Quasi-elliptic space

A projective $ n $- space whose projective metric is defined by an absolute consisting of an imaginary cone (the absolute cone $ Q _ {0} $) with an $ ( n- m - 1 ) $- vertex (the absolute plane $ T _ {0} $) together with an imaginary $ ( n - m - 2 ) $- quadric $ Q _ {1} $ on this $ ( n- m- 1) $- plane (the absolute quadric $ Q _ {1} $); it is denoted by the symbol $ S _ {n} ^ {m} $, $ m < n $. A quasi-elliptic space is of more general projective type in comparison with a Euclidean space and a co-Euclidean space; the metrics of the latter are obtained from those of the former. A quasi-elliptic space is a particular case of a semi-elliptic space. For $ m = 0 $, the absolute cone is a pair of coincident $ ( n- 1) $- planes coinciding with the $ ( n- 1) $- absolute plane $ T _ {0} $, while the absolute coincides with the absolute of Euclidean $ n $- space. For $ m= 1 $, the cone $ Q _ {0} $ is a cone with a point vertex and the absolute in this case is the same as that of the co-Euclidean $ n $- space. When $ m = 1 $, the cone $ Q _ {0} $ is a pair of imaginary $ ( n - 1 ) $- planes. In particular, the cone $ Q _ {0} $ of the quasi-elliptic three-space $ S _ {3} ^ {1} $ is a pair of imaginary two-planes, the line (the $ 1 $- plane) $ T _ {0} $ is the real line of their intersection, while the quadric $ Q _ {1} $ is a pair of imaginary points on $ T _ {0} $.

The distance $ \delta $ between two points $ X $ and $ Y $ is defined in case the line $ X Y $ does not intersect the $ ( n - m - 1 ) $- plane $ T _ {0} $ by the relation

$$ \cos ^ {2} \frac \delta \rho = \ \frac{( \mathbf x ^ {0} E _ {0} \mathbf y ^ {0} ) ^ {2} }{( \mathbf x ^ {0} E _ {0} \mathbf x ^ {0} ) ( \mathbf y ^ {0} E _ {0} \mathbf y ^ {0} ) } , $$

where

$$ \mathbf x ^ {0} = ( x ^ {a} , a \leq m ) ,\ \ \mathbf y ^ {0} = ( y ^ {b} , b \leq m ) $$

are the vectors of the points $ X $ and $ Y $, $ E _ {0} $ is the linear operator defining the scalar product in the space of these vectors and $ \rho $ is a real number; in case $ X Y $ intersects $ T _ {0} $, the distance $ d $ between these points is defined by means of the distance between the vectors of the points $ X $ and $ Y $:

$$ \mathbf x = \mathbf y ^ {1} - \mathbf x ^ {1} , $$

$$ \mathbf x ^ {1} = ( x ^ {a} , a > m ) ,\ \mathbf y ^ {1} = ( y ^ {b} , b > m ) , $$

$$ d ^ {2} = \mathbf a E _ {1} \mathbf a , $$

where $ E _ {1} $ is the linear operator defining the scalar product in the space of these vectors.

The angle between two planes whose $ ( n- 2 ) $- plane of intersection does not intersect the $ ( n - m - 1 ) $- plane $ T _ {0} $ is defined as the (normalized) distance between the corresponding points in the dual quasi-elliptic space $ S _ {n} ^ {n-} m- 1 $, in which the coordinates are numerically equal or proportional to the projective coordinates of the planes in $ S _ {n} ^ {m} $. If the $ ( n - 2 ) $- plane of intersection of two given planes intersects the $ ( n- m- 1 ) $- plane $ T _ {0} $, then the angle between the planes is in this case again defined by the numerical distance. When $ n = 2 $ the angles between the planes are the angles between the lines.

The motions of the quasi-elliptic space $ S _ {n} ^ {m} $ are the collineations of this space that take the cone $ Q _ {0} $ into the plane $ T _ {0} $ and the quadric $ Q _ {1} $ into itself. The group of motions is a Lie group and the motions are described by orthogonal operators. In the quasi-elliptic space $ S _ {2m+} 1 ^ {m} $, which is self-dual, co-motions are defined as the correlations that take each pair of points into two $ 2m $- planes the angle between which is proportional to the distance between the points, and each pair of $ 2 m $- planes into two points the distance between which is proportional to the angle between the planes. The motions and co-motions of $ S _ {2m+} 1 ^ {m} $ form a group, which is a Lie group. The geometry of the $ 2 $- plane $ S _ {2} ^ {0} $ is Euclidean geometry, while the geometry of the $ 2 $- plane $ S _ {2} ^ {1} $ is the same as that of the co-Euclidean plane.

The geometry of the $ 3 $- space $ S _ {3} ^ {1} $ is defined by an elliptic projective metric on lines that is co-Euclidean on planes and Euclidean in bundles of planes. The geometry of the $ 3 $- space $ S _ {3} ^ {0} $ is Euclidean, while the geometry of the $ 3 $- space $ S _ {3} ^ {2} $ is the same as that of the co-Euclidean $ 3 $- space. The space $ S _ {3} ^ {1} $ with radius of curvature $ 1 / 2 $ is isometric to the connected group of motions of the Euclidean $ 2 $- space with a specially introduced metric. The connected group of motions of the quasi-Euclidean space $ S _ {3} ^ {1} $ is isomorphic to the direct product of two connected groups of motions of the Euclidean $ 2 $- plane.

References

[1]	B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian)

References

[a1]	B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian)
[a2]	O. Giering, "Vorlesungen über höhere Geometrie" , Vieweg (1982)