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Teichmüller Space Is Totally Geodesic In Goldman Space

We construct a new Riemannian metric on Goldman space $\mathcal{B}(S)$, the space of the equivalence classes of convex projective structures on the surface $S$, and then prove the new metric, as well as the metric of Darvishzadeh and Goldman, restricts to be the Weil-Petersson metric on Teichm$\ddot{u}$ller space, embedded as a submanifold of Goldman space $\mathcal{B}(S)$. Moreover, Teichm$\ddot{u}$ller space endowed with the Weil-Petersson metric then is totally geodesic in the Riemannian manifold $\mathcal{B}(S)$.