Ah of course, sorry I couldn't explain myself properly there. I'm actually interested in the Euler characteristic, and I was wondering if there's a way to use the Gauss-bonnet theorem to get at that, for which I would need an estimate of local curvature. That's the full story. Thinking about this just made me wonder how much information about the local curvature can be extracted from the estimated Jacobian.

Thanks again. I don't know anything about symplectic manifolds so I am a little confused. Talking simple mindedly, the Hamiltonian $H=\frac{1}{2}(x^2+y^2)$ describes a circle in the phase space. But the circle has curvature right? $\frac {1}{R}$ or $\frac{1}{\sqrt{2} H}$ in this case. As for the context of my question. I'm interested in analyzing geometrical properties of real world dynamical systems, from some experimental data I can reconstruct a phase space (Takens embedding), and can estimate local Jacobians. So I was wondering under what conditions can I also extract the curvature?

@TobiasFritz I see, thank you for the answer (and your patience). I want to understand more what you meant by "since you have not assumed anything about how $F$ relates to the geometry". Can you give an example of how $F$ can relate to the geometry? Like when $F$ is derived from a Hamiltonian?

@PiyushGrover Curvature of the manifold that the trajectories lie on. So for example if the trajectories of $\dot{\textrm{x}}=F(\textrm{x})$ lie on a torus then does the Jacobian evaluated at a point $\textrm{x}_i$ (or some transformation) contain any information about the curvature obtained by doing parallel transport around $\textrm{x}_i$.

@TobiasFritz Thanks for making me think about it a little more. I mean the curvature one would get by doing parallel transport around a point.

I'm wondering whether the Jacobian (or some transformation) contains any information about the local curvature. No, I wasn't aware of that theorem. Looking it up now!