\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 118, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/118\hfil Geodesics of quadratic differentials]
{Geodesics of quadratic differentials on Klein surfaces}

\author[M. Ro\c{s}iu \hfil EJDE-2014/118\hfilneg]
{Monica Ro\c{s}iu}  % in alphabetical order

\address{Monica Ro\c{s}iu \newline
Department of Mathematics, University of Craiova, Craiova 200585,
Romania}
\email{monica\_rosiu@yahoo.com}

\thanks{Submitted February 2, 2014. Published April 22, 2014.}
\subjclass[2000]{30F30, 30F35, 30F50}
\keywords{Klein surface; meromorphic quadratic differential; geodesic}

\begin{abstract}
 The objective of this article is to establish the existence of a local
 Euclidean metric associated  with a quadratic differential on a Klein surface,
 and to describe the shortest curve in the neighborhood of a holomorphic point.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

In this paper we develop a technique, based on similar results for Riemann
surfaces, to determine the geodesics near holomorphic points of a quadratic
differential on a Klein surface. 
Recent results about this topic are due to
Andreian Cazacu \cite{a3},  Bolo\c{s}teanu \cite{b2},  B\^{a}rz\u{a}
\cite{b1} and  Ro\c{s}iu \cite{r1}. The complex double of a Klein
surface is an important tool in connection with the study of the geometric
structure near critical points of a quadratic differential on a Klein
surface. Effectively, we define a metric associated with a quadratic
differential on a Klein surface, which corresponds to a symmetric metric on
its double cover. We introduce special parameters, in terms of which the
quadratic differential has particularly simple representations and we give
explicit descriptions of the geodesics near the holomorphic points of a
quadratic differential.

Throughout this article, by a Klein surface we mean a Klein surface with an
empty boundary, which is not a Riemann surface.

\section{Preliminaries}

Suppose that $X$ is a compact Klein surface, obtained from a compact surface
by removing a finite number of points. We assume that $X$ has hyperbolic
type.
Our study is based on the following theorem (see \cite{b3}) due to Klein.

\begin{theorem} \label{thm1}
Given a Klein surface $X$, its canonical (Riemann) double cover $X_C$
admits a fixed point free symmetry $\sigma $, such that $X$ is
dianalytically equivalent with $X_C/\langle \sigma \rangle $,
where $\langle \sigma \rangle $ is the group generated by $\sigma$. 
Conversely, given a pair $(X_C,\sigma )$ consisting of a Riemann
surface $X$ and a symmetry $\sigma $ , the orbit space 
$X_C/\langle \sigma \rangle $ admits a unique structure of Klein surface,
such that $f:$ $X_C\to X_C/\langle \sigma \rangle $ is a
morphism of Klein surfaces, provided that one regards $X_C$ as a Klein
surface.
\end{theorem}

 Following the next theorem we associate a surface $NEC$ group with
a compact Klein surface $X$. Details can be found in \cite{r1}.

\begin{theorem} \label{thm2}
Let $X$ be a compact Klein surface of algebraic genus $g\geq 2$. Then there
exists  a surface $NEC$ group $\Gamma $, such that $X$ and $H/\Gamma $ are
isomorphic as Klein surfaces. Moreover, the complex double $(X_C,f,\sigma )
$ is isomorphic with $H/\Gamma ^{+}$, where $\Gamma ^{+}$ is the canonical
Fuchsian subgroup of $\Gamma $.
\end{theorem}

Then $\Gamma $ is the group of covering transformations of $X$ and 
$\Gamma ^{+}$ is the group of conformal covering transformations of $X$. 
If $\pi':H\to H/\Gamma \ $and $\pi :H\to H/\Gamma ^{+}$
are the canonical projections, then we note with $\widehat{z}$ the local
parameter near a point $\widehat{P}\in H$, $\widetilde{z}$ the local
parameter near $\widetilde{P}\in X_C$ and $z$ the local parameter near
$P=f(\widetilde{P})\in X$.

Given $A=\{ (U_i,\phi _i)\mid i\in I\} $ the dianalytic atlas
on $X$, we define $U_i'=U_i\times \{ i\} \times
\{ 1\} $ and $U_i''=U_i\times \{
i\} \times \{ -1\} $, $i\in I$. 
Let $U\subset U_i\cap U_{j}\neq \emptyset $ be a connected component 
of $U_i\cap U_{j}$. Then,
we identify $U\times \{ i\} \times $ $\{ \delta \} $
with $U\times \{ j\} \times $ $\{ \delta \} $, for 
$\delta =\pm 1$, if $\phi _{j}\circ \phi _i^{-1}:\phi _i(U)\to
\mathbb{C}$ is analytic and 
$U\times \{ i\} \times \{\delta \} $ with 
$U\times \{ j\} \times \{ -\delta\} $, for $\delta =\pm 1$, if 
$\phi _{j}\circ \phi _i^{-1}:\phi_i(U)\to \mathbb{C}$ is antianalytic. 
As in Alling, $X_C$ is the quotient space of
$X_0=\cup_{i\in I} U_i'\cup \cup_{i\in I} U_i''$, $(i,j)\in I\times I$, with
all the identifications above.

