  We investigate the notion of curvature in the context of Liouville quantum
gravity (LQG) surfaces. We define the Gaussian curvature for LQG, which we
conjecture is the scaling limit of discrete curvature on random planar maps.
Motivated by this, we study asymptotics for the discrete curvature of
$\epsilon$-mated CRT maps. More precisely, we prove that the discrete curvature
integrated against a $C_c^2$ test function is of order $\epsilon^{o(1)},$ which
is consistent with our scaling limit conjecture. On the other hand, we prove
the total discrete curvature on a fixed space-filling SLE segment scaled by
$\epsilon^{\frac{1}{4}}$ converges in distribution to an explicit random
variable.
