Number, emerging level, and divergence measure of the critical points of the curvature field are assessed as indicators of the onset of instability of a two-dimensional stratified gravitational flow. A projection-based Arakawa-C finite-difference method is employed to solve the Navier-Stokes and buoyancy equations governing a flow field which is initiated with a sharp density difference and is inclined with a small slope angle, and then the critical points are obtained from the velocity field. For the well-established critical Richardson numbers the critical points begin to emerge. Then, we monitor the statistical features of the emerging critical points around the sharp interface as pointers of the beginning of the mixing phenomenon. The results show that it is possible to study several critical points to quantitatively predict the onset of instability in gravitational flows.