Co-pseudo-Euclidean space

A space obtained from a pseudo-Euclidean space by applying the duality principle for the projective space of the same dimension. It is denoted by $ {} ^ {l} R _ {n} ^ {*} $. A co-pseudo-Euclidean space is a space with a projective metric, which may be introduced, following the general definition of projective metrics, by specifying the absolute in the projective space of corresponding dimension. A projective metric of a pseudo-Euclidean space $ {} ^ {l} R _ {n} $ is defined by an absolute, which consists of an $ ( n - 1) $- hyperplane and a real $ ( n - 2) $- quadric in that hyperplane; hence a projective metric of the dual co-pseudo-Euclidean space $ {} ^ {l} R _ {n} ^ {*} $ is defined by the dual of the absolute: a real (absolute) second-order cone with a point vertex, the latter being taken as an absolute point. The absolute cone divides $ {} ^ {l} R _ {n} ^ {*} $ into two domains in which the scalar product of a vector with itself is of fixed sign. These domains represent the manifolds of the corresponding hyperplanes of the pseudo-Euclidean space dual to the given $ {} ^ {l} R _ {n} ^ {*} $. Isotropic hyperplanes of the pseudo-Euclidean space represent points of the absolute in $ {} ^ {l} R _ {n} ^ {*} $. Depending on their positions relative to the absolute cone and absolute point (vertex), one distinguishes four types of straight lines: elliptic lines, which intersect the absolute cone at two conjugate complex points; hyperbolic lines, which intersect the absolute cone at two real points; parabolic lines, which pass through the absolute point; and isotropic lines, which are parabolic lines tangent to the absolute cone.

According to the duality principle for a pseudo-Euclidean space, straight lines of the first two types are represented by bundles of hyperplanes intersecting, respectively, in Euclidean and pseudo-Euclidean $ ( n - 2 ) $- hyperplanes; parabolic lines are represented by bundles of parallel hyperplanes; and isotropic lines are represented by bundles of hyperplanes intersecting in isotropic $ ( n - 2) $- hyperplanes.

A distance between the points of a co-pseudo-Euclidean space $ {} ^ {l} R _ {n} ^ {*} $ is defined, in view of the dual nature of this space, with reference to the corresponding pseudo-Euclidean space $ {} ^ {l} R _ {n} $. Let $ X $ and $ Y $ be points in $ {} ^ {l} R _ {n} ^ {*} $, to which correspond planes in $ {} ^ {l} R _ {n} $ with normal equations

$$ ( \mathbf u , \mathbf x ) + u _ {0} = 0; \ \ ( \mathbf v , \mathbf y ) + v _ {0} = 0 , $$

so that

$$ x ^ {0} = \rho u _ {0} ,\ \ x ^ {i} = \rho u _ {i} ; \ \ y ^ {0} = \rho v _ {0} ,\ \ y ^ {i} = \rho v _ {i} ; $$

and moreover

$$ ( \mathbf u E \mathbf u ) = \pm 1,\ \ ( \mathbf v E \mathbf v ) = \pm 1, $$

where $ E $ is the linear operator defining the scalar product in $ {} ^ {l} R _ {n} $. The distance $ \delta $ between two points $ X ( x ^ {0} , x) $ and $ Y ( y ^ {0} , y) $ is defined by

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

where $ \rho $ is an imaginary or real number, called the radius of curvature of $ {} ^ {l} R _ {n} ^ {*} $.

If the hyperplanes in $ {} ^ {l} R _ {n} $ corresponding to the points $ X $ and $ Y $ are parallel, the distance between the points may be defined as the distance $ d $ between these hyperplanes:

$$ d = | y ^ {0} - x ^ {0} | \ \ \textrm{ when } \ ( \mathbf x E \mathbf x ) > 0, $$

$$ d = i | y ^ {0} - x ^ {0} | \ \textrm{ when } ( \mathbf x E \mathbf x ) < 0. $$

A geometry on the different types of straight lines in $ {} ^ {l} R _ {n} ^ {*} $ is defined by the type of the projective metric on them. Thus, a hyperbolic line carries a projective metric of a hyperbolic space; etc.

The angle between two hyperplanes in a co-pseudo-Euclidean space $ {} ^ {l} R _ {n} ^ {*} $ is defined as the normalized distance between the corresponding (dual) points of the pseudo-Euclidean space $ {} ^ {l} R _ {n} $. This angle is equal to the normalized distance between the points of these hyperplanes which are the poles of their $ ( n - 2) $- hyperplane of intersection relative to the quadrics cut out by the absolute cone on the hyperplanes. In all cases the angle defined is that not containing the absolute point. In particular, in $ {} ^ {1} R _ {2} ^ {*} $ the distance between two straight lines is equal to the normalized distance between the two points of these lines which, together with their point of intersection, harmonically separate their points of intersection with the absolute lines.

The transformations of $ {} ^ {l} R _ {n} ^ {*} $ induced by the motions of the dual pseudo-Euclidean space are called the motions of $ {} ^ {l} R _ {n} ^ {*} $. The motions of $ {} ^ {l} R _ {n} ^ {*} $ are described (as are those of $ {} ^ {l} R _ {n} $) by pseudo-orthogonal operators of index $ l $.

In view of the duality governing the properties of the spaces $ {} ^ {1} R _ {2} ^ {*} $ and $ {} ^ {1} R _ {2} $, the geometry of the plane $ {} ^ {1} R _ {2} ^ {*} $ may be derived from that of the plane $ {} ^ {1} R _ {2} $; this applies, in particular, to the geometry of a triangle in $ {} ^ {1} R _ {2} ^ {*} $. The fundamental relations between the lengths of the sides and the magnitudes of angles are expressed in formulas analogous to those for triangles in a co-Euclidean space (see Co-Euclidean space), except that the trigonometric functions appearing in the latter must be replaced by the corresponding hyperbolic functions.

Let $ ABC $ be a triangle whose inner angle $ B $ contains an absolute point. Then the following relations hold for the sides $ a, b, c $ and the values of the angles $ \widehat{A} , \widehat{B} , \widehat{C} $:

$$ b = a + c, $$

$$ \frac{\widehat{A} }{\sinh ( a / \rho ) } = \frac{\widehat{B} }{\sinh ( b / \rho ) } = \frac{\widehat{C} }{\sinh ( c / \rho ) } , $$

$$ \widehat{A} {} ^ {2} = \widehat{B} {} ^ {2} + \widehat{C} {} ^ {2} - 2 \widehat{B} \widehat{C} \cosh \frac{a} \rho . $$

The distance metric on the plane $ {} ^ {1} R _ {2} ^ {*} $ is a hyperbolic projective metric, while the angle metric is parabolic. In the $ 3 $- space $ {} ^ {1} R _ {3} ^ {*} $, the projective metric on a plane is hyperbolic, while that on a straight line is elliptic; the metric in bundles of planes is parabolic (pseudo-Euclidean).

Co-pseudo-Euclidean spaces constitute a limiting case of hyperbolic spaces.

References

[1]	B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian)
[2]	I.M. Yaglom, B.A. Rozenfel'd, E.U. Yasinskaya, "Projective metrics" Uspekhi Mat. Nauk , 19 : 5 (1964) pp. 51–113 (In Russian)

Comments

A co-pseudo-Euclidean space is also called a dual pseudo-Euclidean space.