# Bifurcations of robust features on surfaces in the Minkowski 3-space

Marco Antônio do Couto Fernandes

###### Abstract

We obtain the bifurcation of some special curves on generic 1-parameter
families of surfaces in the Minkowski 3-space. The curves treated here are the
locus of points where the induced pseudo metric is degenerate, the
discriminant of the lines principal curvature, the parabolic curve and the
locus of points where the mean curvature vanishes.

0002020 Mathematics Subject classification: 58K60 53B30 53A35 53A05 58K05
000Key Words and Phrases. Family of surfaces, Minkowski space, Umbilic points,
Special curves on surfaces.

##  1 Introduction

We consider generic 1-parameter families of surfaces in the Minkowski 3-space
$\mathbb{R}^{3}_{1}$. The Lorentzian metric in $\mathbb{R}^{3}_{1}$ induces a
pseudo metric on a surface in $M\subset\mathbb{R}^{3}_{1}$. The locus of
points on $M$ where the induced pseudo metric is degenerate is called the
Locus of Degeneracy and is denoted by the $LD$ for short. The $LD$ separates
the surface into regions where the induced metric is Riemannian or Lorentzian.
On such regions the Gaussian and mean curvature of the surface are well
defined. The zero sets of the Gaussian and mean curvature functions extends to
the $LD$ and are called, respectively, the parabolic curve and mean curvature
null curve ($PC$ and $MCNC$ for short).

The equation of the lines of principal curvature also extends to the $LD$
([14]). In the Riemannian region the discriminant of that equation consists of
the umbilic points. In the Lorentzian region it is the locus of points where
two principal directions coincide and become lightlike and is labelled
Lightlike Principal Locus, $LPL$ for short.

We obtain in this paper the catalogue of all possible local bifurcations of
the $LD$, the $LPL$, the $MCNC$ and the $PC$ when the surface is deformed in
generic 1-parameter families of surfaces. We deal in §4 with the bifurcations
at points in the Lorentzian region and in §5 at points on the $LD$. We give
some preliminaries in §2 and define the notion of genericity and codimension
in §3. We observe that at points on the Riemannian region of the surface the
situation is identical to that of surfaces in the Euclidean space. Families of
such surfaces are studied in, for example, [3, 4, 5, 6, 8, 10, 11].

##  2 Preliminaries

The Minkowski space $(\mathbb{R}_{1}^{3},\langle,\rangle)$ is the vector space
$\mathbb{R}^{3}$ endowed with the 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 $\mathbb{R}^{3}$ (see for example
[7, 15] for a treatment of the geometry of surfaces in $\mathbb{R}^{3}_{1}$).
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$}\rangle|}.$

Let $M$ 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
parametrisation of $M$. We shall simplify notation and write
$\mbox{\boldmath$x$}(U)=M$. Let

$E=\langle{\mbox{\boldmath$x$}}_{u},{\mbox{\boldmath$x$}}_{u}\rangle,\quad
F=\langle{\mbox{\boldmath$x$}}_{u},{\mbox{\boldmath$x$}}_{v}\rangle,\quad
G=\langle{\mbox{\boldmath$x$}}_{v},{\mbox{\boldmath$x$}}_{v}\rangle$

denote the coefficients of the first fundamental form of $M$ with respect to
the parametrisation $x$, where subscripts denote partial derivatives. The
induced (pseudo) metric on $M$ is Lorentzian (resp. Riemannian, degenerate) at
${\rm p}=\mbox{\boldmath$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 $M$ with its pre-image in $U$ by
$x$. Then the $LD$ (in $U$) is given by

$LD=\\{(u,v)\in U\,|\,\delta(u,v)=0\\}.$

A direction $(du,dv)\in T_{\rm p}M$ is lightlike if and only if

$Edu^{2}+2Fdudv+Gdv^{2}=0.$ (1)

Equation (1) has two (resp. no) solutions when ${{\rm p}}$ is in the
Lorentzian (resp. Riemannian) region of $M$. At points on the $LD$, there is a
unique (double) solution of the equation.

###### Theorem 2.1

([14]) There is a local parametrization
$\mbox{\boldmath$x$}:U\subset\mbox{\boldmath$\mathbb{R}$}^{2}\to\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$
of $M$ at $p\in LD$ such that the lightlike directions in $T_{q}M$ is given by
$\mathbb{R}.x_{u}$ for every $q\in LD$, i.e., $E=F=0$ on the $LD$.

If $q\in M$ belongs to the Lorentzian region, there is a local parametrization
of $M$ at $p$ such that $\mathbb{R}.x_{u}$ and $\mathbb{R}.x_{v}$ are the
lightlike directions in $T_{q}M$ for every $q\in\mbox{\boldmath$x$}(U)$, i.e.,
$E=G=0$ on $U$.

At ${{\rm p}}\in M\setminus LD$, 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 ${\rm p}$ is in the Riemannian (resp.
Lorentzian) region of $M$. (See [16] for a definition of an
$\mathbb{R}P^{2}$-valued Gauss map.) The map $A_{{\rm
p}}=-d\mbox{\boldmath$N$}_{{\rm p}}:T_{{\rm p}}M\to T_{{\rm p}}M$ is a self-
adjoint operator on $M\setminus LD$. We denote by

$l=\langle\mbox{\boldmath$N$},\mbox{\boldmath$x$}_{uu}\rangle,\quad
m=\langle\mbox{\boldmath$N$},\mbox{\boldmath$x$}_{uv}\rangle,\quad
n=\langle\mbox{\boldmath$N$},\mbox{\boldmath$x$}_{vv}\rangle$

the coefficients of the second fundamental form on $M\setminus LD$.

When the eigenvalues $\kappa_{1}$ and $\kappa_{2}$ of $A_{p}$ are real, they
are called the principal curvatures and their associated eigenvectors the
principal directions of $M$ at ${{\rm p}}$. There are always two principal
curvatures at each point on the Riemannian part of $M$ but this is not always
true on its Lorentzian part. A point ${\rm p}$ on $M$ is called an umbilic
point if $\kappa_{1}=\kappa_{2}$ at ${\rm p}$ (i.e., if $A_{p}$ is a multiple
of the identity map). It is called a spacelike umbilic point (resp. timelike
umbilic point) if ${\rm p}$ is in the Riemannian (resp. Lorentzian) part of
$M$.

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

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

The discriminant of BDE (2)

$\\{(u,v)\in U\,|\left((En-Gl)^{2}-4(Fn-Gm)(Em-Fl)\right)(u,v)=0\\}$

consists of the umbilic points in the Riemannian region of $M$ and is the
locus of points in the Lorentzian region where two principal directions
coincide and become lightlike. It is labelled Lightlike Principal Locus
$(LPL)$ in [13]. The principal directions are orthogonal when there are two of
them at a given point; in particular one is spacelike and the other is
timelike if the point is in the Lorentzian region.

The Gaussian curvature and mean curvature of $M$ at $p$ are $K=\det(A_{p})$
and $H=-\frac{1}{2}tr(A_{p})$, respectively. We have

$K=\frac{ln-m^{2}}{EG-F^{2}},\qquad H=\frac{lG-2mF+nE}{2(EG-F^{2})}.$

The locus of points where the Gaussian and mean curvature are zero are called
the parabolic curve ($PC$) and mean curvature null curve ($MCNC$),
respectively.

One can extend the lines of principal curvature, the $LPL$, the $PC$ and the
$MCNC$ across the $LD$ as follows ([14]). As equation (2) 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 the equation 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

$(G\bar{m}-F\bar{n})dv^{2}+(G\bar{l}-E\bar{n})dudv+(F\bar{l}-E\bar{m})du^{2}=0$
(3)

is defined at points on the $LD$ and its solutions are the same as those of
equation (2) in the Riemannian and Lorentzian regions of $M$. A lightlike
umbilic point is defined as a point on the $LD$ where all the coefficients of
equation (3) vanish (see [18]). This occurs if and only if the $LD$ is
singular ([18]). 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 $M$ 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 $PC$, note that $K=0$ if and only if $\bar{l}\bar{n}-\bar{m}^{2}=0$.
Thus, the $PC$ extends to points on the $LD$ as the zero set of the function
$\bar{K}=\bar{l}\bar{n}-\bar{m}^{2}$. Similarly, the $MCNC$ extends to the
$LD$ as the zero set of the function $\bar{H}=\bar{l}G-2\bar{m}F+\bar{n}E$.

We use here some concepts from singularity theory (see [1, 19]). Two germs of
functions $f_{i}:(\mathbb{R}^{n},0)\to(\mathbb{R}^{m},0),i=1,2$ are said to be
$\mathcal{R}$-equivalent if $f_{2}=f_{1}\circ h^{-1}$ for some germ of a
diffeomorphism
$h:(\mbox{\boldmath$\mathbb{R}$}^{n},0)\to(\mbox{\boldmath$\mathbb{R}$}^{n},0)$.
Similarly, $f_{1}$ and $f_{2}$ are said to be $\mathcal{L}$-equivalent when
there is a germ of a diffeomorphism
$k:(\mbox{\boldmath$\mathbb{R}$}^{m},0)\to(\mbox{\boldmath$\mathbb{R}$}^{m},0)$
such that $f_{2}=k\circ f_{1}$. Finally, $f_{1}$ and $f_{2}$ are
$\mathcal{A}$-equivalent if there exist germs of diffeomorphisms $h$ and $k$
such that $f_{2}=k\circ f_{1}\circ h^{-1}$.

The simple singularities of germs of functions ($m=1$) are classified by
Arnold [1]. Representatives of their $\mathcal{R}$-orbits when $n=2$ are as
follows

$A_{k}:\pm(x^{2}\pm y^{k+1}),k\geq 1,\,D_{k}:x^{2}y\pm y^{k-1},k\geq
4,\,E_{6}:x^{3}+y^{4},\,E_{7}:x^{3}+xy^{3},\,E_{8}:x^{3}+y^{5}.$

For $n\geq 3$ one adds the quadratic form $\pm x_{3}^{2}\pm\cdots\pm
x_{n}^{2}$ to the above normal forms.

We denote by $j^{k}_{q}f$ the $k$-jet of a germ of a smooth/analytic function
$f:(\mathbb{R}^{n},q)\to\mathbb{R}$, i.e., its Taylor polynomial of order $k$
at $q$, and by $J^{k}(n,1)$ the space of all $k$-jets of germs of functions.
If $q$ is the origin, then we denote the $k$-jet by $j^{k}f$. When $n=2$, we
write the $k$-jet, at the origin, of a function $f$ in the form
$j^{k}f=\sum_{s=0}^{k}\sum_{i=0}^{s}a_{si}x^{s-i}y^{i}$ and identify it with
the coefficients $(a_{si})\in\mbox{\boldmath$\mathbb{R}$}^{(k+1)(k+2)/2}$.

