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 |
Fubini model. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fubini_model&oldid=47003