# Möbius inversion of surfaces in the Minkowski 3-space

Marco Antônio do Couto Fernandes

###### Abstract

We define and present some proprieties of the Möbius inversion of surfaces in
the Minkowski 3-space. We prove that the Möbius inversion preserves the lines
of principal curvature and the locus of points where the metric is degenerate,
but it does not preserve the parabolic set. For ovaloids, we show that it is
possible to translate the surface so that the inversion remains an ovaloid.

0002020 Mathematics Subject classification: 53A35 53A05 53C50 52A15 000Key
Words and Phrases. Möbius Inversion, Minkowski space, Surfaces, Curvature
Lines.

##  1 Introduction

The Möbius inversion of a closed subset $F$ in the Euclidean space
$\mbox{\boldmath$\mathbb{R}$}^{3}$ is given by

$i_{E}(F)=cl\left\\{\frac{p}{\langle p,p\rangle_{E}}:p\in
F\setminus\\{0\\}\right\\},$

where $cl$ denotes the topological closure in
$\mbox{\boldmath$\mathbb{R}$}^{3}$ endowed with the topology induced by the
Euclidean inner product $\langle\cdot,\cdot\rangle_{E}$. Geometrically, points
on the unit Euclidean sphere centered at the origin are fixed points of
$i_{E}$. Moreover, the image of points inside this sphere are points outside
it and vice-versa.

The Möbius inversion preserves the lines of principal curvature on surfaces in
$\mbox{\boldmath$\mathbb{R}$}^{3}$. Ghomi and Howard [2] use this fact to
relate the lines of principal curvature of a closed and convex surface with
one umbilic point removed to the lines of principal curvature of the graph of
an asymptotically constant function. That lead to give the following new
formulation of the Carathéodory Conjecture: the graph of an asymptotically
constant function
$f:\mbox{\boldmath$\mathbb{R}$}^{2}\to\mbox{\boldmath$\mathbb{R}$}$ has at
least one umbilic point.

In this paper, we define the Möbius inversion of surfaces in the Minkowski
3-space $\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ and show that it preserves
their lines of principal curvature. Furthermore, the Möbius inversion
preserves the set of points where the metric is degenerate and the
discriminant of the lines of principal curvature.

We show that the parabolic set is not preserved by the Möbius inversion by
means of a counterexample. However, when we work with ovaloids, which are
closed surfaces with empty parabolic sets, we obtain conditions for the Möbius
inversion of these surfaces to remain ovaloids.

The paper is organized as follows. In Section 2, we present the Minkowski
3-space $\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ and some concepts of the
differential geometry of surfaces in $\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$.
In Section 3, we define the Möbius inversion in
$\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ and give some of its properties.
Finally, in Section 4, we show that the parabolic set is not always preserved
by Möbius inversions, but for ovaloids it is possible to impose conditions in
order to preserve the emptiness of the parabolic set.

##  2 Preliminaries

The Minkowski 3-space $(\mathbb{R}_{1}^{3},\langle,\rangle)$ is the vector
space $\mathbb{R}^{3}$ endowed with the Minkowski metric induced by the
pseudo-scalar product
$\langle\mbox{\boldmath$u$},\mbox{\boldmath$v$}\rangle=u_{0}v_{0}+u_{1}v_{1}-u_{2}v_{2}$,
for any vectors $\mbox{\boldmath$u$}=(u_{0},u_{1},u_{2})$ and
$\mbox{\boldmath$v$}=(v_{0},v_{1},v_{2})$ in $\mathbb{R}^{3}$. A non-zero
vector $\mbox{\boldmath$u$}\in\mathbb{R}^{3}_{1}$ is said to be spacelike if
$\langle\mbox{\boldmath$u$},\mbox{\boldmath$u$}\rangle>0$, lightlike if
$\langle\mbox{\boldmath$u$},\mbox{\boldmath$u$}\rangle=0$ and timelike if
$\langle\mbox{\boldmath$u$},\mbox{\boldmath$u$}\rangle<0$. The norm of a
vector $\mbox{\boldmath$u$}\in\mathbb{R}_{1}^{3}$ is defined by
$\|\mbox{\boldmath$u$}\|=\sqrt{|\langle\mbox{\boldmath$u$},\mbox{\boldmath$u$}\rangle|}.$

If $p$ is a point in $\mbox{\boldmath$\mathbb{R}$}^{3}$, then $\langle
p,p\rangle$ denotes $\langle\vec{0p},\vec{0p}\rangle$, where $\vec{0p}$ is the
vector from the origin to $p$. The set of points
$p\in\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ with $\langle p,p\rangle=0$ is
called the light cone and is denoted by $LC$. The de Sitter sphere, denoted by
$\mathbb{S}^{2}_{1}$, is the set of points
$q\in\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ such that $\langle q,q\rangle=1$.
On the other hand, the hyperbolic plane, denoted by $\mathbb{H}^{2}$, is the
set of the points $q\in\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ where $\langle
q,q\rangle=-1$.

