Fubini model

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

A Fubini model can also be constructed for a three-dimensional hyperbolic space . In this case one uses the Plücker interpretation of the projective space in . The group of motions of is isomorphic to the direct product of the two groups of motions of ; it is represented in the Plücker model by the subgroup of the group of motions of consisting of the motions that map two mutually-polar hyperbolic -planes into themselves. The lines of intersection of these planes with the absolute of form a family of straight line generators of a ruled quadric. The manifold of pairs of polar elliptic lines in is homeomorphic to the topological product of proper domains of the two planes , 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 , 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 given by the quaternions: see [a1].

References

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