Lobachevskii space

A space whose geometry is defined by the axioms of Lobachevskii geometry. In a wider sense a Lobachevskii space is a non-Euclidean hyperbolic space whose definition is connected with concepts of the geometry of a pseudo-Euclidean space. Let $ {} ^ {1} R _ {n+1} $ be the Lorentz–Minkowskian $ ( n + 1 ) $- space with one time-like direction. A sphere of time-like radius is analogous to a hyperboloid of two sheets. One sheet (say the "future" sheet) is isometric to a Lobachevskii $ n $- space $ {} ^ {1} S _ {n} $. This definition of a Lobachevskii space makes it possible to include this space in the projective classification of non-Euclidean spaces. The space $ {} ^ {1} S _ {n} $ in the projective space $ P _ {n} $ is represented by the interior of an oval ( $ n- 1 $)- quadric that is the intersection of an $ n $- sphere of time-like radius with the hyperplane at infinity of the space $ {} ^ {1} R _ {n+1}$ that completes this space to the projective space $ P _ {n+1}$. The points of the oval ( $ n- 1 $)- quadric are the points at infinity of $ {} ^ {1} S _ {n} $, that is, the quadric is the absolute of this space. The outside of this quadric, which completes $ {} ^ {1} S _ {n} $ to the complete space $ P _ {n} $, is called the ideal domain of $ {} ^ {1} S _ {n} $. This interpretation is called the Cayley–Klein projective interpretation. It can also be obtained by projecting an $ n $- sphere of time-like radius in $ {} ^ {1} R _ {n+1}$ from its centre to a tangent $ n $- plane, which is a Euclidean $ n $- space; the space $ {} ^ {1} S _ {n} $ is represented by the inside of an $ n $- ball in this $ n $- plane, and the boundary of the $ n $- ball is the absolute of $ {} ^ {1} S _ {n} $( the latter interpretation of $ {} ^ {1} S _ {n} $ in the Euclidean space $ R _ {n} $ is sometimes called the Beltrami–Klein interpretation).

The projective interpretation of Lobachevskii $ 3 $- space makes it possible to verify the axioms of Lobachevskii geometry, to give a representation of all figures of this geometry, and to establish their properties; in particular, in this interpretation it is easy to establish the geometrical properties of the Lobachevskii $ 2 $- plane that follow from the axioms of Lobachevskii geometry.

When the hyperbolic space $ {} ^ {1} S _ {n} $ is imbedded in the projective space $ P _ {n} $, an $ m $- flat $ ( m < n ) $ is said to be proper if it intersects the absolute in an $ ( m - 1 ) $- quadric; an $ m $- flat that touches the absolute is isotropic; and an $ m $- flat that does not intersect the absolute is ideal. The poles of proper hyperplanes are ideal points, and the proper points are the poles of ideal hyperplanes. More generally, the polar ( $ n - m - 1 $)- flats of proper $ m $- flats of the Lobachevskii space $ {} ^ {1} S _ {n} $ are ideal ( $ n- m- 1 $)- flats, and the polar ( $ n- m- 1 $)- flats of ideal $ m $- flats are proper ( $ n- m- 1 $)- flats.

In the space $ {} ^ {1} S _ {n} $, as coordinates of a point $ X $ one takes the components of the corresponding vector $ \mathbf x $ of this point in $ {} ^ {1} R _ {n+1} $. These Weierstrass coordinates must satisfy the condition

$$ ( x ^ {0} ) ^ {2} - \sum _ { t } ( x ^ {t} ) ^ {2} = 1 ,\ t > 0 . $$

In $ {} ^ {1} S _ {n} $ one may introduce instead coordinates $ u ^ {1} \dots u ^ {n} $, analogous to spherical polar coordinates, which are connected with the coordinates $ x ^ {t} $ by the relations

$$ x ^ {0} = \cosh u ^ {1} \cosh u ^ {2} \dots \cosh u ^ {n} , $$

$$ x ^ {1} = \sinh u ^ {1} \cosh u ^ {2} \dots \cosh u ^ {n} , $$

$$ {\dots \dots \dots } $$

$$ x ^ {t} = \sinh u ^ {t} \cosh u ^ {t+1} \dots \cosh u ^ {n} , $$

$$ {\dots \dots \dots \dots } $$

$$ x ^ {n} = \sinh u ^ {n} , $$

The distance $ \delta $ between two points of $ {} ^ {1} S _ {n} $ is then defined, in terms of their Weierstrass coordinates, by