Let $S$ be a smooth and regular surface in $\mathbb{R}^{3}_{1}$ and let
$\mbox{\boldmath$x$}:U\subset\mathbb{R}^{2}\to\mathbb{R}_{1}^{3}$ be a local
parametrization of $S$. The concepts of differential geometry in
$\mbox{\boldmath$\mathbb{R}$}^{3}$ can be extended to
$\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$, see for example [1]. The pseudo inner
product $\langle\cdot,\cdot\rangle$ in $\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$
induces a pseudo metric on $S$ called the first fundamental form. Let
$E=\langle{\mbox{\boldmath$x$}}_{u},{\mbox{\boldmath$x$}}_{u}\rangle$,
$F=\langle{\mbox{\boldmath$x$}}_{u},{\mbox{\boldmath$x$}}_{v}\rangle$ and
$G=\langle{\mbox{\boldmath$x$}}_{v},{\mbox{\boldmath$x$}}_{v}\rangle$, where
subscripts denote partial derivatives. The induced (pseudo) metric on $S$ is
Lorentzian (resp. Riemannian, degenerate) at $p=\mbox{\boldmath$x$}(u,v)$ if
and only if $\delta(u,v)=(F^{2}-EG)(u,v)>0$ (resp. $<0$, $=0$). The locus of
points on the surface where the metric is degenerate is called the locus of
degeneracy and is denoted by $LD$. We identify the $LD$ on $S$ with its pre-
image in $U$ by $x$.

At $p\in S\setminus LD$, we have a well defined unit normal vector (the Gauss
map)
$\mbox{\boldmath$N$}=\mbox{\boldmath$x$}_{u}\times\mbox{\boldmath$x$}_{v}/||\mbox{\boldmath$x$}_{u}\times\mbox{\boldmath$x$}_{v}||$,
which is timelike (resp. spacelike) if $p$ is in the Riemannian (resp.
Lorentzian) region of $S$. The self-adjoint operator
$A_{p}=-d\mbox{\boldmath$N$}_{p}:T_{p}S\to T_{p}S$ on $S\setminus LD$ defines
a quadratic form $II_{p}(\mbox{\boldmath$v$})=\langle
A_{p}(\mbox{\boldmath$v$}),\mbox{\boldmath$v$}\rangle$ on $T_{p}S$ called the
second fundamental form. We denote by
$l=\langle\mbox{\boldmath$N$},\mbox{\boldmath$x$}_{uu}\rangle$,
$m=\langle\mbox{\boldmath$N$},\mbox{\boldmath$x$}_{uv}\rangle$ and
$n=\langle\mbox{\boldmath$N$},\mbox{\boldmath$x$}_{vv}\rangle$ the
coefficients of the second fundamental form on $S\setminus LD$.

When $A_{p}$ has real eigenvalues $\kappa_{1}$ and $\kappa_{2}$, we call them
the principal curvatures and their associated eigenvectors the principal
directions of $S$ at $p$. There are always two principal curvatures at each
point on the Riemannian part of $S$, but this is not always the case on its
Lorentzian part. A point $p$ on $S$ is called an umbilic point if
$\kappa_{1}=\kappa_{2}$ at $p$ (i.e., if $A_{p}$ is a multiple of the identity
map).

The lines of principal curvature on $S$, which are the integral curves of the
principal directions, are the solutions of the binary differential equation
(BDE)

$(Fn-Gm)dv^{2}+(En-Gl)dvdu+(Em-Fl)du^{2}=0.$ (1)

The discriminant of the BDE (1) consists of the umbilic points in the
Riemannian region of $S$ and is the locus of points in the Lorentzian region
where two principal directions coincide and become lightlike. In the last
case, it is labelled Lightlike Principal Locus $(LPL)$ in [4].