##  3 Genericity and codimension

In this section, we present some concepts used in the rest of the paper. Let
$G(3,1)$ be the group generated by Lorentzian rotations, reflection,
translations and homotheties with the composition operation. Any $\sigma\in
G(3,1)$ preserves the $LD$, $LPL$, $PC$ and $MCNC$ on surfaces in
$\mathbb{R}^{3}_{1}$.

Let $p\in M$ be a point in Lorentzian region. After composing by elements of
$G(3,1)$ if necessary, we can parametrise $M$ locally in Monge-form
$\mbox{\boldmath$x$}:U\subset\mbox{\boldmath$\mathbb{R}$}^{2}\to\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$,
with $\mbox{\boldmath$x$}(x,z)=(x,f(x,z),z)$, so that
$p=\mbox{\boldmath$x$}(0,0)$ and

$j^{k}f=\sum_{i=2}^{k}\sum_{j=0}^{i}a_{ij}x^{i-j}z^{j}.$ (4)

Then, $p$ is an umbilic point if and only if $a_{20}=-a_{22}$ and $a_{21}=0$.

Similarly, if $p\in M$ belongs to $LD$, then after composing by elements of
$G(3,1)$ if necessary, we can parametrise $M$ locally in Monge-form
$\mbox{\boldmath$x$}:U\subset\mbox{\boldmath$\mathbb{R}$}^{2}\to\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$
with $\mbox{\boldmath$x$}(x,y)=(x,y,f(x,y))$, so that
$p=\mbox{\boldmath$x$}(0,0)$ and

$j^{k}f=x+\sum_{i=2}^{k}\sum_{j=0}^{i}a_{ij}x^{i-j}y^{j}.$ (5)

With the above setting, $p\in LPL$ if and only if $a_{20}=0$, and $p$ is an
umbilic point if and only if $a_{20}=a_{21}=0$.

In [2], Bruce described a technique for studying local properties of surfaces
in the Euclidean space $\mathbb{R}^{3}$ called Monge-Taylor map. We follow the
ideas in [2] but, as our study is local, we choose a fixed coordinate system
$\\{{\bf u},{\bf v},{\bf w}\\}$ in a neighbourhood $W$ of a point ${\rm p}\in
M$ with ${\bf w}$ transverse to $M$ at points in $W$. For each ${\rm p}\in W$,
consider the coordinate system $\\{{\bf u},{\bf v},{\bf w}\\}$ with origin at
${\rm p}$ and take a local parametrization of $M$ in Monge form
$(x,y)\mapsto\mbox{\boldmath$x$}_{{\rm p}}(x,y)=(x,y,f_{{\rm p}}(x,y))$. We
define the Monge-Taylor map $\theta:W\to J^{k}(2,1)$ by $\theta({{\rm
p}})=j^{k}f_{{\rm p}}$.

###  3.1 Generic properties of surfaces

Let $\mathcal{G}$ be the set of germs $(M,p)$ where
$M\subset\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ is a smooth surface and $p\in
M$. A germ
$\mbox{\boldmath$x$}:(\mbox{\boldmath$\mathbb{R}$}^{2},0)\to(\mbox{\boldmath$\mathbb{R}$}^{3}_{1},p)$
is a local parametrization if $x$ is smooth, has a continuous inverse and
$d\mbox{\boldmath$x$}_{(0,0)}:\mbox{\boldmath$\mathbb{R}$}^{2}\to\mbox{\boldmath$\mathbb{R}$}^{3}$
is one-to-one. A local parametrization of $M$ is a local parametrization whose
image is contained in $M$. Consider

$\mathcal{F}=\\{\mbox{\boldmath$x$}:(\mbox{\boldmath$\mathbb{R}$}^{2},0)\to(\mbox{\boldmath$\mathbb{R}$}^{3}_{1},p):p\in\mbox{\boldmath$\mathbb{R}$}^{3}_{1},\,\mbox{\boldmath$x$}\textit{
is a local parameterization}\\}$

with the Whitney topology. Note that $\sim_{\mathcal{R}}$ is an equivalence
relation on $\mathcal{F}$ and denote by $[\mbox{\boldmath$x$}]$ the
equivalence class of $\mbox{\boldmath$x$}\in\mathcal{F}$ relative to
$\sim_{\mathcal{R}}$. Therefore, $\mathcal{F}/\sim_{\mathcal{R}}$ is a
topological space with the quotient topology.

The map $f:\mathcal{G}\to\mathcal{F}/\sim_{\mathcal{R}}$ that associates with
$(M,p)\in\mathcal{G}$ the equivalence class
$[\mbox{\boldmath$x$}:(\mbox{\boldmath$\mathbb{R}$}^{2},0)\to(\mbox{\boldmath$\mathbb{R}$}^{3}_{1},p)]$
of a local parameterization of $M$ is well defined and is a bijection.
Therefore, we provide $\mathcal{G}$ with the only topology that makes $f$ a
homeomorphism. A local property $P$ is said to be generic in $\mathcal{G}$
when the subset $\mathcal{P}\subset\mathcal{G}$ formed by the pairs $(M,p)$
such that the point $p$ of $M$ satisfies the property $P$ is open and dense in
$\mathcal{G}$. Given a subset $\tilde{\mathcal{G}}\subset\mathcal{G}$ we
consider the induced topology of $\mathcal{G}$ in $\tilde{\mathcal{G}}$ and
define the concept of generic property in $\tilde{\mathcal{G}}$ as in
$\mathcal{G}$. For example, if

$\tilde{\mathcal{G}}=\\{(M,p)\in\mathcal{G}:p\textit{ is a timelike umbilic
point}\\},$

then the property $P:``LPL\textit{ has a singularity }A_{1}^{-}\textit{ at
}p"$ is generic in $\tilde{\mathcal{G}}$ [9].

A $m$-parameters family of surfaces $M_{t}$ deforming $(M,p)\in\mathcal{G}$ is
a family of surfaces whose parameterizations
$\mbox{\boldmath$x$}_{t}:U\to\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ of $M_{t}$
smoothly depend of
$t=(t_{1},\dots,t_{m})\in(\mbox{\boldmath$\mathbb{R}$}^{m},0)$ and $M_{0}=M$.
Let $z=f(x,y)$ be a Monge parameterization of the surface $M$. Then any
$m$-parameter family of surfaces $M_{t}$ with $M_{0}=M$ can be given by

$z=f(x,y)+h(x,y,t_{1},\dots,t_{m})$

with $h$ differentiable and $h(x,y,0,\dots,0)=0$ for all $(x,y)\in U$. We say
that the family $M_{t}$ is generic when $h$ satisfies generic conditions (open
and dense conditions).

###### Remark 3.1

When the surface $M$ is given in Monge form $x=f(y,z)$ or $y=f(x,z)$, the
definition of a generic family of surfaces is analogous.

We denote by $LD_{t}$ the set of points on the surface $M_{t}$ where the
metric is degenerate. Thus, $LD_{0}$ is the $LD$ of the surface $M$ and the
family of curves $LD_{t}$, with $t\in(\mbox{\boldmath$\mathbb{R}$}^{m},0)$, is
a deformation of the $LD$ of $M$. Similarly, the $LPL$, the $PC$ and the
$MCNC$ of $M$ are deformed along the family of surfaces. We denote the $LPL$,
the $PC$ and the $MCNC$ of $M_{t}$ by $LPL_{t}$, $PC_{t}$ and $MCNC_{t}$,
respectively.

###  3.2 Codimension of a property

Let $M\subset\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ be a surface. A point at
$M$ with a property $(P)$ has codimension $n$ when the submanifold in
$J^{k}(2,1)$, for $k$ sufficiently large, obtained by the conditions imposed
by $(P)$ on the coefficients of the Monge parameterization has codimension
$n+2$. Generically, a $n$-parameter family of surfaces has points with
properties of codimension $\leq n$. We are interested in the bifurcations of
the $LD$, the $LPL$, the $PC$ and the $MCNC$ in generic 1-parameter families
of surfaces, that is, in properties on these curves with codimension 0 and 1.

Properties of codimension 0 with respect to these curves are stable and known.
For example, regular points of the $LD$ have codimension 0. In fact, taking
$M$ in Monge form $z=f(x,y)$, with
$j^{2}f=a_{00}+a_{10}x+a_{11}y+a_{20}x^{2}+a_{21}xy+a_{22}y^{2}$. Then
$p=\mbox{\boldmath$x$}(0,0)$ is a regular point of the $LD$ when

$a_{10}^{2}+a_{11}^{2}=1,\quad a_{11}a_{21}+2a_{10}a_{20}=0,\quad
2a_{11}a_{22}+a_{10}a_{21}\neq 0.$ (6)

As the submanifold of $J^{2}(2,1)$ defined by (6) has codimension 2, regular
points of the $LD$ have codimension 0. Similarly, regular points of the $LPL$,
$PC$ and $MCNC$ have codimension 0. Also, the Morse singularities of the $LPL$
have codimension 0 ([14, 17]).

Given two curves $\gamma_{1}$ and $\gamma_{2}$ on $M$, we denote by
$m(\gamma_{1},\gamma_{2}:p)$ the order of contact between then at $p$. At
points of codimension 0, the $LPL$ has ordinary tangencies with the $LD$ and
with the $PC$ [14], the $LD$ is transversal to the $PC$ (see §5.2) and the
$MCNC$ is transversal to all other curves (see §4.1 and §5.1).

We study here properties of codimension 1. We separete the cases when the
point is considered in Lorentzian region (§4) or on the $LD$ (§5). The case
when the point is in the Riemannian region are already studied elsewhere. At
points in the Lorentzian region, codimension 1 cases can occur as follows:

  * (i)

Degenerate contact between regular relevant curves (§4.1);

  * (ii)

flat timelike umbilic points (§4.2);

  * (iii)

singularities of the relevant curves (§4.3, §4.4).

At points on the $LD$, we get codimension 1 cases when

  * (i)

Degenerate contact between regular relevant curves with $LD$ (§5.1, 5.2);

  * (ii)

singularities of the $LD$ (§5.3).

