A remark on the number of invisible directions for a smooth Riemannian metric

TL;DR

This paper constructs a smooth Riemannian metric on Euclidean space with a specified number of invisible directions, where geodesics passing through a compact set in these directions remain unchanged outside the set.

Contribution

It introduces a novel construction of Riemannian metrics with multiple invisible directions using reflection groups, contrasting with previous billiard obstacle examples.

Findings

Constructed metrics have N = n(n+1) invisible directions.

In the plane, the construction yields three invisible directions.

The method uses reflection groups of the root system An.

Abstract

In this note we give a construction of a smooth Riemannian metric on R^n which is standard Euclidean outside a compact set K and such that it has N = n(n + 1)=2 invisible directions, meaning that all geodesics lines passing through the set K in these directions remain the same straight lines on exit. For example in the plane our construction gives three invisible directions. This is in contrast with billiard type obstacles where a very sophisticated example due to A.Plakhov and V.Roshchina gives 2 invisible directions in the plane and 3 in the space. We use reflection group of the root system An in order to make the directions of the roots invisible.