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On the characteristic curves on a smooth surface
Author(s) -
Oliver J. M.
Publication year - 2011
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdq093
Subject(s) - mathematics , mathematical analysis , homogeneous space , invariant (physics) , elliptic curve , pure mathematics , hessian form of an elliptic curve , surface (topology) , geometry , schoof's algorithm , mathematical physics , quarter period
We study some symmetries between two classical pairs of foliations defined on smooth surfaces in ℝ 3 : the asymptotic curves and the characteristic curves. The asymptotic curves exist in hyperbolic regions of surfaces and have been well studied. The characteristic curves are the analogy of the asymptotic curves in elliptic regions. We produce results on the characteristic curves mirroring those in Uribe‐Vargas ( Mosc. Math. J. 6 (2006) 731–768) on the asymptotic curves. By considering cross‐ratios of Legendrian lines in the manifold of contact elements to the surface, we show that certain properties of the characteristic curves are invariant under projective transformations. We establish an analogy of the Beltrami–Enepper theorem. We exhibit the existence of a curve of points of zero torsion of the characteristic curves and determine its behaviour near cusps of Gauss and umbilic points.