The Jacobian $J$ For a dynamical system $\dot{\textrm{x}}=F(\textrm{x})$ determines the dynamics in the tangent plane at a given point. Intuitively speaking the Jacobian evaluated at a point should contain some information about the curvature at that point but I don't know of any such association. Sorry for the vague phrasing of the question, I lack sufficient training to make this more precise.