Umbilics of Surfaces in the Lorentz–Minkowski 3-Space

Abstract

AbstractIn this paper, we prove several fundamental properties on umbilics of a space-like or time-like surface in the Lorentz–Minkowski space $${{\mathbb {L}}}^3$$ L 3 . In particular, we show that the local behavior of the curvature line flows of the germ of a space-like surface in $${{\mathbb {L}}}^3$$ 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 $$C^{\infty }$$ C ∞ -umbilic (resp. $$C^1$$ C 1 -umbilic) of index $$(3-m)/2$$ ( 3 - m ) / 2 (resp. $$1+m/2$$ 1 + m / 2 ). We also show that the indices of isolated umbilics of time-like surfaces in $${{\mathbb {L}}}^3$$ 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.