Umbilics of surfaces in the Lorentz-Minkowski 3-space

In this paper, we prove several fundamental properties on umbilics of a space-like or time-like surface in the Lorentz-Minkowski space $L^3$. In particular, we show that the local behavior of the curvature line flows of the germ of a space-like surface in $L^3$ 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 $L^3$ that are not the accumulation points ofquasi-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.