Is there a notion of curvature defined for curves in a general metric space, using only the metric and not the derivative of the curve? I've been searching among different books but all I find is curvature of the metric spaces themselves, and nothing about the curves.
The closest thing I found was this expression in Wikipedia, but it has no references:
** Given two points $P$ and $Q$ on $C$, let $s(P, Q)$ be the arc length of the portion of the curve between $P$ and $Q$ and let $d(P, Q)$ denote the length of the line segment from $P$ to $Q$. The curvature of $C$ at $P$ is given by the limit [citation needed] $$ \kappa(P)=\lim _{Q \rightarrow P} \sqrt{\frac{24|s(P, Q)-d(P, Q)|}{s(P, Q)^{3}}} $$ where the limit is taken as the point $Q$ approaches $P$ on $C$. The denominator can equally well be taken to be $d(P, Q)^{3}$. The formula is valid in any dimension. Furthermore, by considering the limit independently on either side of $P$, this definition of the curvature can sometimes accommodate a singularity at $P$. The formula follows by verifying it for the osculating circle. **
I would appreciate any help!
See section 4.1 in
Emil Saucan, Curvature – Smooth, Piecewise-Linear and Metric
and also
Saucan, Emil, Metric curvatures and their applications. I., Geom. Imaging Comput. 2, No. 4, 257-334 (2015). ZBL1357.53003,
which you can find here