A marked hyperbolic surface is a pair $(X,f)$ where $X$ is a hyperbolic surface and $f:S\to X$ is an orientation-preserving homeomorphism. Two marked hyperbolic surfaces $(X,f)$ and $(X^{\text{'}},f\text{'})$ are said to be equivalent if and only if $f\text{'}\circ f^{-\mathrm{1}}:X\to X\text{'}$ is homotopic to an isometry. The Teichmüller space $T(S)$ is defined to be the set of equivalence classes of marked hyperbolic surfaces. In the following，we shall simply denote the equivalence class $[X,f]$ by $X$. It is well known that $T(S)$ is homeomorphic to ${\mathbf{R}}^{-\mathrm{3}(\mathrm{2}-\mathrm{2}g-n)}$.

Recall that $C$ is the set of isotopy classes of simple closed curves on $S$. For each $\alpha \in C$ and $X\in T(S)$，there is a unique geodesic representative of $\alpha$ on the hyperbolic surface $X$. Let $l_{\alpha }(X)$ be the length of this geodesic representative. This defines a function on $T(S)\times C$：

which admits a unique continuous extension：

such that $l_{t\mu }\left(X\right)=t{l}_{\mu }\left(X\right)$ for all $t>\mathrm{0},\mu \in ML(S)$ and $X\in T(S)$. For each measured geodesic lamination $\mu$，$l_{\mu }:T\left(S\right)\to \mathbf{R}$is real analytic （for instance see ［15，Corollary 2.2］）.

Proposition 1   ^{[16    ，Theorem 5.1]} For any hyperbolic surface $X$，the map

embeds $PML(S)$ as the boundary of a convex neighborhood of the origin.

By identifying$PML(S)$ with the subset $\{\lambda \in ML\left(S\right):l_{\lambda }=\mathrm{1}\}$， we obtain a homeomorphism：

In this way，we associate to $ML(S)\bigcup \{\mathrm{0}\}$ a linear structure induced from the linear structure on $T_X^{*}T(S)$.

To see the dependence of the linear structures on the underlying hyperbolic surfaces，we consider the map $D{L}_Y\circ D{L}_X^{-\mathrm{1}}:T_X^{*}T(S)\to T_Y^{*}T(S)$ which is the composition of $D{L}_X^{-\mathrm{1}}:T_X^{*}T(S)\to ML\left(S\right)\bigcup \{\mathrm{0}\}$ and $D{L}_Y:ML\left(S\right)\bigcup \{\mathrm{0}\}\to T_Y^{*}T(S)$. In particular，

Theorem 1   $D{L}_Y\circ D{L}_X^{-\mathrm{1}}:T_X^{*}T(S)\to T_Y^{*}T(S)$ is linear if and only if $Y=X$.

The primary tool we use to prove Theorem 1 is the earthquake deformation.

Given any hyperbolic surface $X\in T(S)$ and $(t,\alpha )\in \mathbf{R}\times C$，the cutting and gluing operation defines a new hyperbolic surface $E_{\alpha }^t(X)$ by cutting $X$ along $\alpha$ and gluing it back with a left twist of （hyperbolic） length $t$ （$t<\mathrm{0}$ means a right twist）.One can extend this operation to the set of weighted simple closed curves by setting $E_{a\alpha }^t\left(X\right)≔E_{\alpha }^{at}(X)$ for any $a>\mathrm{0}$ ，$t\in \mathbf{R}$ and $\alpha \in C$. In ［17］，Kerckhoff showed that the cutting and gluing operation can also be extended to any measured geodesic lamination.

Proposition 2  ^[

The limit $E_{\mu }^t\left(X\right)≔\underset{i\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{E}_{a_i{\alpha }_i}^t$ is said to be the time $t$ left earthquake of $X$along$\mu$. The image $\{E_{\mu }^t{\left(X\right)\}}_{t\in \mathrm{R}}$is called an earthquake line directed by $\mu$.

Proposition 3  ^[

（i）For any two distinct hyperbolic surfaces$X,Y\in T(S)$，there exists a unique earthquake path passing through them.

（ii）For any $X\in T(S)$，and any two measured geodesic laminations $\lambda$ and $\mu$，the length functions $l_{\lambda }:T(S)\to \mathbf{R}$ is convex along the earthquake line $\{E_{\mu }^t{\left(X\right)\}}_{t\in \mathrm{R}}$.Moreover，if $i(\lambda ,\mu )>\mathrm{0}$，then $l_{\lambda }$ is strictly convex along $\{E_{\mu }^t{\left(X\right)\}}_{t\in \mathrm{R}}$.

Let ${\Pi }_{XY}≔D{L}_Y\circ D{L}_X^{-\mathrm{1}}$. If $Y=X$，then ${\Pi }_{XY}$ is the identity map，which is linear.We now consider the converse.In the following，we assume that ${\Pi }_{XY}$ is linear.

