On the number of invisible directions for a smooth Riemannian metric

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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 An in order to make the directions of the roots invisible.

Original languageEnglish
Pages (from-to)48-51
Number of pages4
JournalJournal of Geometry and Physics
StatePublished - 1 Jan 2015


  • Invisible directions
  • Lagrangian graphs
  • Lens rigidity


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