The Gaussian curvature of $S$ at $p$ is given by $K(p)=\det(A_{p})$, that is,

$K(p)=\frac{ln-m^{2}}{EG-F^{2}}(p).$

The locus of points where the Gaussian curvature is zero is called the
parabolic set.

One can extend the lines of principal curvature, the $LPL$ and the parabolic
set across the $LD$ as follows ([5]). As equation (1) is homogeneous in
$l,m,n$, we can multiply these coefficients by
$||\mbox{\boldmath$x$}_{u}\times\mbox{\boldmath$x$}_{v}||$ and substitute them
in equation (1) by

$\bar{l}=\langle\mbox{\boldmath$x$}_{u}\times\mbox{\boldmath$x$}_{v},\mbox{\boldmath$x$}_{uu}\rangle,\quad\bar{m}=\langle\mbox{\boldmath$x$}_{u}\times\mbox{\boldmath$x$}_{v},\mbox{\boldmath$x$}_{uv}\rangle,\quad\bar{n}=\langle\mbox{\boldmath$x$}_{u}\times\mbox{\boldmath$x$}_{v},\mbox{\boldmath$x$}_{vv}\rangle.$

The new equation

$(G\bar{m}-F\bar{n})dv^{2}+(G\bar{l}-E\bar{n})dudv+(F\bar{l}-E\bar{m})du^{2}=0$
(2)

is defined at points on the $LD$ and its solutions are the same as those of
equation (1) in the Riemannian and Lorentzian regions of $S$. The discriminant
of the BDE $(\ref{eq:principalLD})$ extends the $LPL$ to points on the $LD$.
Thus, a point $p=\mbox{\boldmath$x$}(u,v)$ belongs to the $LPL$ of $S$ when

$\tilde{\delta}(u,v)=\left((E\bar{n}-G\bar{l})^{2}-4(F\bar{n}-G\bar{m})(E\bar{m}-F\bar{l})\right)(u,v)=0.$

For the parabolic set, note that $K=0$ if and only if
$\bar{l}\bar{n}-\bar{m}^{2}=0$. Thus, the parabolic set also extends to points
on the $LD$ as the zero set of the function
$\bar{K}=\bar{l}\bar{n}-\bar{m}^{2}$.

##  3 Möbius inversion in $\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$

We define the Möbius inversion of a closed subset
$F\subset\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ as

$i_{M}(F)=cl\left\\{\frac{p}{\langle p,p\rangle}:p\in F\setminus LC\right\\},$

where $cl$ denotes the topological closure in
$\mbox{\boldmath$\mathbb{R}$}^{3}$ with the same topology used to define
$i_{E}$. When necessary, we consider the Möbius inversion as a function
$i_{M}:\mbox{\boldmath$\mathbb{R}$}^{3}_{1}\setminus
LC\to\mbox{\boldmath$\mathbb{R}$}^{3}_{1}\setminus LC$ given by
$i_{M}(p)=p/\langle p,p\rangle$. Consider the following regions of
$\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$:

$\begin{array}[]{l}R_{1}=\\{(x,y,z)\in\mbox{\boldmath$\mathbb{R}$}^{3}_{1}:x^{2}+y^{2}-z^{2}>0\\},\\\
R_{2}=\\{(x,y,z)\in\mbox{\boldmath$\mathbb{R}$}^{3}_{1}:x^{2}+y^{2}-z^{2}<0,\,z>0\\},\\\
R_{3}=\\{(x,y,z)\in\mbox{\boldmath$\mathbb{R}$}^{3}_{1}:x^{2}+y^{2}-z^{2}<0,\,z<0\\}.\end{array}$
(3)

Then $i_{M}$ is a diffeomorphism from $R_{1}$ to $R_{1}$ and from $R_{2}$ to
$R_{3}$ with $i_{M}^{-1}=i_{M}$. Points on the hyperbolic plane
$\mathbb{H}^{2}$ and on the de Sitter sphere $\mathbb{S}^{2}_{1}$ are fixed
points of $i_{M}$. In $R_{1}$, $i_{M}$ is an inversion with respect to
$\mathbb{H}^{2}$, Figure 1 left. On the other hand, in $R_{2}$ and $R_{3}$,
$i_{M}$ is an inversion with respect to the $\mathbb{S}^{2}_{1}$, Figure 1
center and right.

Figure 1: Möbius inversion in Minkowski space.

