A well-known theorem of classical surface theory states that the only doubly ruled surfaces in Euclidean 3-space are planes, 1-sheeted hyperboloids and hyperbolic paraboloids. There are a number of proofs - as discussed on MO at Proofs for doubly ruled surfaces.

Is there a similar result for surfaces ruled by geodesics in hyperbolic 3-space ${\mathbb H}^3$? In particular, what are the doubly ruled surfaces in ${\mathbb H}^3$ that are not totally geodesic?

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    $\begingroup$ Look at the projective mode. $\endgroup$ Feb 21 at 21:36
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    $\begingroup$ A bit more detail to Anton’s comment: in the Klein/projective model of hyperbolic 3-space, geodesic as are straight lines in projective space. The only doubly ruled surfaces in projective 3-space are the projective planes and Clifford tori (in fact, there is a Clifford torus going through any three skew projective lines). This is also a purely local result, so any doubly ruled surface in hyperbolic space will be an intersection of a doubly ruled surface in projective space with hyperbolic space. $\endgroup$
    – Ian Agol
    Feb 22 at 16:14
  • $\begingroup$ Thanks for the replies. I’m guessing that it hasn’t been considered in the literature. It’s of interest because there is a link between doubly ruled surfaces in ${\mathbb R}^3$ and a mean value theorem that holds for solutions of the 4d ultra-hyperbolic equation. This can be extended to ${\mathbb H}^3$ by linking the projective model of the 3-space to the conformal model for its geodesic space and the link between doubly ruled surfaces and the mean value theorem are maintained. Perhaps knowing precisely the doubly ruled surfaces may help with tomography in hyperbolic 3-space. $\endgroup$ Feb 27 at 20:20


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