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Difference between revisions of "Fubini model"

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A model of the manifold of lines in a three-dimensional elliptic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041850/f0418501.png" /> on a pair of two-dimensional elliptic planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041850/f0418502.png" />. Pairs of mutually-polar lines in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041850/f0418503.png" /> are represented in a one-to-one manner by pairs of diametrically-opposite points of two spheres of unit radius in the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041850/f0418504.png" />. When one identifies diametrically-opposite points, one obtains a one-to-one representation of pairs of polar lines in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041850/f0418505.png" /> by points of two elliptic planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041850/f0418506.png" />. The manifold of pairs of polar lines is homeomorphic to the topological product of these two planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041850/f0418507.png" />. A motion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041850/f0418508.png" /> is represented in the Fubini model by independent motions of the two planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041850/f0418509.png" />: Every connected group of motions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041850/f04185010.png" /> is isomorphic to the direct product of two groups of motions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041850/f04185011.png" />; the group of motions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041850/f04185012.png" /> is isomorphic to the direct product of the two groups of motions of the pair of planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041850/f04185013.png" />.
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A Fubini model can also be constructed for a three-dimensional hyperbolic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041850/f04185014.png" />. In this case one uses the [[Plücker interpretation|Plücker interpretation]] of the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041850/f04185015.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041850/f04185016.png" />. The group of motions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041850/f04185017.png" /> is isomorphic to the direct product of the two groups of motions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041850/f04185018.png" />; it is represented in the Plücker model by the subgroup of the group of motions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041850/f04185019.png" /> consisting of the motions that map two mutually-polar hyperbolic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041850/f04185020.png" />-planes into themselves. The lines of intersection of these planes with the absolute of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041850/f04185021.png" /> form a family of straight line generators of a ruled quadric. The manifold of pairs of polar elliptic lines in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041850/f04185022.png" /> is homeomorphic to the topological product of proper domains of the two planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041850/f04185023.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041850/f04185024.png" />, that is, the topological product of two Möbius strips.
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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} $:
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Every connected group of motions of  $  S _ {3} $
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is isomorphic to the direct product of two groups of motions of  $  S _ {2} $;
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the group of motions of  $  S _ {3} $
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is isomorphic to the direct product of the two groups of motions of the pair of planes  $  S _ {2} $.
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A Fubini model can also be constructed for a three-dimensional hyperbolic space $  {}  ^ {2} S _ {3} $.  
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In this case one uses the [[Plücker interpretation|Plücker interpretation]] of the projective space $  P _ {3} $
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in $  {}  ^ {3} S _ {5} $.  
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The group of motions of $  {}  ^ {2} S _ {3} $
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is isomorphic to the direct product of the two groups of motions of $  {}  ^ {1} S _ {2} $;  
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it is represented in the Plücker model by the subgroup of the group of motions of $  {}  ^ {3} S _ {5} $
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consisting of the motions that map two mutually-polar hyperbolic $  2 $-
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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} $,  
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that is, the topological product of two Möbius strips.
  
 
The model was proposed by G. Fubini [[#References|[1]]].
 
The model was proposed by G. Fubini [[#References|[1]]].
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Fubini,  ''Ann. Scuola Norm. Sup. Pisa'' , '''9'''  (1904)  pp. 1–74</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.A. Rozenfel'd,  "Non-Euclidean spaces" , Moscow  (1969)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Fubini,  ''Ann. Scuola Norm. Sup. Pisa'' , '''9'''  (1904)  pp. 1–74</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.A. Rozenfel'd,  "Non-Euclidean spaces" , Moscow  (1969)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The constructions of Fubini follow easily from the decomposition of the orthogonal group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041850/f04185025.png" /> given by the quaternions: see [[#References|[a1]]].
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The constructions of Fubini follow easily from the decomposition of the orthogonal group $  \mathop{\rm SO} ( 4) $
 +
given by the quaternions: see [[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''II''' , Springer  (1987)  pp. Sect. 8.9</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''II''' , Springer  (1987)  pp. Sect. 8.9</TD></TR></table>

Latest revision as of 19:40, 5 June 2020


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
How to Cite This Entry:
Fubini model. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fubini_model&oldid=17226
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article