###### Theorem 3.1

Let $S$ be a compact surface in $\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$. Then
following properties hold:

  1. 1.

$i_{M}(S)=\left\\{\frac{p}{\langle p,p\rangle}:p\in S\setminus LC\right\\}$;

  2. 2.

$i_{M}(S)\cap LC=\emptyset$;

  3. 3.

$i_{M}(S)$ is bounded if and only if $S\cap LC=\emptyset$.

  * Proof

(1) Let $A=\left\\{\frac{p}{\langle p,p\rangle}:p\in S\setminus LC\right\\}$
and $\\{q_{i}\\}_{i\in\mathbb{N}}$ a sequence in $A$ that converges to some
$q\in\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$, with $q_{i}=p_{i}/\langle
p_{i},p_{i}\rangle$ and $p_{i}\in S\setminus LC$, for all $i\in\mathbb{N}$.
Since $S$ is compact, there is a subsequence $\\{p_{i_{k}}\\}$ of
$\\{p_{i}\\}$ that converges to some $p\in S$. If $p\in LC$, then
$\|p_{i_{k}}\|\rightarrow 0$ and $\|q_{i_{k}}\|\rightarrow\infty$, which
contradicts the convergence of $\\{q_{i}\\}$. Thus, $p\in S\setminus LC$ and

$q=\lim_{k\rightarrow\infty}q_{i_{k}}=\lim_{k\rightarrow\infty}\frac{p_{i_{k}}}{\langle
p_{i_{k}},p_{i_{k}}\rangle}=\frac{p}{\langle p,p\rangle}\in A.$

Therefore, $A$ is closed and $i_{M}(S)=cl(A)=A$.

(2) Suppose that $q\in i_{M}(S)\cap LC$. It follows from (1) above that there
exists $p\in S\setminus LC$ such that $q=i_{M}(p)$. Then

$0=\langle q,q\rangle=\left\langle\frac{p}{\langle
p,p\rangle},\frac{p}{\langle p,p\rangle}\right\rangle=\frac{1}{\langle
p,p\rangle},$

which is not possible since $\langle p,p\rangle<\infty$. Therefore, $i_{M}(S)$
does not intersect $LC$.

(3) If $S\cap LC\neq\emptyset$, then there is a sequence
$\\{p_{i}\\}_{i\in\mathbb{N}}$ in $S\setminus LC$ and $p\in S\cap LC$ with
$p_{i}\to p$. Then

$\lim_{i\rightarrow\infty}\|p_{i}\|=\|p\|=0\Rightarrow\lim_{i\rightarrow\infty}\|i_{M}(p_{i})\|=\infty,$

so $i_{M}(S)$ is unbounded. For the converse, if $S\cap LC=\emptyset$, then

$\alpha=\sup\left\\{\frac{1}{|\langle p,p\rangle|},p\in
S\right\\}<\infty\quad{\rm and}\quad\beta=\sup\\{\|p\|,p\in S\\}<\infty.$

Therefore, $i_{M}(S)$ is contained in the Euclidean sphere with center the
origin and radius $\alpha\beta$. $\Box$

An important property of the Möbius inversion in the Euclidean space
$\mbox{\boldmath$\mathbb{R}$}^{3}$ is that it preserves the lines of principal
curvature of surfaces in $\mbox{\boldmath$\mathbb{R}$}^{3}$, that is, if $C$
is a line of principal curvature of a surface
$S\subset\mbox{\boldmath$\mathbb{R}$}^{3}$, then $i_{M}(C)$ is a line of
principal curvature of the surface $i_{M}(S)$. That property also hold for
Möbius inversion in $\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$.

###### Theorem 3.2

The lines of principal curvature of surfaces in
$\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ are preserved by the Möbius inversion.

