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Horocyclic surfaces in hyperbolic 3-space. (English) Zbl 1187.53019

Horocyclic surfaces are surfaces in hyperbolic 3-space that are foliated by horocycles. This notion was introduced by S. Izumiya, K. Saji and M. Takahashi [Horospherical flat surfaces in hyperbolic 3-space. Preprint] in a preprint as an analog to ruled surfaces in the Euclidean 3-space. In the present paper the horocyclic surfaces associated with space-like curves in the light-cone are constructed and their geometric properties are investigated. Furthermore, the homogeneous space-like curves are classified using invariants (the coefficients in their Frenet-Serret-type formulae). In particular, the singularities of horocyclic surfaces are classified using invariants of corresponding space-like curves. The proof depends on the criteria for the recognition of the cuspidal edges. swallowtails, cuspidal beaks and cuspidal lips given by M. Kokubu, W. Rossman, K. Saji; M. Umehara and K. Yamada [Pac. J. Math. 221, No. 2, 303–351 (2005; Zbl 1110.53044)] Kokubu and Izumiya et. al. (Propositions 5.4 and 5.6).

MSC:

53B25 Local submanifolds
57R45 Singularities of differentiable mappings in differential topology

Citations:

Zbl 1110.53044
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