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?