Teichmüller space is totally geodesic in Goldman space (Q254814)

An \(\mathbb RP^{2}\)-structure on a surface is a system of coordinate charts in \(\mathbb RP^{2}\) with transition maps in \(PGL(3,\mathbb R).\) The equivalence classes of such structures on a compact surface \(S\) of genus \(g>1\) form a moduli space \(B(S)\) homeomorphic to an open cell of dimension \(16(g-1).\) Labourie and Loftin gave a correspondence between the deformation space \(B(S)\) and the space of pairs \((\Sigma,U),\) where \(\Sigma\) is a Riemann surface varying in Teichmüller space and \(U\) is a cubic differential on \(\Sigma.\) Teichmüller space \(T(S)\) embeds in \(B(S)\) as the locus of pairs \((\Sigma,0).\)NEWLINENEWLINETheorem 1. The Darvishzadeh and Goldman metric and the Loftin metric both restrict to a constant multiple of the Weil-Petersson metric on Teichmüller space.NEWLINENEWLINETheorem 2. Teichmüller space endowed with the Weil-Petersson metric is totally geodesic in \(B(S)\) endowed with the Loftin metric.