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We undertake a systematic study of the infinitesimal geometry of the Thurstonmetric, showing that the topology, convex geometry and metric geometry of thetangent and cotangent spheres based at any marked hyperbolic surfacerepresenting a point in Teichm{\"u}ller space can recover the marking andgeometry of this marked surface. We then translate the results concerning theinfinitesimal structures to global geometric statements for the Thurstonmetric, most notably deriving rigidity statements for the Thurston metricanalogous to the celebrated Royden theorem.

The infinitesimal and global Thurston geometry of Teichm{\"u}ller space