{"paper":{"title":"A remark on the number of invisible directions for a smooth Riemannian metric","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["nlin.SI"],"primary_cat":"math.DG","authors_text":"Michael (Misha) Bialy","submitted_at":"2013-05-13T10:53:06Z","abstract_excerpt":"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 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.2736","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}