We consider $p:X_0\to X_C$ the canonical projection and
$\widetilde{U_i}=p(U_i'\cup U_i'')$, $i\in I$.

Using Schwarz reflection principle, we can associate an analytic structure
on $X_C$ by $\widetilde{A}=\{ ( \widetilde{U_i},\widetilde{
\phi _i}) : i\in I \} $, where
\[
\widetilde{\phi _i}:\widetilde{U_i}\to \mathbb{C}, \quad
\widetilde{\phi _i}( \widetilde{P})
=\begin{cases}
\phi _i(P), & \text{when }\widetilde{P}\in U_i' \\[4pt]
\overline{\phi _i(P)}, & \text{when }\widetilde{P}\in U_i''.
\end{cases}
\]

\begin{remark} \label{rmk3} \rm
$\widetilde{A}'=\{ ( \widetilde{U}_i,\overline{
\widetilde{\phi _i}}) : i\in I \} $is also an
analytic atlas on $X_C$.
\end{remark}

If $X_C$ $=(X_C,\widetilde{A})$ and
$\overline{X}_C=(X_C, \widetilde{A}')$, then
 $\overline{X}_C$ is $X_C$ endowed with
the second orientation.

Let $f:X_C\to X$, $f( \widetilde{P}) =\{\widetilde{P},\sigma (\widetilde{P})\} $
 be the covering projection, where 
$\sigma :X_C\to\overline{X}_C$ is an
antianalytic involution, without fixed points. If $\widetilde{z}$ is a
parameter near $\widetilde{P}\in X_C$, then $\overline{\widetilde{z}}$ is
a parameter near $\sigma (\widetilde{P})\in \overline{X}_C$.


Because two disjoint neighborhoods $U_i'$\ and $U_i''$ of $X_C$, 
lie over each neighborhood $U_i$ of $X$, we can
make the restriction at $U_i'\cup U_i''$
 in the local study of the quadratic differentials on $X_C$.

As in  Alling and  Greenleaf \cite{a2}, we associate with the
dianalytic atlas $A$ on $X$, the nonzero (holomorphic) quadratic
differential $\varphi =(\varphi _i)_{i\in I}$ on $X$, in the local
parameters $(z_i) _{i\in I}$, such that on each connected
component $U$ of $U_i\cap U_{j}$, the following transformation law
\[
\varphi _i(z_i)( dz_i) ^{2}
=\begin{cases}
\varphi _{j}(z_{j})( dz_{j}) ^{2}, & \text{when }\phi
_{j}\circ \phi _i^{-1}\text{ is analytic on }\phi _i(U) \\[4pt]
\overline{\varphi _{j}(z_{j})}( d\overline{z_{j}}) ^{2}, &
\text{when }\phi _{j}\circ \phi _i^{-1}\text{ is antianalytic on }\phi
_i(U)
\end{cases}
\]
holds whenever $z_i$ and $z_{j}$ are parameter values near the same point
of $X$.

The family $\widetilde{\varphi }=(\widetilde{\varphi _i})_{i\in I}$ of
holomorphic function elements, in the local parameters 
$( \widetilde{z_i}) _{i\in I}$,
\begin{equation}
\widetilde{\varphi _i}(\widetilde{z_i})
=\begin{cases}
\varphi _i(z_i), & \text{when  }\widetilde{P}\in U_i'\\[4pt]
\overline{\varphi _i(z_i)}, & \text{when }\widetilde{P}\in U_i'',
\end{cases} \label{e1}
\end{equation}
where $\widetilde{z_i}$ is a local parameter near $\widetilde{P}$, is a 
(holomorphic) quadratic differential on $X_C$, with respect to the analytic
atlas $\widetilde{A}$ on $X_C$. By analogy, the family 
$\overline{\widetilde{\varphi }}=( \overline{\widetilde{\varphi _i}})
_{i\in I}$ of holomorphic function elements, in the local parameters 
$(\overline{\widetilde{z_i}}) _{i\in I}$ is a (holomorphic) quadratic
differential on $\overline{X}_C$.

\section{The natural parameter near holomorphic points}

As a consequence of the above results, see \cite{r1}, the order of a
nonzero holomorphic quadratic differential $\varphi $ on\ $X$ at a point $P$
is a dianalytic invariant, thus it does make sense to define the zeroes of
the quadratic differential $\varphi $ on a Klein surface. A holomorphic
point is either a regular point or a zero.

