Is there a theorem involving total curvature and some notion of index number for a curve in a metric space, as there is for planar curves? (i.e. total curvature is an integer multiple of $2\pi$.)
I'm guessing "no", at least not with this exact definitions, since they fall apart even in $\mathbb{R}^3$. However, I was wondering if there are any similar theorems/notions for a general metric space.
Edit: I'm thinking of using this as the curvature: (from wikipedia)
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 $$ \kappa(P)=\lim _{Q \rightarrow P} \sqrt{\frac{24(s(P, Q)-d(P, Q))}{s(P, Q)^{3}}} $$