  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 $\mathcal{B}(S)$.