In this section, we extend the notion of natural parameter near a
holomorphic point of a quadratic differential on a Riemann surface to a
Klein surface, which is studied in \cite{s2}.

Let $P$ be a holomorphic point of the quadratic differential $\varphi $ and 
$(U_i,z_i)$, $i\in I$ be a dianalytic chart at $P$. Then, we can take
$U_i$ to be sufficiently small so that a single valued branch of 
$\Phi (z_i)=\int \sqrt{\varphi _i(z_i)}dz_i$ can be chosen. We introduce a
local parameter
\[
w_i=\Phi (z_i)=\int \sqrt{\varphi _i(z_i)}dz_i
\]
in $U_i$, uniquely up to a transformation 
$w_i\to \pm w_i+{\rm const.}$, such that on each connected component $U$ 
of $U_i\cap U_{j}$, the following transformation law
\[
w_i=\begin{cases}
w_{j}, & \text{when }\phi _{j}\circ \phi _i^{-1}\text{ is analytic
on }\phi _i(U)  \\
\overline{w_{j}}, & \text{when }\phi _{j}\circ \phi _i^{-1}
\text{ is antianalytic on }\phi _i(U)
\end{cases}
\]
holds whenever $z_i$ and $z_{j}$ are parameter values near the same point 
$P$ of $X$.

By the definition of a quadratic differential on a Klein surface, we can see
that the local parameter $w_i$ is locally well defined, up to conjugation,
see \cite{a2}.

\begin{remark} \label{rmk4} \rm
The quadratic differential $\varphi $ on the Klein surface $(X,A)$ has, in
terms of the natural parameter $w_i$, the representation identically equal
to one.
\end{remark}

As for a Riemann surface, $w_i$ will be called the natural parameter near 
$P$.
Let $w_i$ be the natural parameter in a neighborhood $U_i$ of a
holomorphic point $P$ of the quadratic differential $\varphi $ on $X$. Then,
using the relation \eqref{e1} it follows that
\[
\widetilde{w_i}=\begin{cases}
w_i, & \text{when  }\widetilde{P}\in U_i' \\
\overline{w_i}, & \text{when }\widetilde{P}\in U_i'',
\end{cases}
\]
is the natural parameter in the corresponding neighborhood of the point
 $\widetilde{P}=p(P)$. The natural parameter on $X_C$ is well defined,
because the family $\widetilde{\varphi }$ is a quadratic differential on 
$X_C$, with respect to the analytic atlas $\widetilde{A}$ on $X_C$.

By analogy, $\overline{\widetilde{w_i}}$ is the natural parameter in the
corresponding neighborhood of the point $\sigma (\widetilde{P})$.

\begin{proposition} \label{prop5}
The quadratic differential $\widetilde{\varphi }$ on the Riemann surface 
$(X_C,\widetilde{A})$ has, in terms of $\widetilde{w_i}$, the
representation identically equal to one.
\end{proposition}

\begin{proof}
If $\widetilde{P}\in U_i'$, the differential $d\widetilde{w_i}$
becomes $dw_i=\sqrt{\varphi (z_i)}dz_i$, therefore by squaring 
$(dw_i) ^{2}=\varphi (z_i)( dz_i) ^{2}$ so, in terms of
$\widetilde{w_i}=w_i$, the quadratic differential $\widetilde{\varphi }$
has the representation identically equal to one. A similar argument applies
if $\widetilde{P}\in U_i''$. The differential 
$d\widetilde{w_i}$ becomes
 $( d\overline{w_i}) ^{2}=\overline{\varphi (z_i)}( d\overline{z_i}) ^{2}$ 
and in terms of $\widetilde{w_i}=\overline{w_i}$, the quadratic differential 
$\widetilde{\varphi }$ has the representation identically equal to one.
\end{proof}

\begin{remark} \label{rmk6} \rm
The quadratic differential $\overline{\widetilde{\varphi }}$ on the Riemann
surface $\overline{X}_C$ has, in terms of $\overline{\widetilde{w_i}}$,
the representation identically equal to one.
\end{remark}

In the special case, when $P$ is a zero of the quadratic differential 
$\varphi $, then as in Strebel \cite[Theorem 6.2]{s2}, we can introduce a
local parameter $\zeta _i$ in the neighborhood $U_i$ of $P$, 
$ P\leftrightarrow \zeta _i=0$, in terms of which the quadratic
differential has the representation
\begin{equation}
(dw_i)^{2}=\varphi (z_i)(dz_i)^{2}=\big( \frac{n+2}{2}\big)
^{2}\zeta_i^{n}d\zeta_i^{2}.  \label{e2}
\end{equation}
As a function of $\zeta_i $, the natural parameter becomes
 $w_i=\Phi (z_i)=(\zeta_i) ^{\frac{n+2}{2}}$and the corresponding natural parameter on 
$X_C$ is
\begin{equation}
\widetilde{w_i}=\begin{cases}
(\zeta_i) ^{\frac{n+2}{2}}, & \text{when  }\widetilde{P}\in U_i' \\[4pt]
(\overline{\zeta_i })^{\frac{n+2}{2}}, & \text{when }\widetilde{P}\in U_i''.
\end{cases} \label{e3}
\end{equation}

\subsection{The $\varphi$-length of a curve}

In this section, we introduce the $\varphi$-metric associated with a
quadratic differential on a Klein surface and we study its relation with the
corresponding $\widetilde{\varphi }$-metric on its canonical (Riemann)
double cover $X_C$, see \cite{s2}.

