Context: Consider the Lorentz-Minkowski space $\Bbb L^3 = (\Bbb R^3, {\rm d}s^2)$, with $${\rm d}s^2 = {\rm d}x^2+{\rm d}y^2 - {\rm d}z^2.$$

Take a differentiable surface $M \subset \Bbb L^3$, orientable and such. We define the mean curvature and the Gaussian curvature by: $$K = \epsilon\,\det(-{\rm d}{\bf N}), \quad H = \frac{\epsilon}{2}\,{\rm tr}(-{\rm d}{\bf N}),$$ where $-{\rm d}{\bf N}$ is the Weingarten map and $\epsilon$ is the causal character of the normal direction to $M$. We have that:

  • $H^2 - \epsilon\,K > 0 \implies -{\rm d}{\bf N}$ diagonalizable.
  • $H^2 - \epsilon\,K < 0 \implies -{\rm d}{\bf N}$ not diagonalizable.
  • $H^2 - \epsilon\,K = 0 \implies -{\rm d}{\bf N}$ diagonalizable for $\epsilon = -1$, and anything can happen for $\epsilon = 1$.

Question: If we consider the De Sitter space $\mathbb{S}^2_1(1)$ we fall in that last case with $\epsilon = 1$ and $-{\rm d}{\bf N}$ is diagonalizable, being a multiple of the identity right off the start. In the text I am using, there is an example for the other situation that uses a lightlike curve and its Frenet Trihedron to build a ${\bf B}$-Scroll. I have no problems with that example, but I would like to see another, that doesn't resort to curves. I've been unable to come up with another example so far. In other words, I want a timelike surface with $H^2 - K = 0$ and $-{\rm d}{\bf N}$ not diagonalizable, that doesn't uses curves in its construction. Help?


I reckon that the original reference for the example you mentioned is the paper by A. Martin, "Indefinite Einstein hypersurfaces with nilpotent shape operators". Hokkaido Math. J. 13 (1984) 241-250 - more precisely, Example 3.1 therein. In this example, the shape operator is nilpotent. More generally, Theorem 4.1 of this paper gives a characterization of indefinite surfaces in Minkowski 3-space with nilpotent shape operator, and it seems that they are constructed in essentially the same way as in Example 3.1: the kernel of the shape operator $-\mathrm{d}\mathbf{N}$ in this case (assumed to be of rank 1 for nontriviality) is an integrable, totally isotropic and parallel 1-dimensional distribution iff $-\mathrm{d}\mathbf{N}$ is nilpotent.


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