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).


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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.