Recent Advances in the Geometry of Submanifolds: Dedicated to the Memory of Franki Dillen (1963–2013) Contemporary Mathematics vol:674 pages:59-73
AMS Special Session on Recent Advances in the Geometry of Submanifolds: Dedicated to the Memory of Franki Dillen (1963–2013) location:Michigan State University, East Lansing, Ml, USA date:March 14–15, 2015
The so-called Clelia curve is a special spherical curve in Euclidean 3-space known already for centuries. A Clelia curve is characterized by the linear dependency of its coordinates when parameterized using spherical coordinates.
Firstly, by analogy we define hyperbolic Clelia curves in Minkowski 3-space as curves on the pseudosphere or on the hyperbolic space with linear dependent coordinates when parameterized using appropriate ‘spherical’ coordinates. For this purpose, distinct appropriate parameterizations of the pseudosphere and of the hyperbolic space are provided.
Secondly, we approach Clelia curves from two different perspectives. On the one hand, we show that these curves emerge quite naturally when studying flat twisted surfaces. On the other hand, we prove that the intersection of the
pseudosphere or the hyperbolic space with an instance of Plücker’s conoid in Minkowski 3-space, is a hyperbolic Clelia curve, hereby generalizing a wellknown result in Euclidean 3-space. Obviously, this result requires extending
the notion of Plücker’s conoids to Minkowski 3-space.