##  4 Bifurcations at points in the Lorentzian region

###  4.1 Bifurcations at $\textbf{LPL}\cap\textbf{MCNC}\cap\textbf{PC}$

In this subsection, we will discuss the intersection between the $LPL$ and the
$PC$ when the $LPL$ is a regular curve (the singular cases are considered in
§4.2 and §5.3). The intersection between the $LPL$ and the $PC$ at points of
the $LD$ are always singularities of the $LPL$ (see §5.3). In the Riemannian
region, the $LPL$ consist of the umbilic points and is singular at such
points. Therefore, the intersection between the $PC$ and regular points of the
$LPL$ occurs only in the Lorentzian region.

###### Theorem 4.1

Let $M\subset\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ be a smooth surface and
$p\in M\setminus LD$. Then $p$ belongs to the intersection of two of three
curves: the $LPL$, the $PC$ and the $MCNC$ if and only if $p$ belongs to the
intersection of the three of then. Furthemore, if $p$ is a regular point on
the $LPL$, then $m(LPL,PC:p)=2m(LPL,MCNC:p)$.

  * Proof

Let $\mbox{\boldmath$x$}:U\to\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ be the
local parametrization of $M$ given by Theorem 2.1 with
$p=\mbox{\boldmath$x$}(0,0)$. Since $E=G=0$ on $U$, then the $LPL$, the $PC$
and the $MCNC$ are given by $\bar{l}\bar{n}=0$, $\bar{m}^{2}-\bar{l}\bar{n}=0$
and $\bar{m}=0$, respectively, and this proves the first part of the
statement.

Consider a local parametrization
$\gamma:(\mbox{\boldmath$\mathbb{R}$},0)\to(\mbox{\boldmath$\mathbb{R}$}^{2},0)$
of the $LPL$ at a regular point $p=\mbox{\boldmath$x$}(0,0)$. We have
$\bar{K}(\gamma(x))=\bar{m}(\gamma(x))^{2}$ and
$\bar{H}(\gamma(x))=-2F(\gamma(x))\bar{m}(\gamma(x))$. Since $F$ does not
vanish on $U$, the order of contact between the $LPL$ and the $PC$ is twice
the order of contact between the $LPL$ and the $MCNC$. $\Box$

Therefore, in this subsection, the intersections between the $LPL$ and the
$MCNC$ will also be considered.

###### Theorem 4.2

Regular points $p$ of the $LPL$ where $m(LPL,PC:p)>2$ are of codimension $\geq
1$.

  * Proof

Take a local parameterization
$\mbox{\boldmath$x$}:U\to\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ of $M$, with
$\mbox{\boldmath$x$}(x,z)=(x,f(x,z),z)$, $p=\mbox{\boldmath$x$}(0,0)$ and

$j^{3}f=a_{00}+a_{10}x+a_{11}z+a_{20}x^{2}+a_{21}xz+a_{22}z^{2}+a_{30}x^{3}+a_{31}x^{2}z+a_{32}xz^{2}+a_{33}z^{3}.$

The point $p$ belongs to the $LPL\cap MCNC$ if and only if

$\left\\{\begin{array}[]{lcc}a_{11}^{2}a_{20}-a_{11}a_{10}a_{21}+a_{22}a_{10}^{2}+a_{22}-a_{20}&=&0,\vspace{0.3cm}\\\
4a_{11}^{2}a_{20}^{2}-4a_{11}a_{10}a_{21}a_{20}+a_{10}^{2}a_{21}^{2}+a_{21}^{2}-4a_{20}^{2}&=&0.\end{array}\right.$
(7)

These curves are transversal at $p$ when

$\begin{array}[]{cclcc}\Lambda_{1}&=&\left(a_{20}-a_{11}^{2}a_{20}+a_{22}a_{10}^{2}+a_{22}\right)(a_{32}a_{31}-9a_{33}a_{30})&&\vspace{0.2cm}\hfil\\\
&&+\left(a_{11}^{2}a_{21}-2a_{11}a_{22}a_{10}-a_{21}\right)\left(a_{31}^{2}-3a_{32}a_{30}\right)&&\vspace{0.2cm}\hfil\\\
&&-\left(a_{10}^{2}a_{21}-2a_{11}a_{10}a_{20}+a_{21}\right)\left(a_{32}^{2}-3a_{33}a_{31}\right)&\neq&0.\end{array}$
(8)

It follows from the Theorem 4.1 that $m(LPL,PC:p)>2$ if $m(LPL,MCNC:p)>1$,
that is, when $\Lambda_{1}=0$. The result follows from the fact that the
system of equations (7), (8) define a submanifold on $J^{3}(2,1)$ of
codimension 3. $\Box$

Let

$\mathcal{G}_{1}=\\{(M,p)\in\mathcal{G}:m(LPL,PC:p)>2\\}.$

Given $(M,p)\in\mathcal{G}_{1}$, take
$\mbox{\boldmath$x$}:U\subset\mbox{\boldmath$\mathbb{R}$}^{2}\to\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$
the local parameterization of $M$ given by Theorem 2.1, with
$p=\mbox{\boldmath$x$}(0,0)$. With this parameterization, we have
$\tilde{\delta}=4F^{2}\bar{l}\bar{n}$, $\bar{K}=\bar{m}^{2}-\bar{l}\bar{n}$
and $\bar{H}=-2F\bar{m}$. We write

$\begin{array}[]{ll}j^{k}F=\sum_{s=0}^{k}\sum_{i=0}^{s}F_{si}u^{s-i}v^{i},&j^{k}\bar{l}=\sum_{s=0}^{k}\sum_{i=0}^{s}l_{si}u^{s-i}v^{i},\vspace{0.2cm}\\\
j^{k}\bar{m}=\sum_{s=0}^{k}\sum_{i=0}^{s}m_{si}u^{s-i}v^{i},&j^{k}\bar{n}=\sum_{s=0}^{k}\sum_{i=0}^{s}n_{si}u^{s-i}v^{i}.\end{array}$

We have $F_{00}\neq 0$ and $m_{00}=0$ because $p\in MCNC$. On the other hand,
$l_{00}=0$ or $n_{00}=0$ for $p\in LPL$, wit as $p$ is a regular point on the
$LPL$. We assume without loss of generality that $l_{00}=0$. Then
$\tilde{\delta}_{u}(0,0)=4F_{00}^{2}l_{10}n_{00}$ and
$\tilde{\delta}_{v}(0,0)=4F_{00}^{2}l_{11}n_{00}$, consequently $l_{10}\neq 0$
or $l_{11}\neq 0$. Suppose that $l_{10}\neq 0$, the other case is similar. We
have $m_{11}=\frac{l_{11}m_{10}}{l_{10}}$ because the $LPL$ and the $MCNC$ are
not transverse at $p$.

It follows from the Implicit Function Theorem that the $LPL$ (in $U$) given by
$\tilde{\delta}(u,v)=0$ is parameterized by $\gamma(v)=(g(v),v)$ in a
neighborhood $V$ of the origin, for some smooth function $g$. We get

$j^{4}g=-\frac{l_{11}}{l_{10}}x-\frac{l_{11}^{2}l_{20}-l_{10}l_{11}l_{21}+l_{10}^{2}l_{22}}{l_{10}^{3}}x^{2}+\frac{c_{3}}{l_{10}^{5}}x^{3}+\frac{c_{4}}{l_{10}^{7}}x^{4},$

with

$\begin{array}[]{ccl}c_{3}&=&-l_{10}^{4}l_{33}+l_{10}^{3}l_{11}l_{32}+l_{10}^{3}l_{21}l_{22}-l_{10}^{2}l_{11}^{2}l_{31}-2l_{10}^{2}l_{11}l_{20}l_{22}-l_{10}^{2}l_{11}l_{21}^{2}+l_{10}l_{11}^{3}l_{30}\vspace{0.2cm}\\\
&&+3l_{10}l_{11}^{2}l_{20}l_{21}-2l_{11}^{3}l_{20}^{2},\vspace{0.3cm}\\\
c_{4}&=&-l_{10}^{6}l_{44}+l_{10}^{5}l_{11}l_{43}+l_{10}^{5}l_{21}l_{33}+l_{10}^{5}l_{22}l_{32}-l_{10}^{4}l_{11}^{2}l_{42}-2l_{10}^{4}l_{11}l_{20}l_{33}-2l_{10}^{4}l_{11}l_{21}l_{32}\vspace{0.2cm}\\\
&&-2l_{10}^{4}l_{11}l_{22}l_{31}-l_{10}^{4}l_{20}l_{22}^{2}-l_{10}^{4}l_{21}^{2}l_{22}+l_{10}^{3}l_{11}^{3}l_{41}+3l_{10}^{3}l_{11}^{2}l_{20}l_{32}+3l_{10}^{3}l_{11}^{2}l_{21}l_{31}\vspace{0.2cm}\\\
&&+3l_{10}^{3}l_{11}^{2}l_{22}l_{30}+6l_{10}^{3}l_{11}l_{20}l_{21}l_{22}+l_{10}^{3}l_{11}l_{21}^{3}-l_{10}^{2}l_{11}^{4}l_{40}-4l_{10}^{2}l_{11}^{3}l_{20}l_{31}-4l_{10}^{2}l_{11}^{3}l_{21}l_{30}\vspace{0.2cm}\\\
&&-6l_{10}^{2}l_{11}^{2}l_{20}^{2}l_{22}-6l_{10}^{2}l_{11}^{2}l_{20}l_{21}^{2}+5l_{10}l_{11}^{4}l_{20}l_{30}+10l_{10}l_{11}^{3}l_{20}^{2}l_{21}-5l_{11}^{4}l_{20}^{3}.\vspace{0.2cm}\end{array}$

###### Theorem 4.3

Let $(M,p)\in\mathcal{G}_{1}$ is generic. Then

(1)

$m(LPL,PC:p)=4$ and $m(LPL,MCNC:p)=2$. There are two possible configurations
for the three curves as shows in Figure 1 middle figures.

(2)

For a generic 1-parameter family of surfaces $M_{t}$, with $M_{0}=M$, the
configurations of the three curves deform as in Figure 1.

Figure 1: Deformations of the $LPL$ (red), the $PC$ (blue) and the $MCNC$
(green) in a generic 1-parameter family of surfaces at points $p$ where
$m(LPL,MCNC)=2$.

