On the number of invisible directions for a smooth Riemannian metric

In this note we give a construction of a C^∞-smooth Riemannian metric on Rⁿ which is standard Euclidean outside a compact set K and such that it has N=n(n+1)/2 invisible directions, meaning that all geodesic 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 A_n in order to make the directions of the roots invisible.