Say I stand at the north pole and talk; in sufficiently frictionless conditions, one imagines that someone standing at the south pole could listen.

More generally, say $M$ is a compact Riemannian manifold, and $N$ is a compact submanifold. I look at the geodesic flow leaving $N$ in all normal directions. For most times $t$, these points form a hypersurface of codimension 1 in $M$.

When can it happen that, at some time $t$, all these geodesics meet in some locus of codimension $> 1$?

(I.e.: when can it happen that the front projection of the time $t > 0$ Reeb flow to a submanifold of the unit conormal bundle is codimension $>1$?)