2
$\begingroup$

Let $(M^n,g)$ be a smooth Riemannian manifold with non-empty boundary $\partial M$. Assume (for simplicity) that $M$ is compact. Let $M$ be locally geodesically convex, i.e. any shortest path in $M$ connecting any two sufficiently close points is a geodesic in $M$ (namely satisfies the standard second order ODE).

Assume in addition that $M$ has non-negative sectional curvature. Is it true that the function on $M$ given by $$x\mapsto dist(x,\partial M)$$ is concave, i.e. its restriction on any shortest path in concave?

A reference would be helpful.

$\endgroup$
2
  • 3
    $\begingroup$ When $M$ is convex this is theorem 8.10 in Cheeger-Ebin book "Comparison theorems in Riemannain geometry". $\endgroup$ Jul 21, 2020 at 16:15
  • $\begingroup$ @IgorBelegradek : Many thanks. It practically answers my question. I have to read the proof. Probably it gives everything. $\endgroup$
    – asv
    Jul 21, 2020 at 17:27

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.