On non-diffractive cones

Abstract

A subject of recent interest in inverse problems is whether a corner must diffract fixed frequency waves. We study the related question of which cones $[0,\infty) \times Y$ do not diffract high frequency waves. We prove that if $Y$ is analytic and does not diffract waves at high frequency then every geodesic on $Y$ is closed with period $2\pi$. Moreover, we show that if $\operatorname{dim} Y=2$, then $Y$ is isometric to either the sphere of radius $1$ or its $\mathbb{Z}^2$ quotient, $\mathbb{RP}^2$.