  * Proof

(1) We observe that

$\left(\bar{K}\circ\gamma\right)(v)=\left(\frac{\Lambda_{2}}{l_{10}^{3}}v^{2}\right)^{2}+O(5)\quad{\rm
and}\quad\left(\bar{H}\circ\gamma\right)(v)=-2F_{00}\frac{\Lambda_{2}}{l_{10}^{3}}v^{2}+O(3),$

where $O(k)$ is a remamber of order $k$, and

$\Lambda_{2}=l_{10}^{3}m_{22}-l_{10}^{2}l_{11}m_{21}-l_{10}^{2}l_{22}m_{10}+l_{10}l_{11}^{2}m_{20}+l_{10}l_{11}l_{21}m_{10}-l_{11}^{2}l_{20}m_{10}.$

We have two cases depending on the sign of $F_{00}l_{10}\Lambda_{2}$. Figure 1
first when $F_{00}l_{10}\Lambda_{2}<0$ and last when
$F_{00}l_{10}\Lambda_{2}>0$.

(2) We take $\mbox{\boldmath$x$}_{t}:U\to\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$
a parameterization of $M_{t}$ given by Theorem 2.1 which depends smoothly on
$t\in(\mbox{\boldmath$\mathbb{R}$},0)$ and
$\mbox{\boldmath$x$}_{0}=\mbox{\boldmath$x$}$. Denote by $E(u,v,t)$,
$F(u,v,t)$, $G(u,v,t)$, $\bar{l}(u,v,t)$, $\bar{m}(u,v,t)$ and
$\bar{n}(u,v,t)$ the coefficients of the first and second fundamental forms of
$M_{t}$ in $(u,v)\in U$ with respect to $\mbox{\boldmath$x$}_{t}$. Thus,
$E(u,v,t)=G(u,v,t)=0$ on $U$.

The point $\mbox{\boldmath$x$}_{t}(u,v)$ is on the $LPL_{t}$ if and only if
$l(u,v,t)=0$. Likewise, $\mbox{\boldmath$x$}_{t}(u,v)\in MCNC_{t}$ when
$m(u,v,t)=0$. Let $\sigma_{t}(u,v)=\left(l(u,v,t),m(u,v,t)\right)$. If
$l(u,v,t)=\bar{l}(u,v)+tl_{2}(u,v,t)$ and
$m(u,v,t)=\bar{m}(u,v)+tm_{2}(u,v,t)$, then

$\sigma_{t}\sim_{\mathcal{A}}\left(u,v^{2}+\frac{l_{10}m_{2}(0,0,0)-m_{10}l_{2}(0,0,0))}{l_{10}}t+O(2)\right),$

where $O(2)$ is a remander of order 2 in $t$. Therefore, when
$l_{10}m_{2}(0,0,0)\neq m_{10}l_{2}(0,0,0)$, the $LPL_{t}$ and the $MCNC_{t}$
intersect at 2 or 0 points depending on the sign of $t$. As the $PC$
intersects these curves at $LPL_{t}\cap MCNC_{t}$, it follows that the
deformation of these curves is as in Figure 1. $\Box$

###  4.2 Bifurcations at a flat timelike umbilic point

A flat umbilic point is an umbilic point where Gaussian curvature is zero.
Thus, flat umbilic points are where the $PC$ passes through a singularity of
the $LPL$. Flat spacelike umbilic points are similar to flat umbilic points on
surfaces in Euclidean space and such points have already been studied in [4].

Let

$\mathcal{G}_{2}=\\{(M,p)\in\mathcal{G}:p{\rm\,\,is\,\,a\,\,flat\,\,timelike\,\,umbilic\,\,point}\\}.$

In this subsection, unless otherwise noted, we use the local parameterization
$\mbox{\boldmath$x$}:U\to\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ of $M$ given as
in (4) with $p=\mbox{\boldmath$x$}(0,0)$. Since $p$ is a flat timelike umbilic
point, it follows that $j^{2}f=0$.

###### Theorem 4.4

Let $(M,p)\in\mathcal{G}_{2}$ is generic. Then

(1)

The branches of the $LPL$, the $PC$ and the $MCNC$ curves are parwise
transverse when they are real. There are three generic configurations as shown
in Figure 2 middle figures.

(2)

For a generic 1-parameter family of surfaces $M_{t}$, with $M_{0}=M$, the
configurations of the three curves deform as in Figure 2.

Figure 2: Generic bifurcations in the $LPL$ (red), the $PC$ (blue) and the
$MCNC$ (green) at a flat timelike umbilic point.

  * Proof

(1) It follows from $j^{3}f=a_{30}x^{3}+a_{31}x^{2}z+a_{32}xz^{2}+a_{33}z^{3}$
that the $LPL$ has a singularity of type $A_{1}^{-}$ at $p$ if and only if
$\Lambda_{3}=-a_{31}^{2}+3a_{30}a_{32}+a_{32}^{2}-3a_{31}a_{33}\neq 0$. The
$PC$ has a singularity of type $A_{1}^{\pm}$ at $p$ if and only if
$\Lambda_{4}=-a_{31}^{2}a_{32}^{2}+4a_{30}a_{32}^{3}+4a_{31}^{3}a_{33}-18a_{30}a_{31}a_{32}a_{33}+27a_{30}^{2}a_{33}^{2}\neq
0$. The $MCNC$ is a regular curve at $p$ if and only if
$\Lambda_{5}=(a_{31}-3a_{33})^{2}+(a_{32}-3a_{30})^{2}\neq 0$.

The tangent lines to the $LPL$ at $p$ are $r_{1}:c_{1}x+d_{1}z=0$ and
$r_{2}:c_{2}x+d_{2}z=0$, where $c_{1}=3a_{30}-2a_{31}+a_{32}$,
$d_{1}=a_{31}-2a_{32}+3a_{33}$, $c_{2}=3a_{30}+2a_{31}+a_{32}$ and
$d_{2}=a_{31}+2a_{32}+3a_{33}$. Also, the tangent line to the $MCNC$ at $p$ is
$r_{3}:c_{3}x+d_{3}z=0$, with $c_{3}=3a_{30}-a_{32}$ and
$d_{3}=a_{31}-3a_{33}$. Thus, the condition for $r_{3}$ to be distinct from
$r_{1}$ and $r_{2}$ are, respectively,

$\Lambda_{6}=a_{31}^{2}-3a_{30}a_{32}-a_{31}a_{32}+a_{32}^{2}+9a_{30}a_{33}-3a_{31}a_{33}\neq
0,$
$\Lambda_{7}=a_{31}^{2}-3a_{30}a_{32}+a_{31}a_{32}+a_{32}^{2}-9a_{30}a_{33}-3a_{31}a_{33}\neq
0.$

If $\Lambda_{4}<0$, the $PC$ has a singularity $A_{1}^{+}$ at $p$. Therefore,
when $\Lambda_{6}\neq 0$ and $\Lambda_{7}\neq 0$, the only possibility is
shown in Figure 2 middle last figure.

When $\Lambda_{4}>0$, the $PC$ has a singularity $A_{1}^{-}$ and divides $U$
into four regions in a neighborhood of the origin. Suppose
$\Lambda_{6}\Lambda_{7}\neq 0$. Let $v_{1}$ and $v_{2}$ be the tangent vectors
to $r_{1}$ and $r_{2}$, respectively, and
$F(x,z)=j^{2}\bar{K}=k_{20}x^{2}+k_{21}xz+k_{22}z^{2}$, with

$k_{20}=4(a_{31}^{2}-3a_{30}a_{32}),\quad
k_{21}=4(a_{31}a_{32}-9a_{30}a_{33}),\quad
k_{22}=4(a_{32}^{2}-3a_{31}a_{33}).$

The relative position of the $LPL$ and of the $PC$ depends on the sign of
$F(v_{1})=\Lambda_{6}^{2}>0$ and $F(v_{2})=\Lambda_{7}^{2}>0$. Thus, the
branches of the $LPL$ are in the same region delimited by the $PC$.

Finally, to the position of the $MCNC$, we define
$G(x,z)=(c_{1}x+d_{1}z)(c_{2}x+d_{2}z)$. Note that $G(x,z)=0$ corresponds to
the lines $r_{1}$ and $r_{2}$. As $F(v_{1})=\Lambda_{6}^{2}>0$ and
$F(v_{2})=\Lambda_{7}^{2}>0$, there are two possible combinations for the sign
of $F$ and $G$, as shown in Figure 3.

Figure 3: Possible combinations for the signs of $F$ and $G$. The blue lines
are $F=0$ and the red lines are $G=0$.

Let $v_{3}$ be the tangent vector to the $MCNC$ at $p$. We have
$F(v_{3})=\Lambda_{6}\Lambda_{7}$ and $G(v_{3})=-4\Lambda_{6}\Lambda_{7}$. If
$F(v_{3})<0$, then $G(v_{3})>0$. Therefore, the configuration in Figure 3
right figure is the only one possible position when we consider the $MCNC$.

For a generic surface $(M,p)\in\mathcal{G}_{2}$, all the conditions above are
satisfied. Therefore, the configurations of the curves as in Figure 2 middle
figures.

(2) Let $M_{t}\subset\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ be a family of
surfaces parameterized by $\mbox{\boldmath$x$}_{t}:U\to M_{t}$, with
$\mbox{\boldmath$x$}_{t}(x,z)=(x,h(x,z,t),z)$ for some $h$ differentiable with
$h(x,z,0)=f(x,z)$.

The $MCNC$ of $M_{t}$ are given by $\tilde{\delta}_{t}=0$ and $\bar{H}_{t}=0$,
respectively. Since $\tilde{\delta}_{t}$ has an $A_{1}^{-}$-singularity for
every $t$, we have $j^{2}\tilde{\delta}_{t}=f_{1}(x,z,t)f_{2}(x,z,t)$, where
$f_{1},f_{2}:\mbox{\boldmath$\mathbb{R}$}^{3}\to\mbox{\boldmath$\mathbb{R}$}$
are degree 1 polynomials in $x$ and $y$, with

$\det\left(\begin{array}[]{cc}\frac{\partial f_{1}}{\partial
x}(0,0,0)&\frac{\partial f_{1}}{\partial y}(0,0,0)\vspace{0.2cm}\\\
\frac{\partial f_{2}}{\partial x}(0,0,0)&\frac{\partial f_{2}}{\partial
y}(0,0,0)\end{array}\right)\neq 0.$

From the Inverse Function Theorem, it follows that $X=f_{1}(x,z,t)$,
$Z=f_{2}(x,z,t)$, $T=t$ is a change of coordinates in a neighborhood of the
origin. Applying this coordinate change to $\tilde{\delta}_{t}$ and
$\bar{H}_{t}$, we get $\tilde{\delta}_{t}\sim_{\mathcal{R}}XZ$ and
$\bar{H}_{t}\sim_{\mathcal{R}}\frac{\Lambda_{6}}{2d_{1}d_{2}}X+\frac{\Lambda_{7}}{2d_{1}d_{2}}Z+\frac{2\Lambda_{8}}{d_{1}d_{2}}T$,
where

$\Lambda_{8}=-3a_{30}a_{32}h_{zzt}+9a_{30}a_{33}h_{xzt}+a_{31}^{2}h_{zzt}-a_{31}a_{32}h_{xzt}-3a_{31}a_{33}h_{xxt}+a_{32}^{2}h_{xxt}\neq
0,$

and the derivatives of $h$ are evaluated in $(0,0,0)$. Since $\Lambda_{6}\neq
0$, $\Lambda_{7}\neq 0$ and $\Lambda_{8}\neq 0$ are generic conditions, in
generic 1-parameter family of surfaces the $LPL$ and the $MCNC$ deforms as in
Figure 4 at a flat timelike umbilic point.

