On non-diffractive cones

TL;DR

This paper investigates conditions under which certain conical geometries do not cause wave diffraction at high frequencies, revealing geometric constraints on the base manifold.

Contribution

It establishes that non-diffractive cones with analytic bases require all geodesics to be closed and characterizes the base manifold when its dimension is two.

Findings

All geodesics on Y are closed with period 2π.

If dim Y=2, Y is isometric to a sphere or a real projective plane.

Non-diffractive cones impose strong geometric restrictions.

Abstract

A subject of recent interest in inverse problems is whether a corner must diffract fixed frequency waves. We generalize this question somewhat and study cones $[0,\infty)\times Y$ which 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 $\dim Y=2$, then $Y$ is isometric to either the sphere of radius 1 or its $\mathbb{Z}^2$ quotient, $\mathbb{R}\mathbb{P}^2$.