Topological Problems of the Theory of Asymptotic Curves

An asymptotic curve on a surface in a three-dimensional Euclidean or projective space is an integral curve of a vector field of asymptotic directions (directions along which the second fundamental form vanishes {reviewer: this means that asymptotic directions are a projective phenomenon, not a metric one}). We prove that the (generic) asymptotic curves on hyperbolic surfaces are precisely the (generic) space curves without flattening points. These curves can also be defined as those curves that are smooth and have smooth dual curves (called rotational curves). Rotational curves have inflection points.ou