Figure 4: Deformations of the $LPL$ (red) and the $MCNC$ (green) in a generic
family of surfaces at a flat timelike umbilic point.

Let $\bar{K}_{t}$ be such that the $PC_{t}$ is defined by $\bar{K}_{t}=0$. We
can assume that $\Lambda_{4}\neq 0$ and $k_{20}\neq 0$. We have

$j^{2}\bar{K}_{t}\sim_{\mathcal{R}^{(2)}}\pm\left(\pm
x^{2}+z^{2}-\frac{\Lambda_{8}^{2}t^{2}}{12|k_{20}\Lambda_{4}|}\right)+t^{3}h(t).$

and the bifurcations in the $PC$ in generic 1-parameter family of surfaces are
generic cone sections as in Figure 2. Therefore, the result follows from
Theorem 4.1. $\Box$

###  4.3 Bifurcations at a none Morse singularity of the LPL

At a timelike umbilic point, the $LPL$ has generically an
$A_{1}^{-}$-singularity. We consider here the case where the $LPL$ has a more
degenerate singularity. Let

$\mathcal{G}_{3}=\\{(M,p)\in\mathcal{G}:p{\rm\,\,is\,\,a\,\,non-
Morse\,\,singularity\,\,of\,\,the\,\,}LPL\\}.$

Given $(M,p)\in\mathcal{G}_{3}$, take
$\mbox{\boldmath$x$}:U\subset\mbox{\boldmath$\mathbb{R}$}^{2}\to\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$
the local parameterization of $M$ given by Theorem 2.1. The $LPL$ in $U$ is
given by $\tilde{\delta}(u,v)=\bar{l}(u,v)\bar{n}(u,v)=0$, with
$j^{2}\bar{l}=\sum_{i=1}^{2}\sum_{j=0}^{i}a_{ij}x^{i-j}y^{j}$ and
$j^{2}\bar{n}=\sum_{i=1}^{2}\sum_{j=0}^{i}b_{ij}x^{i-j}y^{j}$. The $LPL$ has
an $A_{1}^{-}$-singularity at the origin when $a_{10}b_{11}-a_{11}b_{10}\neq
0$.

###### Theorem 4.5

For a generic $(M,p)\in\mathcal{G}_{3}$, we have

(1)

The $LPL$ has a $A^{-}_{3}$ singularity at $p$;

(2)

The bifurcation of the $LPL$ in a generic 1-parameter family of surfaces
$M_{t}$, with $M_{0}=M$, are as in Figure 5.

Figure 5: Bifurcations of the $A^{-}_{3}$ singularity of the $LPL$ on a
generic 1-parameter family of surfaces.

  * Proof

(1) We have $a_{10}b_{11}-a_{11}b_{10}=0$ as the $LPL$ has singularity more
degenerate than $A_{1}^{-}$. Suppose that $a_{10}\neq 0$ or $b_{10}\neq 0$.
Then $\tilde{\delta}\sim_{\mathcal{R}^{(4)}}\pm(u^{2}-\Lambda_{9}^{2}v^{4})$,
with

$\Lambda_{9}=-a_{11}^{2}a_{20}b_{10}+a_{10}a_{11}a_{21}b_{10}-a_{10}^{2}a_{22}b_{10}+a_{10}a_{11}^{2}b_{20}-a_{10}^{2}a_{11}b_{21}+a_{10}^{3}b_{22}.$

Generically, $\Lambda_{9}\neq 0$ and the singularity is of type $A_{3}^{-}$.

(2) We take $\mbox{\boldmath$x$}_{t}:U\to\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$
a parameterization of $M_{t}$ given by Theorem 2.1 which depends smoothly on
$t\in(\mbox{\boldmath$\mathbb{R}$},0)$ and
$\mbox{\boldmath$x$}_{0}=\mbox{\boldmath$x$}$. Denote by $E(u,v,t)$,
$F(u,v,t)$, $G(u,v,t)$, $\bar{l}(u,v,t)$, $\bar{m}(u,v,t)$ and
$\bar{n}(u,v,t)$ the coefficients of the first and second fundamental forms of
$M_{t}$ in $(u,v)\in U$ with respect to $\mbox{\boldmath$x$}_{t}$. Thus,
$E(u,v,t)=G(u,v,t)=0$ on $U$ and
$\tilde{\delta}_{t}(u,v)=\bar{l}(u,v,t)\bar{n}(u,v,t)$.

The deformation of the $A_{3}^{-}$ singularity of the $LPL$ (see §6) induced
by the family $M_{t}$ satisfy

$\tilde{\delta}_{t}(u,v)\sim_{\mathcal{R}}\pm(x^{2}-\Lambda_{9}^{2}y^{4}+\gamma_{1}(t)y^{2}+\gamma_{2}(t)y+\gamma_{3}(t)),$

where $\gamma_{1}$, $\gamma_{2}$ and $\gamma_{3}$ are germs of differentiable
functions. We get a germ of a differentiable curve
$\gamma(t)=(\gamma_{1}(t),\gamma_{2}(t),\gamma_{3}(t))$ in
$\mbox{\boldmath$\mathbb{R}$}^{3}$ with
$\gamma^{\prime}(0)=\left(\frac{\Lambda_{9}}{a_{10}b_{10}},0,0\right)$.

A point $q=\mbox{\boldmath$x$}_{t}(u,v)\in M_{t}$ is a singularity of the
$LPL$ if and only if $q$ is an umbilic point, that is,
$\sigma_{t}(u,v)=(0,0)$, where
$\sigma_{t}(u,v)=\left(\bar{l}(u,v,t),\bar{n}(u,v,t)\right)$. We have

$\sigma_{t}(u,v)\sim_{\mathcal{A}}\left(u,\frac{a_{10}^{2}(a_{10}\bar{n}_{t}(0,0,0)-b_{10}\bar{l}_{t}(0,0,0))t}{\Lambda_{9}}+v^{2}\right),$

provided that
$a_{10}^{2}(a_{10}\bar{n}_{t}(0,0,0)-b_{10}\bar{l}_{t}(0,0,0))\neq 0$.
Therefore, $\sigma_{t}$ can have either $0$ or $2$ zeros when $t\neq 0$,
depending on the sign of
$A=\Lambda_{9}(a_{10}\bar{n}_{t}(0,0,0)-b_{10}\bar{l}_{t}(0,0,0))t$.

When $A<0$, the only possibility is that $\gamma$ belongs to the stratum
$(IV)$ of Figure 10 left figure, because there are two singularities. On the
other hand, when $A>0$, $\gamma$ must be in the $(IX)$ or $(X)$ stratum, since
$\gamma$ is regular. Therefore, the deformation of the $LPL$ with singularity
$A_{3}^{-}$ is as shown in Figure 5. $\Box$

###  4.4 Bifurcations at a singularity of the PC and MCNC

Regular points on the $PC$ and $MCNC$ have codimension 0. Therefore, in this
subsection, we will study the singularities of these curves at points of
codimension 1.

Consider $MCNC$ in Lorentzian region, the Riemannian case is analogous. Let

$\mathcal{G}_{4}=\\{(M,p)\in\mathcal{G}:p{\rm\,\,is\,\,a\,\,singularity\,\,of\,\,the\,\,}MCNC{\rm\,\,in\,\,Lorentzian\,\,region}\\}.$

###### Theorem 4.6

For a generic $(M,p)\in\mathcal{G}_{4}$, we have

(1)

The $MCNC$ has a $A_{1}$-singularity at $p$;

(2)

In generic 1-parameter family of surfaces, the $A_{1}$-singularity of the
$MCNC$ at $p$ is $\mathcal{R}$-versally deformed.

  * Proof

(1) Given $(M,p)\in\mathcal{G}_{4}$, take
$\mbox{\boldmath$x$}:U\subset\mbox{\boldmath$\mathbb{R}$}^{2}\to\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$
the local parameterization of $M$ given as in (4) by
$\mbox{\boldmath$x$}(x,z)=(x,f(x,z),z)$ with $p=\mbox{\boldmath$x$}(0,0)$.
Since $p$ is a singularity of $MCNC$, we have $a_{22}=a_{20}$,
$a_{32}=3a_{30}$ and $a_{31}=3a_{33}$. This is a Morse singularity when
$\Lambda_{10}\neq$ with

$\begin{array}[]{ccl}\Lambda_{10}&=&-16\left(4a_{20}^{3}-a_{20}a_{21}^{2}-6a_{40}+a_{42}\right)\left(4a_{20}^{3}-a_{20}a_{21}^{2}-a_{42}+6a_{44}\right)\\\
&&+\left(8a_{20}^{2}a_{21}-2a_{21}^{3}-6a_{41}+6a_{43}\right)^{2}.\end{array}$

(2) Let $M_{t}\subset\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ be a 1-parameter
family of surfaces parameterized by $\mbox{\boldmath$x$}_{t}:U\to M_{t}$, with
$\mbox{\boldmath$x$}_{t}(x,y)=(x,h(x,z,t),z)$ for some $h$ differentiable with
$h(x,z,0)=f(x,z)$ and $f$ as in item (1). The $MCNC_{t}$ is formed by the
points $\mbox{\boldmath$x$}_{t}(x,y)$ where $\bar{H}_{t}(x,y)=0$. Suppose
$\Lambda_{11}=4a_{20}^{3}-a_{20}a_{21}^{2}-6a_{40}+a_{42}\neq 0$. We have
$j^{2}\tilde{H}_{t}\sim_{\mathcal{R}^{(2)}}\rho(t)+x^{2}+\frac{\Lambda_{10}}{8\Lambda_{11}}y^{2}$
and $j^{1}\rho=(h_{xx}(0,0,0)-h_{yy}(0,0,0))t$. Therefore, the $MCNC_{t}$ is a
$\mathcal{R}-$versal deformation of the $MCNC$ of $M$. $\Box$

The $PC$ of a surface is independent of the metric because it can be defined
using the surface contact with the tangent plane [12]. Therefore, the results
obtained in the Lorentzian region can be extended to the Riemannian region.
Let

$\mathcal{G}_{5}=\\{(M,p)\in\mathcal{G}:p{\rm\,\,is\,\,a\,\,singularity\,\,of\,\,the\,\,}PC{\rm\,\,in\,\,Lorentzian\,\,region}\\}.$

###### Theorem 4.7

For a generic $(M,p)\in\mathcal{G}_{5}$, we have

(1)

The $PC$ has a $A_{1}$-singularity at $p$;

(2)

In generic 1-parameter family of surfaces, the $A_{1}$-singularity of the $PC$
at $p$ is $\mathcal{R}$-versally deformed.