  * Proof

Let $S\subset\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ be a surface and
$\varphi:U\subset\mbox{\boldmath$\mathbb{R}$}^{2}\rightarrow\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$
be a local parameterization of $S$. Then $\tilde{\varphi}=i_{M}\circ\varphi$
is a local parameterization of $S_{M}=i_{M}(S)$, where $E$, $F$ and $G$ are
the coefficients of the first fundamental form of $S$ and $E_{M}$, $F_{M}$ and
$G_{M}$ are those of $S_{M}$. It follows that

$E_{M}=\frac{E}{\langle\varphi,\varphi\rangle^{2}},\quad
F_{M}=\frac{F}{\langle\varphi,\varphi\rangle^{2}},\quad
G_{M}=\frac{G}{\langle\varphi,\varphi\rangle^{2}}.$ (4)

The coefficients $\bar{l}_{M}$, $\bar{m}_{M}$ and $\bar{n}_{M}$ of the second
fundamental form of $S_{M}$ with denominator removed are expressed as follows
in terms of the coefficients $\bar{l}$, $\bar{m}$ and $\bar{n}$ of $S$:

$\bar{l}_{M}=\frac{1}{\langle\varphi,\varphi\rangle^{3}}(\bar{l}+\alpha
E),\qquad\bar{m}_{M}=\frac{1}{\langle\varphi,\varphi\rangle^{3}}(\bar{m}+\alpha
F),\qquad\bar{n}_{M}=\frac{1}{\langle\varphi,\varphi\rangle^{3}}(\bar{n}+\alpha
G),$

with $\alpha=-2\langle N,\tilde{\varphi}\rangle$.

The equation of the lines of principal curvature of $S_{M}$ is given by

$(\bar{m}_{M}E_{M}-\bar{l}_{M}F_{M})dx^{2}+(\bar{n}_{M}E_{M}-\bar{l}_{M}G_{M})dxdy+(\bar{n}_{M}F_{M}-\bar{m}_{M}G_{M})dy^{2}=0.$
(5)

Substituting the coefficients $E_{M}$, $F_{M}$, $G_{M}$, $\bar{l}_{M}$,
$\bar{m}_{M}$ and $\bar{n}_{M}$ by their expressions in terms of $E$, $F$,
$G$, $\bar{l}$, $\bar{m}$ and $\bar{n}$, it follows that (5) is satisfied if
and only if

$(\bar{f}E-\bar{e}F)dx^{2}+(\bar{g}E-\bar{e}G)dxdy+(\bar{g}F-\bar{f}G)dy^{2}=0,$

which is precisely the equation of the lines of principal curvatures of $S$.
$\Box$

###### Corollary 3.3

The LPL and the LD of a surface in the Minkowski 3-space are preserved by the
Möbius inversion.

  * Proof

The proof for the preservation of the LPL is consequence of the Theorem 3.2.
As for the LD, observe from (4) that $F_{M}^{2}-E_{M}G_{M}=0$ if and only if
$F^{2}-EG=0$. $\Box$

##  4 Möbius inversion and the parabolic set

Let $p_{0}=(a,b,c)\in\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ and

$S(p_{0},r)=\\{p\in\mbox{\boldmath$\mathbb{R}$}^{3}_{1}:\langle
p-p_{0},p-p_{0}\rangle_{E}=r^{2}\\},$

so $S(p_{0},r)$ is the Euclidean sphere with center $p_{0}$ and radius $r>0$
considered as a set of points in $\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$.
Denote $S_{M}(p_{0},r)=i_{M}(S(p_{0},r))$.

###### Theorem 4.1

The surface $S_{M}(p_{0},r)=i_{M}(S(p_{0},r))$ is homeomorfic to $S(p_{0},r)$
if and only if $(\sqrt{a^{2}+b^{2}}-|c|)^{2}>2r^{2}$.

  * Proof

From item (3) of Theorem 3.1 and from the fact that
$i_{M}:\mbox{\boldmath$\mathbb{R}$}^{3}_{1}\setminus
LC\to\mbox{\boldmath$\mathbb{R}$}^{3}_{1}\setminus LC$ is a diffeomorphism, it
follows that $S_{M}(p_{0},r)$ is homeomorfic to $S(p_{0},r)$ if and only if
$S(p_{0},r)\cap LC=\emptyset$. Note that $S(p_{0},r)\cap LC=\emptyset$ when
$d(p_{0},LC)>r$, where $d(p_{0},LC)$ is the Euclidean distance between $p_{0}$
and $LC$.

