We obtain the topological configurations of the lines of curvature, the asymptotic and characteristic curves on a cross-cap, in the domain of a parametrisation of this surface as well as on the surface itself.
Farid Tari. "On pairs of geometric foliations on a cross-cap." Tohoku Math. J. (2) 59 (2) 233 - 258, 2007. https://doi.org/10.2748/tmj/1182180735