Basic invariants for time-like surfaces in ${\mathbb{R}}_1^3$ with real asymptotic lines

Abstract

The geometrically defined wide class of time-like surfaces in ${\mathbb{R}}^3$, admitting real asymptotic lines is considered. A fundamental theorem of Bonnet-type is obtained for these surfaces. It states that a surface in this class is determined (up to a motion) by four invariant functions, satisfying some natural PDEs. Then canonical parameters are defined for these surfaces and it is proved that such a surface is determined (up to a motion) in canonical parameters with only two invariant functions (which in particular can be the Gauss and the mean curvature), satisfying a partial differential equation, equivalent to the Gauss equation.

13 pages