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.
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1$\begingroup$ What kind of curvature are you referring to, in the absence of a metric or a connection? And do you know the theorem stating that every nonzero vector field is locally a coordinate vector field? $\endgroup$– Tobias FritzCommented Sep 4, 2017 at 10:39
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$\begingroup$ 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! $\endgroup$– user9563Commented Sep 4, 2017 at 11:06
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$\begingroup$ One reference for the theorem that every nonzero vector field is locally a coordinate field is an early chapter of F. Warner's book Introduction to Differentiable Manifolds and Lie Groups $\endgroup$– NealCommented Sep 4, 2017 at 12:21
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$\begingroup$ @user9563: But what do you actually mean by curvature? The usual notion of curvature is the curvature of a connection, but it's not clear which connection you'd be referring to in your context. $\endgroup$– Tobias FritzCommented Sep 4, 2017 at 14:33
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2$\begingroup$ curvature of what ? $\endgroup$– Piyush GroverCommented Sep 4, 2017 at 15:20
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