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Fubini model

From Encyclopedia of Mathematics
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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
How to Cite This Entry:
Fubini model. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fubini_model&oldid=47003
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article