$$ \cosh \delta = x ^ {0} y ^ {0} - \sum x ^ {t} y ^ {t} . $$

The angle $ \phi $( $ \leq \pi / 2 $) between two intersecting hyperplanes $ X _ {0} x ^ {0} + \sum X _ {t} x _ {t} = 0 $ and $ Y _ {0} x ^ {0} + \sum Y _ {t} x ^ {t} = 0 $( where $ - X _ {0} ^ {2} + \sum X _ {t} ^ {2} = 1 = - Y _ {0} ^ {2} + \sum Y _ {t} ^ {2} $) can be identified with the space-like distance between their poles $ ( - X _ {0} , X _ {1} \dots X _ {n} ) $ and $ ( - Y _ {0} , Y _ {1} \dots Y _ {n} ) $, and is thus given by

$$ \cos \phi = | - X _ {0} Y _ {0} + \sum X _ {t} Y _ {t} | . $$

Similarly, the distance $ \delta $ between two ultraparallel hyperplanes is given by

$$ \cosh \delta = | - X _ {0} Y _ {0} + \sum X _ {t} Y _ {t} | . $$

The distance between points and the values of the angles between planes admit expressions in terms of the cross ratios (cf. Cross ratio) of points, using points of the absolute.

In the Lobachevskii space $ {} ^ {1} S _ {n} $ one can define spheres (balls), equi-distant surfaces, horospheres (horocycles for $ n= 2 $, cf. Horocycle), $ m $- simplexes, etc.

The classification of motions of the Lobachevskii space $ {} ^ {1} S _ {n} $ as collineations that take points of the absolute (oval quadric) into itself reduces to the classification of motions fixing one point of the Lorentz–Minkowskian space $ {} ^ {1} R _ {n+1} $ without interchanging "future" and "past" . (This is a Lie group.) In order to specify a motion of $ {} ^ {1} S _ {n} $ it is sufficient to give the images of $ n+ 1 $ points that do not lie in one hyperplane.

There are several conformal interpretations of a Lobachevskii space, one of which is the Poincaré model. It is also possible to have a conformal interpretation of the space on one of its hyperplanes. Apart from these there are interpretations in complex spaces. In particular, for the space $ {} ^ {1} S _ {n} $ one can construct the Kotel'nikov interpretation of manifolds of lines.

By means of projective interpretations, quadrics in $ {} ^ {1} S _ {n} $, and particularly in the $ 2 $- plane $ {} ^ {1} S _ {2} $, can be classified more completely.

The space $ {} ^ {1} S _ {n} $ is a Riemannian $ n $- space of constant negative curvature $ - 1 / \sigma ^ {2} $, where $ \sigma i $ is the radius of curvature of the space. The geometry of a Lobachevskii space in sufficiently small neighbourhoods of points is close to the geometry of the Euclidean space of the same dimension.

In the large, the space $ {} ^ {1} S _ {n} $ is homeomorphic to the space $ R _ {n} $; it extends indefinitely in all directions. Any $ m $- flat of $ {} ^ {1} S _ {n} $, $ m < n $, is a space $ {} ^ {1} S _ {m} $. Also, straight lines of $ {} ^ {1} S _ {n} $ are geodesics, and $ m $- flats are totally geodesic $ m $- surfaces of this space.

In the projective classification of metrics of non-Euclidean spaces a Lobachevskii space is also classified with respect to the metrics on lines, pencils of planes and $ m $- flats. In particular, on a $ 2 $- flat of a Lobachevskii space the projective metric on a line is hyperbolic, and the metric in pencils of lines is elliptic.

References

[1]	N.V. Efimov, "Höhere Geometrie" , Deutsch. Verlag Wissenschaft. (1960) (Translated from Russian)
[2]	F. Klein, "Vorlesungen über Nicht-Euklidische Geometrie" , Springer (1928)
[3]	V.F. Kagan, "Foundations of geometry" , 1–2 , Moscow-Leningrad (1949–1956) (In Russian)
[4]	B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian)
[a1]	B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) pp. Chapt. 6 (Translated from Russian)
[a2]	A.A. Robb, "Geometry of time and space" , Cambridge Univ. Press (1936) pp. 406
[a3]	H.S.M. Coxeter, "Non-Euclidean geometry" , Univ. Toronto Press (1965) pp. 209
[a4]	M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French)
[a5]	H. Busemann, P.J. Kelly, "Projective geometry and projective metrics" , Acad. Press (1953)