Let $w_i$ be the natural parameter in a neighborhood $U_i$ of a
holomorphic point $P$ of the quadratic differential $\varphi $ on $X$. We
define the length element of the $\varphi $-metric,
 $|dw_i|$, such that on each connected component $U$ of $U_i\cap U_{j}$, the
 transformation law
\[
|dw_i| =\begin{cases}
|dw_{j}| , & \text{when }\phi _{j}\circ \phi _i^{-1}
\text{ is analytic on }\phi _i(U)  \\
|d\overline{w_{j}}| , & \text{when }\phi _{j}\circ \phi
_i^{-1}\text{ is antianalytic on }\phi _i(U) 
\end{cases}
\]
holds whenever $z_i$ and $z_{j}$ are parameter values near the same point 
$P$ of $X$.

\begin{remark} \label{rmk} \rm
The length element of the $\varphi $-metric is a dianalytic invariant,
namely $|dw_i| =|d\overline{w_i}| =\sqrt{|\varphi (z_i)| }|dz_i| $.
\end{remark}

The length element of the $\widetilde{\varphi }$ -metric is defined by
\begin{equation}
|d\widetilde{w_i}| =\begin{cases}
|dw_i| , & \text{when }\widetilde{P}\in U_i' \\[4pt]
|d\overline{w_i}| , & \text{when }\widetilde{P}\in U_i''.
\end{cases}  \label{e4}
\end{equation}

Let $U_i$ be a neighborhood of a regular point $P$ of the quadratic
differential $\varphi $ on $X$ and $f^{-1}(U_i)=U_i'$\ $\cup $\
$U_i''$\ . Let $\gamma $ be a piecewise smooth curve in $
U_i$. The curve $\gamma $ has exactly two liftings at $f^{-1}(U_i)$ . If
$\gamma $ has the initial point $P$ and $\widetilde{\gamma }$ is the lifting
of $\gamma $ at $\widetilde{P}\in U_i'$, then $\sigma (\widetilde{
\gamma })$ is the lifting of $\gamma $ at $\widetilde{P}\in U_i''$.

\begin{remark} \label{rmk8} \rm
Because $\sigma :$ $X_C$ $\to $ $\overline{X}_C$ is an
antianalytic involution, without fixed points, we get that if $\widetilde{
\gamma }$ is the lifting of $\gamma $ at $\widetilde{P}\in U_i''$, then 
$\sigma (\widetilde{\gamma })$ is the lifting of $\gamma $
at $\widetilde{P}\in U_i'$.
\end{remark}

The curves $\widetilde{\gamma }$ and $\sigma (\widetilde{\gamma })$ are
called symmetric on $X_C$.
Next, we will identify $\widetilde{\gamma }$ and $\sigma (\widetilde{\gamma })$, 
with their images in the complex plane, from the corresponding charts.

The curve $\gamma $ is mapped by a branch of 
$\Phi (z_i)=\int \sqrt{\varphi _i(z_i)}dz_i$ onto\ a curve $\gamma _0$ 
in the $w$-plane.
The Euclidean length of $\gamma _0$ does not depend on the branch of $\Phi $
which we have chosen. The $\varphi $-length of $\gamma $ can be computed, in
terms of the natural parameter $w_i$, by
\[
l_{\varphi }(\gamma )=\int_{\gamma_0} |dw_i| =
\int_{\gamma} \sqrt{|\varphi _i(z_i)| }|
dz_i| ,
\]
then the $\varphi $-length of $\gamma $ is the Euclidean length of $\gamma
_0$.

