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If $f:[0,1]\rightarrow \mathbb{R}^d$ is a smooth curve, then is there are relationship between the total variation of $f$ and the geodesic curvature of $f$?

I expect they both should be zero iff $f$ is a line but how to make this clear?

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  • $\begingroup$ No, the total variation of a function $f$ is $0$ if only if the function is constant. Intuitively, total variation is about $\|f'\|$ and curvature is about $\|f''\|$, so I don't know what you're looking for or expecting. $\endgroup$ – Ted Shifrin May 14 '16 at 16:57
  • $\begingroup$ So the total variation of the derivative is 0 if and only if the curvature is zero? $\endgroup$ – AIM_BLB May 15 '16 at 2:39
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    $\begingroup$ Assuming a constant-speed parametrization, yes. $\endgroup$ – Ted Shifrin May 15 '16 at 2:40
  • $\begingroup$ How would the formal proof of this look? $\endgroup$ – AIM_BLB May 15 '16 at 2:48
  • $\begingroup$ Could you define "variation"? $\endgroup$ – rrogers May 17 '16 at 20:26

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