We need to obtain the minimum value of the function

$f(x,y,z)=(x-a)^{2}+(y-b)^{2}+(z-c)^{2}$

restricted to the light cone $x^{2}+y^{2}=z^{2}$. Using the Lagrange
Multipliers method, we find the minimum point and it follows that

$d(p_{0},LC)=\frac{\sqrt{(\sqrt{a^{2}+b^{2}}-|c|)^{2}}}{\sqrt{2}}.$

Therefore, $S(p_{0},r)\cap LC=\emptyset$ if and only if
$(\sqrt{a^{2}+b^{2}}-|c|)^{2}>2r^{2}$. $\Box$

The surface $S_{M}(p_{0},r)$ with $p_{0}=(2,0,0)$ and $r=1$ is closed (ie,
compact without boundary) by Theorem 4.1 because the point
$\left(\frac{3}{4},\frac{\sqrt{15}}{12},0\right)\in S_{M}(p_{0},r)$, for
example, belongs to the parabolic set, see Figure 2. Thus, unlike the lines of
principal curvature, the $LD$ and the $LPL$, the parabolic set is not
preserved by the Möbius inversion. We have the following conditions on $p_{0}$
and $r$ for $S_{M}(p_{0},r)$ to be convex.

Figure 2: The surface $S_{M}(p_{0},r)$ with $p_{0}=(2,0,0)$ and $r=1$.

###### Theorem 4.2

The parabolic set of $S_{M}(p_{0},r)$ is empty if and only if
$\sqrt{a^{2}+b^{2}}>2r+\sqrt{c^{2}+r^{2}}$ or
$|c|>2r+\sqrt{a^{2}+b^{2}+r^{2}}$.

  * Proof

Let $L=S(p_{0},r)\cap LC$. Take a local parameterization
$\varphi:U\to\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ of $S(p_{0},r)$ given by
$\varphi(u,v)=\left(r\cos(u)\sin(v)+a,r\sin(u)\sin(v)+b,r\cos(v)+c\right)$,
with $U=(\epsilon_{0},\epsilon_{0}+2\pi)\times(0,\pi)$ for some
$\epsilon_{0}>0$ small enough. Then $\tilde{\varphi}:U\setminus L\to
S_{M}(p_{0},r)$ is a local parameterization of $S_{M}(p_{0},r)$ with
$\tilde{\varphi}(u,v)=i_{M}(\varphi(u,v))$. Calculating the Gaussian curvature
of $S_{M}(p_{0},r)$, the point $\tilde{\varphi}(u,v)$ belongs to the parabolic
set of $S_{M}(p_{0},r)$ if and only if $f(u,v)=0$ or $g(u,v)=0$, with

$\begin{array}[]{ccl}f(u,v)&=&-a^{2}-b^{2}+c^{2}+r^{2}+4cr\cos(v)+2r^{2}\cos(v)^{2},\vspace{0.2cm}\\\
g(u,v)&=&-a^{2}-b^{2}+c^{2}+4cr\cos(v)^{3}+3r^{2}\cos(2v)-4rh(u)\sin(v)^{3},\end{array}$

and $h(u)=a\cos(u)+b\sin(u)$.

Note that $f(u,v)=0$ if and only if

$\cos(v)=\frac{-2c\pm\sqrt{2}\sqrt{a^{2}+b^{2}+c^{2}-r^{2}}}{2r}.$ (6)

Equation (6) has no solutions when $a^{2}+b^{2}+c^{2}<r^{2}$,
$|c|<-2r+\sqrt{a^{2}+b^{2}+r^{2}}$ or $|c|>2r+\sqrt{a^{2}+b^{2}+r^{2}}$.

The image of the function $h$ is the closed interval
$\left[-\sqrt{a^{2}+b^{2}},\sqrt{a^{2}+b^{2}}\right]$, so $g_{min}(v)\leq
g(u,v)\leq g_{max}(v)$ for all $(u,v)$ in $U$, with

$\begin{array}[]{ccl}g_{min}(v)=-a^{2}-b^{2}+c^{2}+4cr\cos(v)^{3}+3r^{2}\cos(2v)-4r\sqrt{a^{2}+b^{2}}\sin(v)^{3},\vspace{0.2cm}\\\
g_{max}(v)=-a^{2}-b^{2}+c^{2}+4cr\cos(v)^{3}+3r^{2}\cos(2v)+4r\sqrt{a^{2}+b^{2}}\sin(v)^{3}.\end{array}$

Computing the critical values of $g_{min}$ in $[0,\pi]$, we get that
$g_{min}>0$ when $|c|>2r+\sqrt{a^{2}+b^{2}+r^{2}}$. Likewise, $g_{max}<0$ when
$\sqrt{a^{2}+b^{2}}>2r+\sqrt{c^{2}+r^{2}}$. Hence, $g(u,v)=0$ has no solutions
if and only if $\sqrt{a^{2}+b^{2}}>2r+\sqrt{c^{2}+r^{2}}$ or
$|c|>2r+\sqrt{a^{2}+b^{2}+r^{2}}$.