Thus, the natural parameter is the local isometry between the $\varphi $-
metric and the Euclidean metric.

\begin{remark} \label{rmk9} \rm
By the definition of the quadratic differential $\varphi $ on $X$, we obtain
that the $\varphi $-length of $\gamma $ is a dianalytic invariant.
\end{remark}

From \eqref{e4}, we deduce that the $\widetilde{\varphi }$-length of 
$\widetilde{\gamma }$ is
\[
l_{\widetilde{\varphi }}(\widetilde{\gamma })
=\begin{cases}
l_{\varphi }(\gamma ), &\text{when } \widetilde{\gamma }\in U_i' \\[4pt]
l_{\overline{\varphi }}(\gamma ), &\text{when }  \widetilde{\gamma }\in U_i'',
\end{cases}
\]
where $l_{\overline{\varphi }}(\gamma )=\int_{\gamma } 
\sqrt{|\overline{\varphi _i(z_i)}| }|d\overline{z_i}| $.

\begin{proposition} \label{prop10}
The symmetric curves $\widetilde{\gamma }$ and $\sigma (\widetilde{\gamma })$
have the same $\widetilde{\varphi }$-length, namely 
$l_{\widetilde{\varphi }}(\widetilde{\gamma })
=l_{\widetilde{\varphi }}(\sigma (\widetilde{\gamma
}))$ $=l_{\varphi }(\gamma )$. Therefore, the $\widetilde{\varphi }$-metric
is a symmetric metric on $X_C$.
\end{proposition}

\begin{proof}
We assume without loss of generality, that
 $\widetilde{\gamma }\in U_i'$. Then, $\sigma (\widetilde{\gamma })\in U_i''$.
 By definition, the $\widetilde{\varphi }$-length of $\widetilde{\gamma }$ is
\[
l_{\widetilde{\varphi }}(\widetilde{\gamma })=l_{\varphi }(\gamma )
=\int_{\gamma _0} |dw_i| =\int_{\gamma }\sqrt{|\varphi _i(z_i)| }|dz_i|
\]
and the $\widetilde{\varphi }$-length of $\sigma (\widetilde{\gamma })$ is
\[
l_{\widetilde{\varphi }}(\sigma (\widetilde{\gamma }))
=l_{\overline{\varphi }}(\gamma )
=\int_{\gamma_0} |d\overline{w_i}| 
=\int_{\gamma} \sqrt{|\overline{\varphi _i(z_i)}| }|d\overline{z_i}|
\]
where $\gamma _0$ is the image of $\gamma $ by a branch of $\Phi $. Thus, 
$l_{\widetilde{\varphi }}(\widetilde{\gamma })=l_{\widetilde{\varphi }
}(\sigma (\widetilde{\gamma }))$.
\end{proof}

Next, we consider a point from the surface as being its image through the
corresponding local chart.
The $\varphi $-distance between two points $z_1$ and $z_2$ in $U_i$
is, by definition,
\[
d_{\varphi }(z_1,z_2)=\inf_{\gamma} l_{\varphi }(\gamma)
=\inf_{\gamma} \int_{\gamma} \sqrt{|\varphi _i(z_i)| }|dz_i|
\]
where $\gamma $ ranges over the piecewise smooth curves in $U_i$ joining 
$z_1$ and $z_2$. The $\varphi $-distance depends of the domain which is
selected.

Because $X_C$ is compact, any two points $\widetilde{z_1}$ and 
$\widetilde{z_2}$ can be connected with a shortest curve whose length is
the $\widetilde{\varphi }$-distance between $\widetilde{z_1}$ and 
$\widetilde{z_2}$. We note this distance with 
$d_{\widetilde{\varphi }}(\widetilde{z_1},\widetilde{z_2})$.

Using the definition of the $\widetilde{\varphi }$-length of a curve, we can
observe that
\[
d_{\widetilde{\varphi }}( \widetilde{z_1},\overline{\widetilde{z_2}}
) =d_{\widetilde{\varphi }}( \overline{\widetilde{z_1}},
\widetilde{z_2}) \quad \text{and}\quad 
d_{\widetilde{\varphi }}(\widetilde{z_1},\widetilde{z_2}) 
=d_{\widetilde{\varphi }}(
\overline{\widetilde{z_2}},\overline{\widetilde{z_1}}) .
\]

A piecewise smooth curve is called geodesic if it is locally shortest. The
similar notion defined on Riemann surfaces is studied in \cite{s2}.

A straight arc with respect to the quadratic differential $\varphi $  is a
smooth curve $\gamma $ along  which
\[
\arg( dw) ^{2}=\arg\varphi (z)( dz) ^{2}=\theta =const.,\quad 0\leq \theta <2\pi .
\]

\begin{remark} \label{rmk11} \rm
A straight arc only contains regular points of $\varphi $.
\end{remark}

\subsection{The $\varphi $-metric near holomorphic points}

Given a holomorphic quadratic differential $\varphi $ on a Klein surface 
$X$, a $\varphi $-disk is a region which is mapped homeomorphically onto a disk
in the complex plane by a branch of $\Phi $. The $\varphi $-radius of a 
$\varphi $-disk is the Euclidean radius of the corresponding disk in the
complex plane.

