Subprojective space

One of the generalizations of spaces of constant curvature (of projective space). One defines a -fold projective space with an affine connection and expresses its geodesic lines in some coordinate system by a system of equations of which exactly are linear. When , the geodesic lines are planar, and are situated in two-dimensional Euclidean planes, and the space is said to be subprojective if all these two-dimensional Euclidean planes pass through a common point or are parallel in one direction (the common point is infinitely distant).

Let be an -dimensional subprojective space with a torsion-free affine connection. With respect to a projective coordinate system of , the coefficients of the connection take the form

where are the Kronecker symbols and

In this coordinate system, all two-dimensional Euclidean planes on which the geodesic lines of are situated pass through the coordinate origin.

In general, in a subprojective space there exists a canonical coordinate system in which the coefficients of the connection take the simplest form

A Riemannian subprojective space is defined in the same way; its metric reduces to one of three possible forms:

where

1) ,

2) , ; here is an arbitrary function in the coordinates , is a function of the variable , is a quadratic form in the , in 1) is a linear form and in 2) it is the square root of a quadratic form that is not a complete square.

3) The exceptional case

where , , is a homogeneous function of degree one in and , and and are functions related by

The functions and are not homogeneous of the first degree.

All three cases can be reduced to a uniform expression by the choice of coordinates :

1) ,

2) , ,

3) , . All Riemannian subprojective spaces are conformal Euclidean spaces (cf. Conformal Euclidean space). Riemannian subprojective spaces belong to the class of semi-reducible Riemannian spaces and their metrics have a special structure.

Tensor criteria for conformal Euclidean subprojective spaces exist, distinguishing them from the class of all conformal Euclidean spaces. Every subprojective space (apart from the case 3)) can be realized as a hypersurface in a Euclidean space in the case 1), or as a hypersurface of rotation in in the case 2). The converse is also true: Every hypersurface of rotation around a non-isotropic axis in a Euclidean space , , is a Riemannian subprojective space with a metric of the form 2).

Motions in Riemannian subprojective spaces are defined in the usual way. The subprojective spaces are characterized by the fact that if is not a space of constant curvature, then it permits a maximal intransitive group of motions of order , and, conversely, every Riemannian space that permits a maximal intransitive group of order is a subprojective space. Riemannian subprojective spaces are maximally-mobile non-Einsteinian spaces (spaces of constant curvature occupy the same position among the Einstein spaces).

The concept of a subprojective space permits the following generalizations: A space with an affine connection is called a generalized subprojective space if its geodesic lines lie in Euclidean planes , , that pass through a fixed plane (at a finite or infinite distance).

References

[1]	V.F. Kagan, "Subprojective spaces" , Moscow (1961) (In Russian)

References

[a1]	J.A. Schouten, "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer (1954) (Translated from German)