The parameterization $\varphi$ does not include the points $(0,0,\pm r)\in
S(p_{0},r)$. However, using other parameterization of $S(p_{0},r)$, it follows
that $i_{M}(0,0,\pm r)$ does not belong to the parabolic set of
$S_{M}(p_{0},r)$ when $\sqrt{a^{2}+b^{2}}>2r+\sqrt{c^{2}+r^{2}}$ or
$|c|>2r+\sqrt{a^{2}+b^{2}+r^{2}}$. $\Box$

A surface $S$ has a contact $A_{1}^{+}$ with $T_{p}S$ at $p\in S$ when the
height function $h$ has a Morse singularity of index 0 or 2 at $p$. A closed
surface $S$ is an ovaloid if it is strictly convex, that is,it has a contact
$A_{1}^{+}$ with the tangent plane $T_{p}S$, for all $p\in S$. Note that the
definition of ovaloid is independent of the metric (Euclidean or Lorentzian)
in $\mbox{\boldmath$\mathbb{R}$}^{3}$. For example, the Euclidean spheres are
ovaloids.

###### Remark 4.3

A closed surface is an ovaloid if and only if its parabolic set is empty.
Indeed, an ovaloid consists of eliptic points.

###### Corollary 4.4

$S_{M}(p_{0},r)$ is an ovaloid if and only if
$\sqrt{a^{2}+b^{2}}>2r+\sqrt{c^{2}+r^{2}}$ or
$|c|>2r+\sqrt{a^{2}+b^{2}+r^{2}}$.

  * Proof

The proof follows from the Theorems 4.1, 4.2 and Remark 4.3. $\Box$

Given an ovaloid $S$, let $N_{E}:S\to\mathbb{S}^{2}$ be the Gauss map of $S$
viewed as a surface on Euclidean 3-space and take $N_{E}$ pointing to the
interior of $S$. For $p\in S$ and $r>0$, define $S_{p}(r)$ to be the Euclidean
sphere with center $p+rN_{E}(p)$ and radius $r$. Thus, $S_{p}(r)$ and $S$ are
tangent at $p$ for any $r>0$.

###### Lemma 4.5

Let $S\subset\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ be an ovaloid. Then there
exist $R>0$ such that $S\setminus\\{p\\}$ is contained inside $S_{p}(R)$, for
all $p\in S$.

  * Proof

Let $p$ be a point in $S$ and $d$ the Euclidean distance in
$\mbox{\boldmath$\mathbb{R}$}^{3}$. Denote by $\theta(q)\in[0,\pi]$ the angle
between the vectors $q-p$ and $N_{E}(p)$, for all $q\in S\setminus\\{p\\}$.

Since the contact between $S$ and $T_{p}S$ in $p$ is $A_{1}^{+}$, there is a
neighborhood $U$ of $p$ in $\mbox{\boldmath$\mathbb{R}$}^{3}$ such that, in
$U$, no point of the interior of $S$ belongs to $T_{p}S$ (considering $T_{p}S$
as the plane that passes through $p$). Suppose some $q\in S\setminus\\{p\\}$
belongs to $T_{p}S$. For sufficiently small $\epsilon>0$, the point
$(1-\epsilon)p+\epsilon q$ belong to $U$ and $T_{p}S$, since the segment
$[p,q]$ is contained in $T_{p}S$. Therefore, $[p,q]$ is not contained within
$S$, which contradicts the convexity of $S$. Therefore, $S\cap T_{p}S=\\{p\\}$
and $\theta(q)\in[0,\pi/2)$ for all $q\in S\setminus\\{p\\}$.

