On the characteristic curves on a surface in R^4

Jorge Luiz Deolindo-Silva

Journal of Singularities
volume 22 (2020), 28-39

Received: 28 February 2019. Received in revised form: 3 June 2020.

DOI: 10.5427/jsing.2020.22c

Add a reference to this article to your citeulike library.

Abstract:

We study some robust features of characteristic curves on smooth surfaces in R^4. These curves are analogous to the asymptotic curves in the elliptic region. A P_3(c)-point is an isolated special point at which the unique characteristic (or asymptotic) direction is tangent to the parabolic curve. At this point, by considering the cross-ratio invariant, we show that the 2-jet of the curve formed by the inflections of the characteristic curves is projectively invariant. In addition, we exhibit the possible configurations of the characteristic curves at a P_3(c)-point.

2010 Mathematical Subject Classification:

Primary: 57R45, 53A05, 53A20, 34A09; Secondary: 34A09, 37C10

Key words and phrases:

Characteristic curves, singularities, binary differential equations, projective invariants, surface in R^4

Author(s) information:

Jorge Luiz Deolindo-Silva
Departamento de Matemática
Universidade Federal de Santa Catarina
Blumenau-SC, Brazil.
email: jorge.deolindo@ufsc.br