Invariants and quasi-umbilicity of timelike surfaces in Minkowski space R3,1R3,1

We first describe the numerical invariants and the curvature hyperbola attached to the second fundamental form of a timelike surface in four-dimensional Minkowski space: beside the four natural invariants, a new invariant appears at some special points of the surface, which are said to be quasi-umbilic; at such quasi-umbilic points, the curvature hyperbola degenerates to a line with one point removed. We then study the asymptotic lines on a timelike surface, and characterize the quasi-umbilic points of the surface as the points where the asymptotic directions degenerate to a double lightlike line. We also give an interpretation of the new invariant at a quasi-umbilic point, using the Gauss map of the surface. We finally describe the timelike surfaces which are quasi-umbilic at every point.