TY - JOUR

T1 - On the number of invisible directions for a smooth Riemannian metric

AU - Bialy, Misha

N1 - Publisher Copyright:
© 2014 Elsevier B.V.

PY - 2015/1/1

Y1 - 2015/1/1

N2 - 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.

AB - 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.

KW - Invisible directions

KW - Lagrangian graphs

KW - Lens rigidity

UR - http://www.scopus.com/inward/record.url?scp=85028108901&partnerID=8YFLogxK

U2 - 10.1016/j.geomphys.2014.08.005

DO - 10.1016/j.geomphys.2014.08.005

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AN - SCOPUS:85028108901

VL - 87

SP - 48

EP - 51

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

ER -