implies that the curvature of $\gamma$ is $\kappa=1$. Then we obtain
$\tau=0$ and the Frenet frame is

$\left\\{\begin{array}[]{l}T(s)=(0,1,2\alpha s,2\alpha s),\\\\[5.69054pt]
N(s)=\left(0,-s,-\alpha s^{2}+\alpha+\frac{1}{4\alpha},-\alpha
s^{2}+\alpha-\frac{1}{4\alpha}\right),\\\\[5.69054pt]
B(s)=(1,0,0,0).\end{array}\right.$

######  Example 5.3.

Let $\vec{a}=(1,0,0,0)$ and $r>0$. Consider the curve
$\gamma:(-\frac{\pi}{2},\frac{3\pi}{2})\rightarrow\mathbb{H}^{3}$ defined by

$\gamma(s)=\left(0,\frac{\cos(\phi(s))}{1+\sin(\phi(s))},r-\frac{1}{2r(1+\sin(\phi[s)))},r+\frac{1}{2r(1+\sin(\phi[s)))}\right),$

where $\phi$ is a smooth function satisfying the differential equation
$\phi^{\prime}(s)=1+\sin(\phi(s))$. It is straightforward to show that
$\gamma$ is a unit speed conformal trajectory with $q=1$ having constant
curvature $\kappa=1$ and zero torsion.

######  Proposition 5.4.

If $\gamma$ is conformal trajectory on $\mathbb{H}^{3}$ with zero torsion,
then $\vec{a}$ must be a space-like vector.

###### Proof.

The condition $\tau=0$ yields $a_{1}=a_{2}=0$; see Thm. 5.1. Moreover, it
follows that $\langle\vec{a},\gamma(s)\rangle=0$, for any $s$. Making use of
the Cauchy–Bunyakovsky–Schwarz inequality (in $\mathbb{R}^{3}$), we obtain
that $\langle\vec{a},\vec{a}\rangle\geq 0$ and
$\langle\vec{a},\vec{a}\rangle=0$ if and only if $\vec{a}=\vec{0}$. ∎

To show graphics of conformal trajectories, let us fix $\vec{a}=(1,0,0,0)$ in
(6). Then

$V=(1+x^{2})\partial_{x}+xy\partial_{y}+xz\partial_{z}+xt\partial_{t}.$

If $\gamma(s)=(x(s),y(s),z(s),t(s))$ and fix $q=1$, then (2) writes as

$\left.\begin{split}x^{\prime\prime}-x&=0\\\
y^{\prime\prime}-y-zt^{\prime}+tz^{\prime}&=0\\\
z^{\prime\prime}-z-ty^{\prime}+yt^{\prime}&=0\\\
t^{\prime\prime}-t+yz^{\prime}-zy^{\prime}&=0.\end{split}\right\\}$ (20)

In Fig. 3 we show the graphic of a conformal trajectory $\gamma$ of $V$ in the
Poincaré ball (left) and in the upper half-space model (right). In order to
have a curve with non-zero torsion, we need $\langle\gamma(s),\vec{a}\rangle$
is not $0$ at the initial condition. For this, it is enough $x(0)\not=0$. Let
$\gamma(0)=(1,0,0,\sqrt{2})$. Since in the system (20) the curve $\gamma$ is
parametrized by arc-length, then we take $\gamma^{\prime}(0)$ of modulus $1$,
for example $\gamma^{\prime}(0)=(0,1,0,0)$.

Figure 3. A conformal trajectory for $V$ with $\vec{a}=(1,0,0,0)$ and initial
conditions $\gamma(0)=(1,0,0,\sqrt{2})$, $\gamma^{\prime}(0)$. Here $q=1$.
Left: in the model of the Poincaré ball. Right: in the upper half-space model.

## Acknowledgements

The authors would like to thank Prof. Sebastián Montiel (Granada) for
discussions on conformal vector fields in space forms. Rafael López is a
member of the IMAG and of the Research Group “Problemas variacionales en
geometría”, Junta de Andalucía (FQM 325). This research has been partially
supported by MINECO/MICINN/FEDER grant no. PID2020-117868GB-I00, and by the
“María de Maeztu” Excellence Unit IMAG, reference CEX2020-001105- M, funded by
MCINN/AEI/10.13039/501100011033/ CEX2020-001105-M. Marian Ioan Munteanu is
thankful to Romanian Ministry of Research, Innovation and Digitization, within
Program 1 – Development of the national RD system, Subprogram 1.2 –
Institutional Performance – RDI excellence funding projects, Contract
no.11PFE/30.12.2021, for financial support.

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