We want to determine the shortest curve (in the $\varphi $-metric) near
holomorphic points. In this case the $\varphi $-length of a curve $\gamma $
has sense and is finite, even if $\gamma $ goes through a zero of $\varphi $.
Further, we consider a point from the surface $X$ or $X_C$ as being its
image through the corresponding local charts.

The following theorem is an extension of Strebel's Theorem 5.4., see 
\cite{s2}, from Riemann surfaces to Klein surfaces.

\begin{theorem} \label{thm12}
Let $P$ be a regular point $P$ of $\varphi $. Then there exists a neighborhood 
$U$ of $P$ such that any two points $P_1$ and $P_2$ in $U$ can be joined
by a uniquely determined shortest curve $\gamma $ (in the $\varphi $-metric)
in $U$. The geodesic $\gamma $ is the pre-image of a straight line segment
under a branch of $\Phi $.
\end{theorem}

\begin{proof}
Let $U_0$ be the largest $\varphi $-disk around a regular point $P$.
Choose a branch $\Phi _0$ of $\Phi $ in $U_0$, such that $\Phi _0(P)=0$. 
If the radius of $U_0$ is $r$, let $V$ be the disk $|w| <\frac{r}{2}$ and 
$U=\Phi _0^{-1}(V)$. Then $w=\Phi _0(z)=\int \sqrt{\varphi _0(z)}dz$ is 
the natural parameter in $U$, where $\varphi_0(z)( dz) ^{2}$ is the local
 representation of the quadratic differential on $X$.

Let $P_1$ and $P_2$ be two points from $U$ and $w_i=\Phi _0(P_i)$,
$i=1,2$. We lift $P_1$ and $P_2$ to $X_C$. Let $\widetilde{P_i}$
and $\sigma (\widetilde{P_i})$ be the two points of $X_C$ which lie
over the same point $P_i$ of $X$, $i=1,2$.

We  notice that $\gamma $ does not pass through any zeroes of $\varphi $.
If $\gamma $ is an arbitrary curve joining $P_1$ and $P_2$ in $U$, then
either $\widetilde{\gamma }$ preserves the orientation or 
$\widetilde{\gamma}$ changes the orientation in a point of it.

In the first case, either $\widetilde{\gamma }$ is contained in $U'$
or $\widetilde{\gamma }$ is contained in $U''$.

If $\widetilde{\gamma }$ is contained in $U'$, by Strebel's 
Theorem 5.4., see \cite{s2}, $\widetilde{P_1}$ and $\widetilde{P_2}$ can be
joined by a uniquely determined shortest arc $\widetilde{\gamma }$ on $X_C$
(in the $\widetilde{\varphi }$-metric). The arc $\widetilde{\gamma }$ is
the pre-image of the straight line segment joining $w_1$ and $w_2$ in
the $w$-plane, under $\Phi _0$.
Then $l_{\varphi }(\gamma )=l_{\widetilde{\varphi }}(\widetilde{\gamma })=
\int_{\gamma} \sqrt{|\varphi _0(z)| }|
dz|=|w_2-w_1| $.

Similarly, if $\widetilde{\gamma }$ is contained in $U''$,
both $\sigma ( \widetilde{P_1}) $ and 
$\sigma ( \widetilde{P_2}) $ are in $U''$ and the uniquely determined
shortest arc $\widetilde{\gamma }$ is the pre-image of the straight line
segment joining $\overline{w_1}$ and $\overline{w_2}$ in the $w$-plane,
under $\overline{\Phi _0}$.
Then $l_{\varphi }(\gamma )=l_{\widetilde{\varphi }}(\sigma (\widetilde{
\gamma }))=\int_{\gamma} \sqrt{|\overline{\varphi _0(z)
}| }|d\overline{z}| =|\overline{w_2}-\overline{w_1}| $.

In the second case, because $\gamma $ does not pass through a zero of 
$\varphi $, then we consider the direct analytic continuation of 
$\Phi_0^{-1}$ along the straight line segment joining $w_1$ and 
$\overline{w_2}$ in the $w$-plane. Thus, the points $\widetilde{P_1}$ 
and $\sigma( \widetilde{P_2}) $ are joined by the shortest curve 
$\widetilde{\gamma }$ with respect to the $\widetilde{\varphi }$-metric on 
$X_C$, which is composed of straight arcs of the above type and the length
of $\widetilde{\gamma }$ is the sum of the lengths of the component straight
arcs. Applying the above results, we obtain 
$l_{\varphi }(\gamma )=l_{\widetilde{\varphi }}(\widetilde{\gamma })
=|\overline{w_2}-w_1| $.