  * Proof

Similar to the proof of the Theorem 4.6. $\Box$

##  5 Bifurcations at points on the LD

###  5.1 Bifurcations at $\textbf{LD}\cap\textbf{LPL}\cap\textbf{MCNC}$

In this section, we study the intersections between the $LD$ and the $LPL$ in
the case where these curves are regular (singular case is considered in §5.3).
It is show next that $LD\cap LPL$ coincide with $LD\cap MCNC$. Therefore,
every point in $LD\cap LPL$ is also in $MCNC$.

###### Theorem 5.1

Let $M\subset\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ be a smooth surface and
$p\in LD$. Then $p\in LPL$ if and only if $p\in MCNC$. Furthemore, if $p$ is a
regular point on the $LD$, then $m(LD,LPL:p)=2m(LD,MCNC:p)$. Consequently,
$m(LD,LPL:p)$ is always even.

  * Proof

We take $\mbox{\boldmath$x$}:U\to\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ a local
parameterization of $M$ as in Theorem 2.1 with $p=\mbox{\boldmath$x$}(0,0)$.
Thus, $\bar{H}(0,0)=\bar{l}(0,0)G(0,0)$ and
$\tilde{\delta}(0,0)=l(0,0)^{2}G(0,0)^{2}$. Therefore, $\bar{H}(0,0)=0$ if and
only if $\tilde{\delta}(0,0)=0$.

Let
$\gamma:(\mbox{\boldmath$\mathbb{R}$},0)\to(\mbox{\boldmath$\mathbb{R}$}^{2},0)$
be a parametrization of the $LD$ at a regular point $p$. Since $E=F=0$ along
the $LD$, it follows that $E\circ\gamma$ and $F\circ\gamma$ are identically
zero. Therefore,

$\bar{H}(\gamma(x))=(\bar{l}G-2\bar{m}F+\bar{n}E)(\gamma(x))=\bar{l}(\gamma(x))G(\gamma(x)),$
$\tilde{\delta}(\gamma(x))=((\bar{l}G-\bar{n}E)^{2}-4(\bar{m}G-\bar{n}F)(\bar{l}F-\bar{m}E))(\gamma(x))=\left(\bar{l}(\gamma(x))G(\gamma(x))\right)^{2}$

and the result follows. $\Box$

Points where the $LD$ and the $MCNC$ are transversal are stable. Let

$\mathcal{G}_{6}=\\{(M,p)\in\mathcal{G}:m(LD,LPL:p)>2\\}.$

Therefore, the $LD$ and the $MCNC$ of $M$ are tangent at $p$ for every
$(M,p)\in\mathcal{G}_{6}$. Given $(M,p)\in\mathcal{G}_{6}$, take
$\mbox{\boldmath$x$}:U\to\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ the local
parameterization of $M$ as in (5) with $p=\mbox{\boldmath$x$}(0,0)$. As $p\in
LPL$ and is not an umbilic point, $a_{20}=0$ and $a_{21}\neq 0$. Applying a
homothety, consider $a_{21}=1$. As $m(LD,LPL:p)>2$, it follows that
$a_{30}=-\frac{1}{3}$.

###### Theorem 5.2

Let $(M,p)\in\mathcal{G}_{6}$ is generic. Then

(1)

$m(LD,LPL:p)=4$ and $m(LD,LPL:MCNC)=2$. There are two possible configurations
for the three curves as shows in Figure 6 middle figures.

(2)

For a generic 1-parameter family of surfaces $M_{t}$, with $M_{0}=M$, the
configurations of the three curves deform as in Figure 6.

Figure 6: Deformations of the $LD$ (black), the $LPL$ (red) and the $MCNC$
(green) in a generic 1-parameter family of surfaces at points $p$ where
$m(LD,LPL)=4$.

  * Proof

(1) Similar to the proof of Theorem 4.3 (1).

(2) Let $M_{t}\subset\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ be a family of
surfaces parameterized by $\mbox{\boldmath$x$}_{t}:U\to M_{t}$, with
$\mbox{\boldmath$x$}_{t}(x,z)=(x,y,h(x,y,t))$ for some $h$ differentiable with
$h(x,y,0)=f(x,y)$. Thus, the $LD_{t}$ is given by $\delta_{t}(x,y)=0$, with
$\delta_{t}(x,y)=h_{x}(x,y,t)^{2}+h_{y}(x,y,t)^{2}-1$. For $t=0$, we have that

$\frac{\partial\delta_{0}}{\partial y}(0,0)=2\neq 0.$

So, for $t$ small enough, the $LD_{t}$ is a regular curve and there is
$y_{t}:(\mbox{\boldmath$\mathbb{R}$},0)\to(\mbox{\boldmath$\mathbb{R}$},0)$
such that $\delta_{t}(x,y_{t}(x))=0$, for every $x$ close to $0$. If
$g_{t}(x)=\bar{H}_{t}(x,y_{t}(x))$, then

$g_{t}\sim_{\mathcal{R}}\Lambda_{12}x^{2}+\left(2h_{yt}(0,0,0)-2(2a_{22}+a_{31})h_{xt}(0,0,0)+h_{xxt}(0,0,0)\right)t+\rho(t),$

with $j^{1}\rho=0$ and $\Lambda_{12}=4a_{22}+7a_{31}+12a_{40}$. Therefore, the
$MCNC_{t}$ and the $LD_{t}$ intersect at 0 or 2 points, depending on the sign
of $t$. As $\bar{K}(0,0)\neq 0$, since $p$ does not belong to the $PC$, the
$LPL_{t}$ does not intersect with the $PC_{t}$ for small values of $t$, that
is, the only intersections of the $LPL_{t}$ and the $MCNC_{t}$ occur along the
$LD_{t}$. $\Box$

###  5.2 Bifurcations at $\textbf{LD}\cap\textbf{PC}$

The intersections between the $LD$ and the $PC$ are the simplest cases dealt
with in this papper. The transversal intersections between these curves have
codimension 0. We consider here the case where the $LD$ and the $PC$ are
tangent. So, let

$\mathcal{G}_{7}=\\{(M,p)\in\mathcal{G}:m(LD,PC:p)>1\\}.$

###### Theorem 5.3

Let $(M,p)\in\mathcal{G}_{7}$ be generic. Then

(1)

$m(LD,PC:p)=2$.

(2)

For a generic 1-parameter family of surfaces $M_{t}$, with $M_{0}=M$, the
configurations of the two curves deform as in Figure 7.

Figure 7: Deformations of the $LD$ (black) and the $PC$ (blue) in a generic
family of surfaces.

  * Proof

(1) Similar to the proof of Theorem 4.3 (1). (2) Similar to the proof of the
Theorem 5.2 (2). $\Box$

###  5.3 Bifurcations at a lightlike umbilic point

Lightlike umbilic points are singularities of the $LD$ [18]. We prove below
that the $LPL$ (extended to the $LD$ as the discriminant of equation (3))
passes through lightlike umbilic points and is also singular at that point.

###### Proposition 5.4

Let $M$ be a smooth surface in $\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ and
$p\in LD$.

(1)

$p$ is a lightlike umbilic point if and only if the $LPL$ is singular at $p$

(2)

If $p$ is a lightlike umbilic point, then $p$ belongs to the $PC$.

(3)

If $p$ belongs to the $LPL$ and the $PC$, then $p$ is an umbilic point.

(4)

$p$ is a lightlike umbilic point if and only if $p$ belongs to the $PC$ and
the $MCNC$.

  * Proof

Consider $\mbox{\boldmath$x$}:U\to\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ the
local parameterization of $M$ given by the Theorem 2.1 with
$p=\mbox{\boldmath$x$}(q)$. Thus, $E(q)=F(q)=0$, $E_{u}(q)=\lambda\bar{l}(q)$
and $E_{v}(q)=\lambda\bar{m}(q)$, for some $\lambda\neq 0$. The curvature
lines BDE is $\bar{m}(q)dv^{2}+\bar{l}(q)dvdu=0$.

(1) It follows that

$\begin{array}[]{lcl}\tilde{\delta}(q)&=&G(q)\bar{l}(q),\vspace{0.2cm}\\\
\tilde{\delta}_{u}(q)&=&2G(q)\bar{l}(q)\left(-2F_{u}(q)\bar{m}(q)+G_{u}(q)\bar{l}(q)+G(q)\bar{l}_{u}(q)-\lambda\bar{l}(q)\bar{n}(q)+2\lambda\bar{m}(q)^{2}\right),\vspace{0.2cm}\\\
\tilde{\delta}_{v}(q)&=&2G(q)\left(\bar{l}(q)\left(G(q)\bar{l}_{v}(q)-\bar{m}(q)(2F_{v}(q)+\lambda\bar{n}(q))+G_{v}(q)\bar{l}(q)\right)+2\lambda\bar{m}(q)^{3}\right).\end{array}$

Since $G(q)\neq 0$, $p$ is a singular point of the $LPL$ if and only if
$\bar{l}(q)=\bar{m}(q)=0$.

(2) If $p$ is a umbilic point, then $\bar{l}(q)=\bar{m}(q)=0$. Therefore,
$\bar{K}(q)=0$ and $p$ belongs to the $PC$.

(3) If $p\in LPL$, then $\bar{l}(q)=0$. Thus, $\bar{K}(q)=\bar{m}(q)^{2}$. As
$p$ belongs to the $PC$, it follows that $\bar{K}(q)=0$. Therefore,
$\bar{m}(q)=0$ and $p$ is a umbilic point.

