3
$\begingroup$

Assume M is an oriented manifold and $W \subset M$ is a smooth compact embedded hypersurface without boundary.

Is there an example that the outer equidistants $W_t$ is not smooth for all $t$ in some oriented $M$ with non-positive sectional curvature $K_M \leq 0$?

And if we consider the case $M = \mathbb{R}^n$, why the outer equidistants $W_t$ is qualitively close to a perfectly round standard sphere for $t \gg 0$?

Thank you very much

1 Answers 1