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Let $M$ be a smooth Riemannian manifold. Let $X\subset M$ be a closed path connected subset which has curvature bounded below in the sense of Alexandrov with respect to the induced intrinsic metric. Let $\gamma\colon [a,b]\to X$ be a shortest path in $X$ (parameterized by its length). Let $\iota\colon X\to M$ denote the natural imbedding.

Question. Is it true that $\iota\circ\gamma\colon [a,b]\to M$ has one sided derivatives everywhere?

Remark. The question has positive answer at almost every point of $[a,b]$ since $\iota\circ \gamma$ is a Lipschitz map, and such maps have first derivative almost everywhere.

ADDED: In the special case when $M=\mathbb{R}^n$ is the Euclidean space, and $X$ is a convex hypersurface, the answer is positive and due to I.M. Liberman (1941).

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No. If $M = R^2$ and $X$ is the sawtooth below, then the map has no derivative at the central point. Note that the induced intrinsic metric on $X$ is just proportional to distance on the $x$-axis, and in particular is flat. Similarly if $M = R^3$ and $X$ is the sawtooth below crossed with $R$, then the induced intrinsic metric is still flat, and there is still no derivative at any of the central points. sawtooth

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