(4) Follows from items 2, 3 and the Theorem 4.1. $\Box$

Let

$\mathcal{G}_{8}=\\{(M,p)\in\mathcal{G}:p{\rm\,\,is\,\,a\,\,lightlike\,\,umbilic\,\,point}\\}.$

For a $(M,p)\in\mathcal{G}_{8}$, we use the local parameterization
$\mbox{\boldmath$x$}:U\to\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ of $M$ given as
in (5) with $p=\mbox{\boldmath$x$}(0,0)$ and $j^{2}f=x+a_{22}y^{2}$.

###### Theorem 5.5

Let $(M,p)\in\mathcal{G}_{8}$ is generic. Then

(1)

The $LD$ and the $LPL$ have $A_{1}^{\pm}$ and $A_{3}^{\pm}$ singularities at
$p$, respectively, and the $PC$ and the $MCNC$ are regular curves. There are
six generic configurations as shown in Figure 8 middle figures.

(2)

For a generic 1-parameter family of surfaces $M_{t}$ , with $M_{0}=M$, the
configurations of the $LD$, $LPL$, $PC$ and $MCNC$ deform as in Figure 8

Figure 8: Deformation of the curves $LD$ (black), $LPL$ (red), $PC$ (blue) and
$MCNC$ (green) in generic families of surfaces.

  * Proof

(1) Let $\delta=F^{2}-EG$, then
$j^{2}\delta=6a_{30}x^{2}+4a_{31}xy+(4a_{22}^{2}+2a_{32})y^{2}$ and the $LD$
is given by $\delta=0$. Thus, the $LD$ has a Morse singularity at $p$ if and
only if $\Lambda_{13}=6a_{22}^{2}a_{30}+3a_{30}a_{32}-a_{31}^{2}\neq 0$.

On the other hand, if $a_{30}\neq 0$, we have
$j^{4}\tilde{\delta}\sim_{\mathcal{R}^{(4)}}x^{2}-\frac{32\Lambda_{13}^{3}}{27a_{30}^{3}}y^{4}$.
Therefore, the $LPL$ has a singularity $A_{3}^{\pm}$ at $p$. For the $PC$ and
the $MCNC$, it suffices to note that $j^{1}\bar{K}=4a_{22}(3a_{30}x+a_{31}y)$
and $j^{1}\bar{H}=2(3a_{30}x+a_{31}y)$. Suppose $\Lambda_{13}\neq 0$,
$a_{30}\neq 0$ and $a_{22}\neq 0$. Note that the $LPL$, the $PC$ and the
$MCNC$ are tangent at $p$ and the tangent line is $3a_{30}x+a_{31}y=0$.

If the $LPL$ has a $A_{3}^{+}$ singularity, then we have two possibilities,
one for each singularity of the $LD$ ($A_{1}^{+}$ or $A_{1}^{-}$).

When the $LPL$ has a $A_{3}^{-}$ singularity, after coordinate changes it
follows that

$\tilde{\delta}\sim_{\mathcal{R}}x^{2}-\frac{32\Lambda_{13}^{3}}{27a_{30}^{3}}y^{4}=\left(x-\sqrt{\frac{32\Lambda_{13}^{3}}{27a_{30}^{3}}}y^{2}\right)\left(x+\sqrt{\frac{32\Lambda_{13}^{3}}{27a_{30}^{3}}}y^{2}\right).$
(9)

Therefore, the $LPL$ is locally formed by the graphs of the functions
$x=\pm\sqrt{\frac{32\Lambda_{13}^{3}}{27a_{30}^{3}}}y^{2}$. Applying the same
coordinate changes to $\bar{K}$ and $\bar{H}$, then

$\begin{array}[]{l}j^{2}\bar{K}=-2a_{22}x+\frac{\Lambda_{13}}{9a_{30}^{2}}x^{2}-\frac{2\left(-12a_{22}^{2}a_{30}a_{31}+2a_{31}^{3}-9a_{32}a_{30}a_{31}+27a_{33}a_{30}^{2}\right)}{9a_{30}^{2}}xy+\frac{4\Lambda_{13}\left(6a_{22}^{2}a_{30}+\Lambda_{13}\right)}{9a_{30}^{2}}y^{2},\vspace{0.2
cm}\\\
j^{2}\bar{H}=x-\frac{2a_{22}}{3a_{30}}x^{2}-\frac{8a_{22}\Lambda_{13}}{3a_{30}}y^{2}.\end{array}$

From the Implicit Function Theorem, there are differentiable functions
$x_{K},x_{H}:(\mbox{\boldmath$\mathbb{R}$},0)\to(\mbox{\boldmath$\mathbb{R}$},0)$
such that $\bar{K}(x_{K}(y),y)=0$ and $\bar{H}(x_{H}(y),y)=0$, for all $y$
close to $0$, with
$x_{K}^{\prime\prime}(0)=\frac{4\Lambda_{13}\left(6a_{22}^{2}a_{30}+\Lambda_{13}\right)}{9a_{22}a_{30}^{2}}$
and $x_{H}^{\prime\prime}(0)=\frac{16a_{22}\Lambda_{13}}{3a_{30}}$.

As the $LPL$, the $PC$ and the $MCNC$ are graphs of functions with 1-jet null,
the position of these curves depends on the coefficient of $y^{2}$. First, we
have that

$\left|\frac{x_{K}^{\prime\prime}(0)}{2}\right|>\sqrt{\frac{32\Lambda_{13}^{3}}{27a_{30}^{3}}}\quad\Leftrightarrow\quad
3a_{30}a_{32}-a_{31}^{2}\neq 0,$

and generically the $PC$ is never between the two branches of the $LPL$.

Furthemore, note that

$\left|\frac{x_{H}^{\prime\prime}(0)}{2}\right|>\sqrt{\frac{32\Lambda_{13}^{3}}{27a_{30}^{3}}}\quad\Leftrightarrow\quad\frac{3a_{30}a_{32}-a_{31}^{2}}{a_{30}}<0.$

Thus, the position between the $LPL$ and the $MCNC$ depends on the sign of
$\frac{3a_{30}a_{32}-a_{31}^{2}}{a_{30}}$. The $MCNC$ is between the branches
of the $LPL$ if and only if $\frac{3a_{30}a_{32}-a_{31}^{2}}{a_{30}}>0$, the
result follows in this case.

Suppose $\frac{3a_{30}a_{32}-a_{31}^{2}}{a_{30}}<0$. Since
$x_{K}^{\prime\prime}(0)$ and $x_{H}^{\prime\prime}(0)$ have the same sign,
the $PC$ and the $MCNC$ are in the same region bounded by the $LPL$. The
$MCNC$ is contained between the $PC$ and the $LPL$ when
$|x_{K}^{\prime\prime}(0)|>|x_{H}^{\prime\prime}(0)|$, being that

$\left|x_{K}^{\prime\prime}(0)\right|>\left|x_{H}^{\prime\prime}(0)\right|\quad\Leftrightarrow\quad(3a_{30}a_{32}-a_{31}^{2})(18a_{22}^{2}a_{30}+\Lambda_{13})<0.$
(10)

Since $\Lambda_{13}$ and $a_{30}$ have the same sign, because $LPL$ has a
singularity $A_{3}^{-}$, so does $18a_{22}^{2}a_{30}+\Lambda_{13}$. However,
$a_{30}$ and $3a_{30}a_{32}-a_{31}^{2}$ have opposite signs, so the inequality
(10) is always true.

For a generic surface $(M,p)\in\mathcal{G}_{8}$, all the conditions above are
satisfied. Therefore, the configurations of the curves as in Figure 8 middle
figures.

(2) Let $M_{t}\subset\mbox{\boldmath$\mathbb{R}$}^{3}_{1}$ be a 1-parameter
family of surfaces parameterized by $\mbox{\boldmath$x$}_{t}:U\to M_{t}$, with
$\mbox{\boldmath$x$}_{t}(x,y)=(x,y,h(x,y,t))$ for some $h$ differentiable with
$h(x,y,0)=f(x,y)$. The $LD_{t}$ is formed by the points
$\mbox{\boldmath$x$}_{t}(x,y)$ where $\delta_{t}(x,y)=0$. We have
$j^{2}\delta_{t}\sim_{\mathcal{R}^{(2)}}\rho(t)+a_{30}x^{2}+\frac{\Lambda_{13}}{a_{30}}y^{2}$
and $j^{1}\rho=-2h_{xt}(0,0,0)t$. Therefore, the $LD_{t}$ is a
$\mathcal{R}-$versal deformation of the $LD$ of $M$ if and only if
$h_{xt}(0,0,0)\neq 0$.

The deformation of the $LPL$ induced by $M_{t}$ is equivalent to

$x^{2}\pm(y^{4}+w_{1}(t)y^{2}+w_{2}(t)y+w_{3}(t)).$ (11)

Define the differentiable curve
$\gamma:(\mbox{\boldmath$\mathbb{R}$},0)\to\mbox{\boldmath$\mathbb{R}$}^{3}$,
given by $\gamma(t)=(w_{1}(t),w_{2}(t),w_{3}(t))$, with $\gamma(0)$ being the
origin.

We will consider only the linear part of $\gamma$ to calculate tangent vector
to $\gamma$. Therefore, take $h(x,y,t)=f(x,y)+t\tilde{h}(x,y)$ for some smooth
function $\tilde{h}$. The $LPL$ of the surfaces $M_{t}$ is given by
$\tilde{\delta}_{t}=0$, with

$j^{4}\tilde{\delta}_{t}\sim_{\mathcal{R}}x^{2}\pm
y^{4}-h_{xt}(0,0,0)\sqrt{\frac{32|\Lambda_{13}|}{3|a_{30}|}}ty^{2}.$ (12)

Thus, the tangent vector to the $\gamma$ at $t=0$ is $\left(\pm
h_{xt}(0,0,0)\sqrt{\frac{32|\Lambda_{13}|}{3|a_{30}|}},0,0\right)$.

Since the $LD$ has a singularity $A_{1}$, it follows that $p$ is a stable
umbilic point [9]. Therefore, $\tilde{\delta}_{t}$ has a singularity for every
$t$, because umbilic points are singularities of the $LPL$, and the curve
$\gamma$ is contained in the swallowtail.

As $\gamma$ is regular, it follows from the Theorem 6.2 that one part of
$\gamma$ lies in the stratum $II$ and the other part in the stratum $VI$, see
Figure 10 left figure. Therefore, the $A_{3}^{\pm}$-singularity of the $LPL$
deforms as in Figure 9.

(a) $A_{3}^{+}$ Singularity.

(b) $A_{3}^{-}$ Singularity.

Figure 9: Deformation of the $LPL$ in a generic family of surfaces at a
lightlike umbilic point.

