Umbilics of Surfaces in the Lorentz–Minkowski 3-Space

Abstract

In this paper, we prove several fundamental properties on umbilics of a space-like or time-like surface in the Lorentz–Minkowski space {{mathbb {L}}}^3. In particular, we show that the local behavior of the curvature line flows of the germ of a space-like surface in {{mathbb {L}}}^3 is essentially the same as that of a surface in Euclidean space. As a consequence, for each positive integer m, there exists a germ of a space-like surface with an isolated C^{infty }-umbilic (resp. C^1-umbilic) of index (3-m)/2 (resp. 1+m/2). We also show that the indices of isolated umbilics of time-like surfaces in {{mathbb {L}}}^3 that are not the accumulation points of quasi-umbilics are always equal to zero. On the other hand, when quasi-umbilics accumulate, there exist countably many germs of time-like surfaces which admit an isolated umbilic with non-zero indices.