In conclusion, if $|w_2-w_1| \leq |\overline{w_2}-w_1| $, 
the geodesic $\gamma $ is the pre-image of the straight
line segment $[w_1,w_2]$  under $\Phi _0$ and if 
$|\overline{w_2}-w_1| \leq |w_2-w_1| $, the geodesic $\gamma $
is the pre-image of the straight line segment $[w_1,\overline{w_2}]$
under a branch of $\Phi $.
\end{proof}

\begin{corollary} \label{coro13}
The $\varphi$-distance between $P_1$ and $P_2$ ,
\[
d_{\varphi }(z_1,z_2)=\min(|w_2-w_1| ,|\overline{w_2}-w_1| )
\]
thus $d_{\varphi }(z_1,z_2)=\min\big( d_{\widetilde{\varphi }
}( \widetilde{z_1},\widetilde{z_2}) ,d_{\widetilde{\varphi }
}( \widetilde{z_1},\overline{\widetilde{z_2}}) \Big) $.
\end{corollary}

The following theorem is an extension of Strebel's Theorem 8.1., 
see \cite{s2}, from Riemann surfaces to Klein surfaces.

\begin{theorem} \label{thm14}
Let $P$ be a zero of $\varphi $ of order $n$. Then there exists a
neighborhood $U$ of $P$ such that any two points $P_1$ and $P_2$ in $U$
can be joined by a uniquely determined shortest curve in $U$ 
(in the $\varphi $-metric). The geodesic $\gamma $ is either the pre-image of a
straight line segment under a branch of $\Phi $ or is composed of the
pre-images of two radii under branches of $\Phi $.
\end{theorem}

\begin{proof}
Let $P$ be zero of $\varphi $ of order $n$. Using \eqref{e2}, in the neighborhood 
$U $ of $P$, there is a parameter $\zeta$, $P\leftrightarrow \zeta =0$
such that the local representation of $\varphi $ is 
$\varphi (z)(dz) ^{2}=( \frac{n+2}{2}) ^{2}\zeta ^{n}( d\zeta) ^{2}$. 
Thus the corresponding natural parameter is 
$w=\Phi (\zeta)=\zeta ^{\frac{n+2}{2}}$. The function $\Phi $ maps each 
one of the sector
\[
\{\zeta \in C|\frac{2\pi }{n+2}k\leq arg\zeta \leq \frac{2\pi }{n+2}
( k+1) ,k=0,1,\dots ,n+1\}
\]
onto an upper or lower half-plane.

The $\varphi $-length of the radius of the circle 
$|\zeta |=\rho $ is equal to $\rho ^{\frac{n+2}{2}}$. 
Let $V$ be the disk $|w| <\frac{1}{2}\rho ^{\frac{n+2}{2}}$ 
and $U=\Phi _0^{-1}(V)$.

Let $P_1$ and $P_2$ be two points in $U$ and $w_i=\Phi _0(P_i)$, 
$i=1,2$. We lift $P_1$ and $P_2$ to $X_C$. Let $\widetilde{P_i}$ and
$\sigma (\widetilde{P_i})$ be the two points of $X_C$ which lie over
the same point $P_i$ of $X$, $i=1,2$.

If $\gamma $ is an arbitrary curve joining $P_1$ and $P_2$ in $U$, then
either $\widetilde{\gamma }$ preserves the orientation or 
$\widetilde{\gamma}$ changes the orientation in a point of it.
We may assume, without loss of generality, that $z_1\neq 0$ and
$\arg z_1=0 $.

In the first case, either $\widetilde{\gamma }$ is contained in $U'$
or $\widetilde{\gamma }$ is contained in $U''$.

If $\widetilde{\gamma }$ is contained in $U'$, by Strebel's 
Theorem 8.1., see \cite{s2} and \eqref{e3}, $\widetilde{P_1}$ and
$\widetilde{P_2}$ can be joined by a uniquely determined shortest curve 
$\widetilde{\gamma }$ on $X_C$ (in the $\widetilde{\varphi }$-metric). 
Furthermore,

(a) if $|\arg\widetilde{\zeta }-\arg\widetilde{\zeta _1}| \leq
\frac{2\pi }{n+2}$ for any $\widetilde{\zeta }\in \widetilde{\gamma }$,
 $ \widetilde{\zeta }\neq 0$, then $\widetilde{\gamma }$ is the pre-image of
the straight line segment joining $w_1$ and $w_2$ in the $w$-plane,
under $\Phi _0$. Hence 
$l_{\varphi }(\gamma )=l_{\widetilde{\varphi }}(
\widetilde{\gamma })$ $=\int_{\gamma} \sqrt{|\varphi
_0(z)| }|dz| =|w_2-w_1| $.

(b) If there is a $\widetilde{\zeta }\in \widetilde{\gamma }$ such that 
$|\arg\widetilde{\zeta }-\arg\widetilde{\zeta _1}| >\frac{2\pi }{n+2}$, 
then the curve $\widetilde{\gamma }$ is composed, in terms of 
$\widetilde{\zeta }$, of two radii enclosing angles $\geq \frac{2\pi }{n+2}$.
Hence $l_{\varphi }(\gamma )=l_{\widetilde{\varphi }}(\widetilde{\gamma })
=|w_1| +|w_2| $.