Finally, it remains to obtain the intersections between these curves along the
$M_{t}$ family. Let $\theta_{\mbox{\boldmath$x$}_{t}}:U\to J^{3}(2,1)$ be the
Monge-Taylor map with respect to $\mbox{\boldmath$x$}_{t}$. Put

$\theta_{\mbox{\boldmath$x$}_{t}}(x,y)=\left(\theta_{00}(x,y,t),\theta_{10}(x,y,t),\theta_{11}(x,y,t),\dots,\theta_{32}(x,y,t),\theta_{33}(x,y,t)\right),$

where
$\theta_{ij}:U\times(\mbox{\boldmath$\mathbb{R}$},0)\to\mbox{\boldmath$\mathbb{R}$}$
is the coefficient of $x^{i-j}y^{j}$. A point $\mbox{\boldmath$x$}_{t}(x,y)$
belongs to the intersection of the $LD$ with the $MCNC$ if and only if
$\sigma_{t}(x,y)=(0,0)$, where

$\sigma_{t}(x,y)=(\theta_{10}^{2}+\theta_{11}^{2}-1,\theta_{10}^{2}\theta_{20}+\theta_{10}\theta_{11}\theta_{21}+\theta_{11}^{2}\theta_{22})$

with $\theta_{ij}$ evaluated in $(x,y,t)$. Note that

$\sigma_{t}\sim_{\mathcal{A}}\left(\frac{\Lambda_{13}}{a_{30}}y^{2}+2h_{xt}(0,0,0)t+\rho(t),x\right),$

with $j^{1}\rho=0$. Therefore, the $LD_{t}$ and the $MCNC_{t}$ intersect at 2
or 0 points when $\Lambda_{14}<0$ or $\Lambda_{14}>0$, respectively, with
$\Lambda_{14}=a_{30}h_{xt}(0,0,0)\Lambda_{13}t$. The $LD_{t}$ and the
$MCNC_{t}$ are tangent at $\mbox{\boldmath$x$}_{t}(x,y)$ when $T(x,y,t)=0$,
where

$\begin{array}[]{ccl}T&=&2\theta_{10}\theta_{11}^{2}\theta_{20}\theta_{31}-6\theta_{10}\theta_{11}^{2}\theta_{20}\theta_{33}-3\theta_{10}\theta_{11}^{2}\theta_{21}\theta_{30}-\theta_{10}\theta_{11}^{2}\theta_{21}\theta_{32}+6\theta_{11}\theta_{22}\theta_{30}\\\
&&-4\theta_{20}\theta_{21}\theta_{22}+8\theta_{10}\theta_{11}\theta_{20}^{2}\theta_{22}-2\theta_{10}\theta_{11}\theta_{20}\theta_{21}^{2}-8\theta_{10}\theta_{11}\theta_{20}\theta_{22}^{2}-2\theta_{10}\theta_{20}\theta_{31}\\\
&&+2\theta_{10}\theta_{11}\theta_{21}^{2}\theta_{22}+4\theta_{11}^{3}\theta_{20}\theta_{32}-2\theta_{11}^{2}\theta_{21}^{3}-\theta_{11}^{3}\theta_{21}\theta_{31}-3\theta_{11}^{3}\theta_{21}\theta_{33}-6\theta_{11}^{3}\theta_{22}\theta_{30}\\\
&&+2\theta_{11}^{3}\theta_{22}\theta_{32}+8\theta_{11}^{2}\theta_{20}\theta_{21}\theta_{22}+\theta_{21}^{3}-4\theta_{11}\theta_{20}\theta_{32}+\theta_{11}\theta_{21}\theta_{31}+4\theta_{10}\theta_{11}^{2}\theta_{22}\theta_{31}\\\
&&+3\theta_{10}\theta_{21}\theta_{30}.\end{array}$

If $\eta_{t}(x,y)=(\sigma_{t}(x,y),T(x,y,t))$, then
$\eta_{t}\sim_{\mathcal{A}}(2h_{xt}(0,0,0)t+\rho(t),x,y)$, with $j^{1}\rho=0$.
Therefore, the $LD_{t}$ and the $MCNC_{t}$ are transversal, because
$\eta_{t}(x,y)\neq(0,0,0)$ for $t\neq 0$.

Similarly, we obtain the intersections between the $LD$ and the $PC$ and
between the $MCNC$ and the $PC$. For the intersections between the $LD$ and
the $LPL$, use the Theorem 5.1. As $LD_{t}$ is regular for $t\neq 0$, the
intersections between the $LPL_{t}$ and the $PC$ occur outside the $LD_{t}$
and the result follows from the Theorem 4.1.

Given a point $q$ in $LPL_{t}$, it follows from the Theorems 4.1 and 5.1 that
$q$ belongs to the $MCNC_{t}$ if and only if $q$ belongs to the $PC$ or the
$LD_{t}$. Hence, the number of intersection points is $0$, $2$ or $4$. As the
$LPL_{t}$ is tangent to the $LD_{t}$ and the $PC$, but the $MCNC_{t}$ is
transversal to the $LD_{t}$ and the $PC$, it follows by transitivity that the
$MCNC_{t}$ is transversal to the $LPL_{t}$ at the intersection points.

The result follows from the combination between the deformations of the curves
alone and the possible intersections. $\Box$

##  6 Appendix

We consider some regular curves on the discriminant of an $A_{3}$-singularity.
The discriminant of the $\mathcal{R}-$versal unfolding
$F(x,y,u,v,w)=x^{2}\pm(y^{4}+uy^{2}+vy+w)$ of the $A_{3}^{\pm}$-singularity
$f(x,y)=x^{2}\pm y^{4}$ is given by

$D_{F}=\left\\{(u,v,w)\in(\mbox{\boldmath$\mathbb{R}$}^{3},0):\exists
y\in(\mbox{\boldmath$\mathbb{R}$},0):y^{4}+uy^{2}+vy+w=0,\,4y^{3}+2uy+v=0\right\\}.$

It is a swallowtail and is parameterized by
$\varphi(u,y)=(u,-4y^{3}-2uy,3y^{4}+uy^{2})$.

Figure 10 left presents a Whitney stratification of the swallowtail. The
configuration of the curve $x^{2}\pm(y^{4}+uy^{2}+vy+w)=0$ depends only on the
stratum in which the parameters $(u,v,w)$ are taken. Figure 10 presents these
configurations.

Figure 10: Whitney stratification of the swallowtail and deformations of
$A_{3}$ singularity ($A_{3}^{+}$ middle, $A_{3}^{-}$ right) according to the
stratum where the parameters (u,v,w) are taken.

Let $\eta:(-\epsilon,\epsilon)\to\mbox{\boldmath$\mathbb{R}$}^{3}$ be a smooth
and regular curve on the swallowtail with $\eta(0)=(0,0,0)$. Then
$\eta^{\prime}(0)=(1,0,0)$ and we can write $\eta(t)=(t,v(t),w(t))$, where
$v(t)=-4y(t)^{3}-2ty(t)$ and $w(t)=3y(t)^{4}+ty(t)^{2}$, for some function
$y:(-\epsilon,\epsilon)\to\mbox{\boldmath$\mathbb{R}$}$.

###### Lemma 6.1

The function $y(t)$ is smooth at $t=0$.

  * Proof

Since $\eta$ is smooth, so are $v$ and $w$. Also, $v(t)=t^{2}\tilde{v}(t)$ and
$w(t)=t^{2}\tilde{w}(t)$, with $\tilde{v}$ and $\tilde{w}$ smooth, because
$v(0)=w(0)=v^{\prime}(0)=w^{\prime}(0)=0$. Let
$P_{t}(y)=y^{4}+ty^{2}+v(t)y+w(t)$ and $Q_{t}(y)=4y^{3}+2ty+v(t)$. Since
$\eta$ is a curve on the swallowtail, the resultant

$r(t)=256w(t)^{3}-128t^{2}w(t)^{2}+16t^{4}w(t)+144tv(t)^{2}w(t)-27v(t)^{4}-4t^{3}v(t)^{2}$

of the polynomials $P_{t}$ and $Q_{t}$ with respect to $y$ is identically
zero. It follows that $j^{6}r=16\tilde{w}(0)(4\tilde{w}(0)-1)^{2}t^{6}\equiv
0$, so $\tilde{w}(0)=0$ or $\tilde{w}(0)=1/4$.

Let $Y(t)=y(t)^{2}$, then $t^{2}\tilde{w}(t)=3Y(t)^{2}+tY(t)$ and
$Y(t)=-t\left(\frac{1\pm\sqrt{1+12\tilde{w}(t)}}{6}\right)$. Since $Y(t)\geq
0$ for all $t\in(-\epsilon,\epsilon)$, it follows that $\tilde{w}(0)=0$ and
$Y(t)=-t\tilde{Y}(t)$, with
$\tilde{Y}(t)=\frac{1-\sqrt{1+12\tilde{w}(t)}}{6}$.

On the other hand, $t^{2}\tilde{v}(t)=-4Y(t)y(t)-2ty(t)$. Therefore,
$y(t)=\frac{t\tilde{v}(t)}{4\tilde{Y}(t)-2}$. Since $4\tilde{Y}(0)-2=-2\neq
0$, it follows that $y$ is smooth. $\Box$

###### Theorem 6.2

If $\eta$ is a smooth and regular curve on the swallowtail with
$\eta(0)=(0,0,0)$, then one part of $\eta$ lies in the stratum $(II)$ and the
other part in the stratum $(VI)$.

  * Proof

In Figure 10 left figure, the strata $(III)$ and $(IV)$ are curves
parameterized by $\gamma_{1}(t)=(-6t^{2},8t^{3},-3t^{4})$ and
$\gamma_{2}(t)=(-2t^{2},0,t^{4})$, respectively. The parabolas $(-6t^{2},t)$
and $(-2t^{2},t)$ in the $uy$ plane are the pre-images of $\gamma_{1}$ and
$\gamma_{2}$ by the parameterization $\varphi$ of the swallowtail, see Figure
11.

Figure 11: Partition of the $uy$ plane induced by the swallowtail
stratification.

Since $\eta(t)=(t,v(t),w(t))$ and from the Lemma 6.1, it follows that the pre-
image of $\eta$ by $\varphi$ is the smooth curve $(t,y(t))$ in the $uy$ plan.
Therefore, $\varphi^{-1}(\eta)$ is transversal to the parabolas in Figure 11,
that is, $\eta$ passes from the stratum $(II)$ to the stratum $(VI)$ when $t$
changes sign. $\Box$

  * Acknowledgments.

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

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