Local Rigidity of Teichm\"uller space with Thurston metric.

We show that every $\mathbb R$-linear surjective isometry between the cotangent spaces to the Teichm\"uller space equipped with the Thurston norm is induced by some isometry between the underlying hyperbolic surfaces, which is an analogue of Royden's theorem concerning the Teichm\"uller metric. Consequently, we obtain the local rigidity theorem of the Thurston metric, as well as a new proof of the global rigidity theorem which was first proved by Walsh.