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My best answer so far is that at every time ##t## the acceleration (2nd derivative) is parallel to the velocity (1st derivative), i.e. ##\ddot{\phi}(t) = g(t)\cdot\dot{\phi}(t)##, for some function ##g(t)## (which likely should better not be zero anywhere).

Is this a valid description (necessary and sufficient) of a straight line? Are there different ones?

And a very similar question for a 2-dimensional surface, i.e. now we have ##\phi:\mathbb{R}^2\to\mathbb{R}^n##. Assuming the above is true for the straight line description. Is there a similar condition for the surface?