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On MSE there is the question "Why are only singly and doubly ruled non-planar surfaces found? Why not triply ruled?" and in the reference (Fuks,Tabachnikov) given in the comments they characterise doubly ruled surfaces in $\mathbb{R}^3$. In particular the only doubly rules surfaces are the plane, the hyperbolic paraboloid, and the single-sheeted hyperboloid.

Maybe I am missing something obvious but what about the part of the question which asks if there are "3D hyper-surfaces in a 4D+ space allow for triply ruled surfaces?". More generally, are there any $d$-ruled hypersurfaces in $\mathbb{R}^n$ for $d<n$ other than those three mentioned above?

Shorty
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  • In higher dimensions, start by deciding what a ruling should be. – Ted Shifrin Aug 12 '21 at 04:14
  • Should have clarified that by $d$-ruled I meant that there are $d$-many distinct lines passing through each point of the set, that are contained inside the set. – Shorty Aug 12 '21 at 10:52
  • You need to think/read about how rulings will be by higher-dimensional linear spaces when you move to higher dimensions. So there will in general be infinitely many lines through each point. – Ted Shifrin Aug 12 '21 at 18:39
  • You are right - I was not really thinking of ruled surfaces in terms of rulings but by simply the existence of lines property. Would you suggest any references for higher dimensional rulings? – Shorty Aug 13 '21 at 08:55
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    Unfortunately, I do not know any elementary references. If you know about Grassmannians, you can define ruled submanifolds/subvarieties in terms of submanifolds of appropriate Grassmannians (e.g., $k$-planes in $\Bbb R^n$ or $\Bbb P^n$). You can define generalizations of the quadrics in $3$-space by the Segre embeddings of $\Bbb P^m\times\Bbb P^n $ in $\Bbb P^{m+n+1}$. See, for example, Harris's Algebraic Geometry, Griffiths/Harris's "Algebraic Geometry and Local Differential Geometry," or various papers by J.M. Landsberg. – Ted Shifrin Aug 14 '21 at 22:33
  • Thanks, will check those out! – Shorty Aug 16 '21 at 22:39

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