The closed non-self-intersecting geodesics on the surface of a three-dimensional simplex are studied. It is proved that every geodesic on an arbitrary simplex can be realized on a regular simplex. This enables us to obtain a complete classification of all geodesics and describe their structure. Conditions for the existence of geodesics are obtained for an arbitrary simplex. It is proved that a simplex has infinitely many essentially different geodesics if and only if it is isohedral. Estimates for the number of geodesics are obtained for other Simplexes. © 2007 RAS(DoM) and LMS.
Closed geodesics on the surface of a simplex
Protasov, Vladimir
2007-01-01
Abstract
The closed non-self-intersecting geodesics on the surface of a three-dimensional simplex are studied. It is proved that every geodesic on an arbitrary simplex can be realized on a regular simplex. This enables us to obtain a complete classification of all geodesics and describe their structure. Conditions for the existence of geodesics are obtained for an arbitrary simplex. It is proved that a simplex has infinitely many essentially different geodesics if and only if it is isohedral. Estimates for the number of geodesics are obtained for other Simplexes. © 2007 RAS(DoM) and LMS.File in questo prodotto:
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