Tohoku Mathematical Journal
2016
June
SECOND SERIES VOL. 68, NO. 2

Tohoku Math. J.
68 (2016), 293-328

Title ON THE GEOMETRY OF THE CROSS-CAP IN THE MINKOSWKI 3-SPACE AND BINARY DIFFERENTIAL EQUATIONS

Author Fabio Scalco Dias and Farid Tari

(Received November 2, 2012, revised October 30, 2014)
Abstract. We initiate in this paper the study of the geometry of the cross-cap in Minkowski 3-space $\mathbb{R}^3_1$. We distinguish between three types of cross caps according to their tangential line being spacelike, timelike or lightlike. For each of these types, the principal plane which is generated by the tangential line and the limiting tangent direction to the curve of self-intersection of the cross-cap plays a key role. We obtain special parametrisations for the three types of cross-caps and consider their affine properties. The pseudo-metric on the cross-cap changes signature along a curve and the singularities of this curve depend on the type of the cross-cap. We also study the binary differential equations of the lightlike curves and of the principal curves in the parameters space and obtain their topological models as well as the configurations of their solution curves.

Mathematics Subject Classification. Primary 51B20; Secondary 34A09, 58K05.
Key words and phrases. Cross-cap, Minkowski space, Lightlike lines, Lines of principal curvature.

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