Totally quasi-umbilic timelike surfaces in $\mathbb{R}^{1,2}$

For a regular surface in Euclidean space $\mathbb{R}^3$, umbilic points are precisely the points where the Gauss and mean curvatures $K$ and $H$ satisfy $H^2 = K$; moreover, it is well-known that the only totally umbilic surfaces in $\mathbb{R}^3$ are planes and spheres. But for timelike surfaces in Minkowski space $\mathbb{R}^{1,2}$, it is possible to have $H^2 = K$ at a non-umbilic point; we call such points $quasi-umbilic$, and we give a complete classification of totally quasi-umbilic timelike surfaces in $\mathbb{R}^{1,2}$.

Keywords

Timelike surfaces, quasi-umbilic, method of moving frames

2010 Mathematics Subject Classification

Primary 51B20, 53C42. Secondary 53A55, 53C10.

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Published 6 April 2012