Given $q\in S\setminus\\{p\\}$, let $q^{\prime}$ be the orthogonal projection
of $q$ to the line passing through $p$ with direction $N_{E}(p)$. We have

$\begin{array}[]{ccl}d(q,p+rN_{E}(p))^{2}&=&d(q,q^{\prime})^{2}+d(q^{\prime},p+rN_{E}(p))^{2}\\\
&=&d(q,q^{\prime})^{2}+\left(d(p,p+rN_{E}(p))-d(p,q^{\prime})\right)^{2}\\\
&=&r^{2}-2d(q)r\cos(\theta(q))+d(q)^{2}.\end{array}$ (7)

If $2r>d(q)^{2}/\cos(\theta(q))$, it follows from (7) that
$d(q,p+rN_{E}(p))<r$. Since $S$ is compact,

$d_{max}=\sup\\{d(p,q):q\in S\\}<\infty\quad{\rm and}\quad
c=\inf\\{\cos(\theta(q)):q\in S\\}>0$

because $\theta(q)\in[0,\pi/2)$. Let $R(p)=d_{max}^{2}/2c$. For all $r>R(p)$
the set $S\setminus\\{p\\}$ is inside $S_{p}(r)$. Finally, let
$R=\sup\\{R(p):p\in S\\}+1$, which is finite because $S$ is compact.
Therefore, $S\setminus\\{p\\}$ is inside $S_{p}(R)$ for all $p$. $\Box$

###### Theorem 4.6

Given an ovaloid $S\subset\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$, there is a
translation
$T:\mbox{\boldmath$\mathbb{R}$}^{3}_{1}\to\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$
such that $i_{M}(T(S))$ is an ovaloid.

  * Proof

Take $R>0$ given by the Lemma 4.5. From the Corollary 4.4 it is possible to
obtain a translation
$T:\mbox{\boldmath$\mathbb{R}$}^{3}_{1}\to\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$
such that the Möbius inversion in $\mbox{\boldmath$\mathbb{R}$}_{1}^{3}$ of
$T(S_{p}(R))$ is an ovaloid for all $p\in S$.

It follows from item (3) of Theorem 3.1 that $T(S_{p}(R))$ does not intersect
$LC$ for all $p\in S$ and also $T(S)$ do not intersect $LC$. Thus,
$i_{M}(T(S))$ is a closed surface.

Given $p\in S$, $i_{M}$ is a diffeomorphism between the interior of
$T(S_{p}(R))$ and the interior of $i_{M}(T(S_{p}(R)))$ because
$T(S_{p}(R))\cap LC=\emptyset$. So $i_{M}(T(S\setminus\\{p\\}))$ is contained
inside $i_{M}(T(S_{p}(R)))$ and $i_{M}(T(S))$ is tangent to
$i_{M}(T(S_{p}(R)))$ in $i_{M}(T(p))$. Since $i_{M}(T(S_{p}(R)))$ is an
ovaloid, the contact between $i_{M}(T(S_{p}(R)))$ and its tangent plane at
$i_{M}(T(p))$ is a $A_{1}^{+}$, then the contact of $i_{M}(T(S))$ and its
tangent plane at $i_{M}(T(p))$ is also $A_{1}^{+}$. Therefore, $i_{M}(T(S))$
is an ovaloid. $\Box$

To exemplify the Theorem 4.6, consider again the Euclidean sphere $S(p_{0},r)$
with $p_{0}=(2,0,0)$ and $r=1$, then $i_{M}(S(p_{0},r))$ is not an ovaloid as
noted earlier (see Figure 2). However, taking the translation
$T(x,y,z)=(x,y,z)+(2,0,0)$, then $i_{M}(T(S(p_{0},r)))$ is an ovaloid (see
Figure 3).

Figure 3: The surface $S_{M}(p_{0},r)$ with $p_{0}=(2,0,0)$ and $r=1$.

  * Acknowledgments.

This work was financed by the doctoral grant Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001, and was
supervised by Farid Tari.

## References

  * [1] I. T. Couto and A. Lymberopoulos, Introduction to Lorentz Geometry: Curves and Surfaces. Chapman and Hall/CRC, 2021.
  * [2] M. Ghomi and R. Howard, Normal curvatures of asymptotically constant graphs and Carathéodory’s conjecture. Proc. Amer. Math. Soc. 140 (2012), 4323–4335.
  * [3] S. Izumiya, M. D. C. R. Fuster, M. A. S. Ruas, F. Tari, Differential geometry from a singularity theory viewpoint. World Scientific Publishing Co. Pte. Ltd., 2016.
  * [4] S. Izumiya, M. Takahashi and F. Tari, Folding maps on spacelike and timelike surfaces and duality. Osaka J. Math. 47 (2010), 839–862.
  * [5] S. Izumiya and F. Tari, Self-adjoint operators on surfaces with singular metrics. J. Dyn. Control Syst. 16 (2010), 329–353.

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