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Let $(X,d)$ be a metric space. Given a continuous curve $\gamma_t : [0,1] \rightarrow X$, the metric speed is defined by $$ |\gamma_t^\prime | := \lim_{s\rightarrow t} \frac{d(\gamma_s, \gamma_t)}{|t-s|}. $$

Is there an analogous notion of "metric acceleration" or "metric curvature" $|\gamma_t^{\prime \prime} |$?

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    $\begingroup$ For the overwhelming majority of continuous curves, this definition has no meaning (in an arcwise connected component of a metric space), because the limit does not exist. $\endgroup$
    – Daniel Asimov
    Commented Aug 30 at 0:47
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    $\begingroup$ @DanielAsimov for Lipschitz curves it is defined almost everywhere; moreover integraiting it, you get the length of curve. $\endgroup$
    – Anton Petrunin
    Commented Aug 30 at 2:12
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    $\begingroup$ You can have a look at the works by Nicola Gigli on second-order differential calculus on some special metric spaces. $\endgroup$
    – Akira
    Commented Aug 30 at 9:05

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Check "Extrinsic curvature of..." by S. Alexander and R. Bishop. They define curvature as the limit of $\sqrt{24{\cdot}(s-r)/r^3}$ as $s\to0$, where $s$ is the length of an arc and $r$ is the distance between its ends.

While this definition makes sense in an arbitrary metric space, it behaves nicely only if the space is nice.

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