Suppose to the contrary that $Y\ne X$. By Proposition 3，there exists a unique measured lamination $\delta$ such that $E_{\delta }^{\mathrm{1}}\left(X\right)=Y$. Let $\lambda$ be a measured foliation which intersects $\delta$.Then there exists a unique $\mu \in ML(S)$ such that

By assumption，${\Pi }_{XY}$ is linear，then

Since $\lambda$ intersects $\delta$，it follows from Proposition 3 that $l_{\lambda }$ is strictly convex along the earthquake line $\{E_{\delta }^t{\left(X\right)\}}_{t\in \mathrm{R}}$.Combing with that $l_{\delta }$ is a constant on the earthquake line $\{E_{\delta }^t{\left(X\right)\}}_{t\in \mathrm{R}}$，we see that$l_{\mu }$ is not a constant on the earthquake line $\{E_{\delta }^t{\left(X\right)\}}_{t\in \mathrm{R}}$.Therefore， $\mu$ intersects $\delta$.This in turn implies that both $l_{\lambda }$ and $l_{\mu }$ are strictly convex along the earthquake line $\{E_{\delta }^t{\left(X\right)\}}_{t\in \mathrm{R}}$.Let ${\dot{E}}_{\delta }$ be the tangent vector field along the earthquake line $\{E_{\delta }^t{\left(X\right)\}}_{t\in \mathrm{R}}$. Then

In particular，

On the other hand， it follows from （1） and （2） that

Contradiction！

Remark 1   If $\lambda$ and $\mu$ are disjoint measured laminations，then $\lambda +\mu$ is also a measured lamination.In this case，$d_X{l}_{\lambda +\mu }=d_X{l}_{\lambda }+d_X{l}_{\mu }$ holds at each point $X\in T(S)$.In particular，for each point $Y\in T(S)$，we have

We guess that the converse is also true，namely，$D{L}_Y\circ D{L}_X^{-\mathrm{1}}\left(d_X{l}_{\lambda }+d_X{l}_{\mu }\right)=d_Y{l}_{\lambda }+d_Y{l}_{\mu }$ holds only when $\lambda$ and $\mu$ are disjoint.

A marked conformal structure on $S$ is a pair $(X,f)$ where $X$ is a Riemann surface and $f:S\to X$ is an orientation-preserving homeomorphism.Two marked conformal structures $(X,f)$ and $(X^{\text{'}},f\text{'})$ are said to be equivalent if and only if $f\text{'}\circ f^{-\mathrm{1}}:X\to X\text{'}$ is homotopic to a conformal map.The Teichmüller space $T(S)$ can also be defined to be the set of equivalence classes of marked conformal structures.In the following，we shall simply denote the equivalence class $[X,f]$ by $X$.

Let $\alpha$ be an isotopy class of simple closed curve，and $X$ be a （marked） Riemann surface.A conformal metric on $X$ is a metric which can be expressed as $\rho \left(z\right)|\mathrm{d}z|$ locally.The extremal length of $\alpha$ on $X$ is defined by：

where the sup ranges over all the conformal metrics on $X$，$\mathrm{A}\mathrm{r}\mathrm{e}\mathrm{a}\left(\rho \right)$ is the area of $X$ endowed with the metric $\rho$， and $l_{\rho }\left(\alpha \right)≔\mathrm{i}\mathrm{n}\mathrm{f}{\int }_{\alpha \text{'}}^{}\rho \left|dz\right|,$where the infimum is taken over all representatives $\alpha \text{'}$ in $\alpha$.It is clear that

for any positive constant $a$.The extremal length is a conformal invariant.Kerckhoff^{［18， Proposition 3］} extended the definition of extremal length from weighted simple closed curves to measured laminations：

such that $Ex{t}_{t\mu }\left(X\right)=t^{\mathrm{2}}Ex{t}_{\mu }\left(X\right)$ for all $t>\mathrm{0},\mu \in ML(S)$ and $X\in T(S)$. Moreover，for each measured lamination $\lambda$， the extremal length function $Ex{t}_{\lambda }:T(S)\to \mathbf{R}$ is differentiable and positive.

Proposition 4  ^{[19    ，Theorem 4.1］} For any complex structure $X$，the map

embeds $PML(S)$ as the boundary of a convex neighborhood of the origin.

By identifying$PML(S)$ with the subset $\{\lambda \in ML\left(S\right):Ex{t}_{\lambda }(X)=\mathrm{1}\}$，we obtain a homeomorphism：

In this way，we associate to $ML(S)\bigcup \{\mathrm{0}\}$ another linear structure induced from the linear structure on $T_X^{*}T(S)$. We consider the map

which is the composition of $D{E}_X^{-\mathrm{1}}:T_X^{*}T(S)\to ML\left(S\right)$ and $D{E}_Y:ML\left(S\right)\to T_Y^{*}T(S)$.In particular，

Theorem 2 $D{E}_X\circ D{E}_Y^{-\mathrm{1}}$ is linear if and only if $Y=X$.

Proof   For each $\mu \in ML(S)$，there exists a unique holomorphic quadratic differential $Q(\mu ,X)$ on $X$ such that for every isotopy class $\alpha$ of simple closed curve，the geometric intersection number between $\alpha$ and the horizontal measured lamination of $Q(\mu ,X)$ is the same as the intersection number between $\alpha$ and $\mu$.Let ${\mu }_X^{\perp }$ be the vertical measured lamination of $Q(\mu ,X)$.By Gardiner's formula^{［20，Theorem 8］}，（see also ［21，Theorem 1.2］），we see that

Similarly，we have

Since $D{E}_X\circ D{E}_Y^{-\mathrm{1}}:T_Y^{*}T(S)\to T_X^{*}T(S)$ is linear，it follows that

Therefore， ${\mu }_X^{\perp }={\mu }_Y^{\perp }$. Consequently， $X=Y$.

Remark 2   Using the theory of lines of minima^[