Fubini model

A model of the manifold of lines in a three-dimensional elliptic space $ S _ {3} $ on a pair of two-dimensional elliptic planes $ S _ {2} $. Pairs of mutually-polar lines in $ S _ {3} $ are represented in a one-to-one manner by pairs of diametrically-opposite points of two spheres of unit radius in the Euclidean space $ \mathbf R ^ {3} $. When one identifies diametrically-opposite points, one obtains a one-to-one representation of pairs of polar lines in $ S _ {3} $ by points of two elliptic planes $ S _ {2} $. The manifold of pairs of polar lines is homeomorphic to the topological product of these two planes $ S _ {2} $. A motion of $ S _ {3} $ is represented in the Fubini model by independent motions of the two planes $ S _ {2} $: Every connected group of motions of $ S _ {3} $ is isomorphic to the direct product of two groups of motions of $ S _ {2} $; the group of motions of $ S _ {3} $ is isomorphic to the direct product of the two groups of motions of the pair of planes $ S _ {2} $.

A Fubini model can also be constructed for a three-dimensional hyperbolic space $ {} ^ {2} S _ {3} $. In this case one uses the Plücker interpretation of the projective space $ P _ {3} $ in $ {} ^ {3} S _ {5} $. The group of motions of $ {} ^ {2} S _ {3} $ is isomorphic to the direct product of the two groups of motions of $ {} ^ {1} S _ {2} $; it is represented in the Plücker model by the subgroup of the group of motions of $ {} ^ {3} S _ {5} $ consisting of the motions that map two mutually-polar hyperbolic $ 2 $- planes into themselves. The lines of intersection of these planes with the absolute of $ {} ^ {3} S _ {5} $ form a family of straight line generators of a ruled quadric. The manifold of pairs of polar elliptic lines in $ {} ^ {2} S _ {3} $ is homeomorphic to the topological product of proper domains of the two planes $ {} ^ {1} S _ {2} $, that is, to the topological product of two discs, and the manifold of pairs of polar hyperbolic lines is homeomorphic to the product of ideal domains of the two planes $ {} ^ {1} S _ {2} $, that is, the topological product of two Möbius strips.

The model was proposed by G. Fubini [1].

References

[1]	G. Fubini, Ann. Scuola Norm. Sup. Pisa , 9 (1904) pp. 1–74
[2]	B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian)

Comments

The constructions of Fubini follow easily from the decomposition of the orthogonal group $ \mathop{\rm SO} ( 4) $ given by the quaternions: see [a1].

References

[a1]	M. Berger, "Geometry" , II , Springer (1987) pp. Sect. 8.9