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I've been reading Fuchs and Tabachnikov's Mathematical Omnibus from 2007, freely available here: http://www.math.psu.edu/tabachni/Books/taba.pdf

In chapter 16, they prove that any doubly ruled surface that is not planar has to be either a one sheeted hyperboloid or a hyperbolic paraboloid. This is culminated in section 16.8, although you require 16.5 through 16.7 for it.

The thing is, this is only a local result. In a remark below the theorem they say, though, that if you add the hypothesis of connection, you can get a global result. But I can't see how to get this extension.

If you have a regular surface that is doubly ruled and you know already that locally it is one of such surfaces, and you know it's connected, can you deduce that globally it has to be one of them?

I've thought maybe the fact that it is ruled (and hence contains full straight lines) would allow us to get that extension, but I can't see how.

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