Let $\Sigma$ be a smooth convex surface in Euclidean 3-space and $\gamma$ be a unit speed minimizing geodesic in $\Sigma$. Assume that for some $a < b < c$, we have $$\gamma'(a)=\gamma'(b)=\gamma'(c).$$
Is it true that $\gamma'$ is constant on one of two intervals $[a,b]$ or $[b,c]$?
Comments
- This is a simplification-variation of an other question I heard from D. Burago.
- It is not hard t construct an example of a minimizing geodesic such that $\gamma'(a)=\gamma'(b)$, but $\gamma'$ is not a constant on $[a,b]$. (See the example below.)
- I would be also interested in the analog for $n$ points.
- This paper: Total curvature and spiralling shortest paths by Barany, Kuperberg, Zamfirescu is relevant.
Example. I will construct a convex polyhedron, but it is easy to smooth. Consider polyhedron defined by 5 inequlaities: $$z\ge 0,\ \ |x+\alpha{\cdot}y|\le \alpha\ \ \text{and}\ \ z\pm\beta{\cdot}y\le \beta$$ and look at the minimizing geodesic between points $(0,1-\epsilon, \beta{\cdot}\epsilon)$ and $(0,-(1-\epsilon), \beta{\cdot}\epsilon)$. For appropriately chousen $\alpha$, $\beta$ and $\epsilon$ this minimizing geodesic will pass through the faces in this order $$\{z+\beta{\cdot}y= \beta\},\ \ \{x+\alpha{\cdot}y= \alpha\},\ \ \{z=0\},\ \ \{x+\alpha{\cdot}y= -\alpha\},\ \ \{z-\beta{\cdot}y= \beta\}$$ and it will have the same velocity vector $(1,0,0)$ on both faces $\{z+\beta{\cdot}y= \beta\}$ and $\{z-\beta{\cdot}y= \beta\}$.