Analogously, for $\widetilde{\gamma }$ contained in $U''$, we have:

(c) If $|arg\widetilde{\zeta }-arg\widetilde{\zeta _1}| \leq
\frac{2\pi }{n+2}$ for any $\widetilde{\zeta }\in \widetilde{\gamma }$,
 $ \widetilde{\zeta }\neq 0$, then $\widetilde{\gamma }$ is the pre-image of
the straight line segment joining $\overline{w_1}$ and $\overline{w_2}$
in the $w$-plane, under $\overline{\Phi _0}$. Hence 
$l_{\varphi }(\gamma )=l_{\widetilde{\varphi }}(\widetilde{\gamma })
=\int_{\gamma} \sqrt{|\overline{\varphi _0(z)}| }|d\overline{z}| 
=|\overline{w_2}-\overline{w_1}| $.

(d) If there is a $\widetilde{\zeta }\in \widetilde{\gamma }$ such that 
$|\arg\widetilde{\zeta }-\arg\widetilde{\zeta _1}| 
>\frac{2\pi }{n+2}$, then the curve $\widetilde{\gamma }$ is composed, 
in terms of $\widetilde{\zeta }$, of two radii enclosing angles 
greater than or equal $\frac{2\pi }{n+2}$.
Hence $l_{\varphi }(\gamma )=l_{\widetilde{\varphi }}(\widetilde{\gamma }
) =|\overline{w_1}| +|\overline{w_2}| $.

In the second case, we have:

(e) If $|\arg\widetilde{\zeta }-\arg\widetilde{\zeta _1}| \leq
\frac{2\pi }{n+2}$ for any $\widetilde{\zeta }\in \widetilde{\gamma }$, 
$\widetilde{\zeta }\neq 0$ we consider the direct analytic continuation of 
$\Phi _0^{-1}$ along the straight line segment joining $w_1$ and 
$\overline{w_2}$ in the $w$-plane. The points $\widetilde{P_1}$ and 
$\sigma ( \widetilde{P}_2) $ can be joined by a uniquely
determined shortest curve $\widetilde{\gamma }$ with respect to the 
$\widetilde{\varphi }$-metric on $X_C$ which is composed of straight arcs
of the above type. Then, 
$l_{\varphi }(\gamma )=l_{\widetilde{\varphi }}(
\widetilde{\gamma })=|\overline{w_2}-w_1| $.

(f) If there is a $\widetilde{\zeta }\in \widetilde{\gamma }$ such that 
$|\arg\widetilde{\zeta }-\arg\widetilde{\zeta _1}| >\frac{2\pi }{n+2}$, 
then the curve $\widetilde{\gamma }$ is composed, in terms of 
$\widetilde{\zeta }$, of two radii enclosing angles greater than or equal to
$\frac{2\pi }{n+2}$.
Hence $l_{\varphi }(\gamma )=l_{\widetilde{\varphi }}(\widetilde{
\gamma })=|w_1| +|\overline{w_2}| $.

In conclusion, if $|\arg\widetilde{\zeta }-\arg\widetilde{\zeta _1}
| \leq \frac{2\pi }{n+2}$, then the geodesic $\gamma $ is the
pre-image of one of the straight line segments, $[w_1,w_2]$ or 
$[w_1, \overline{w_2}]$, namely the one that has the smallest Euclidean
length, under a branch of $\Phi $ and if there is a 
$\widetilde{\zeta }\in \widetilde{\gamma }$ such that
 $|\arg\widetilde{\zeta }-\arg\widetilde{\zeta _1}| >\frac{2\pi }{n+2}$, 
then the geodesic $\gamma $ is composed either of the pre-images of the 
radii $[0,w_1]$ and $[0,w_2]$
or of the pre-images of the radii $[0,w_1]$ and $[0,\overline{w_2}]$,
namely those that have the smallest sum of the Euclidean lengths, under
branches of $\Phi $.
\end{proof}

\begin{corollary} \label{coro15}
The $\varphi $-distance between the points $P_1$ and $P_2$,
\[
d_{\varphi }(z_1,z_2)=\min\big(|w_2-w_1| ,|\overline{
w_2}-w_1| ,|w_1| +|w_2| \big),
\]
thus $d_{\varphi }(z_1,z_2)=\min( d_{\widetilde{\varphi }}(
\widetilde{z_1},\widetilde{z_2}) ,d_{\widetilde{\varphi }}(
\widetilde{z_1},\overline{\widetilde{z_2}}) ) $
\end{corollary}

\subsection*{Acknowledgements} 
This work was supported by the
University of Craiova, [grant number 41C/2014].

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\end{document}