View source for Subprojective space

One of the generalizations of spaces of constant curvature (of projective space). One defines a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s0910101.png" />-fold [[Projective space|projective space]] with an [[Affine connection|affine connection]] and expresses its geodesic lines in some coordinate system by a system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s0910102.png" /> equations of which exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s0910103.png" /> are linear. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s0910104.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s0910105.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s0910106.png" />-dimensional subprojective space with a torsion-free affine connection. With respect to a projective coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s0910107.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s0910108.png" />, the coefficients of the connection take the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s0910109.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101010.png" /> are the Kronecker symbols and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101011.png" /></td> </tr></table>

In this coordinate system, all two-dimensional Euclidean planes on which the geodesic lines of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101012.png" /> are situated pass through the coordinate origin.

In general, in a subprojective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101013.png" /> there exists a canonical coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101014.png" /> in which the coefficients of the connection take the simplest form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101015.png" /></td> </tr></table>

A Riemannian subprojective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101016.png" /> is defined in the same way; its metric reduces to one of three possible forms:

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where

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101018.png" />,

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101020.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101021.png" /> is an arbitrary function in the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101023.png" /> is a function of the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101025.png" /> is a quadratic form in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101027.png" /> 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101028.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101031.png" /> is a homogeneous function of degree one in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101033.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101035.png" /> are functions related by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101036.png" /></td> </tr></table>

The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101038.png" /> are not homogeneous of the first degree.

All three cases can be reduced to a uniform expression by the choice of coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101039.png" />:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101040.png" />,

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101042.png" />,

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101044.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101045.png" />. All Riemannian subprojective spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101046.png" /> are conformal Euclidean spaces (cf. [[Conformal Euclidean space|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101047.png" /> (apart from the case 3)) can be realized as a hypersurface in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101048.png" /> in the case 1), or as a hypersurface of rotation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101049.png" /> in the case 2). The converse is also true: Every hypersurface of rotation around a non-isotropic axis in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101051.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101052.png" /> are characterized by the fact that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101053.png" /> is not a space of constant curvature, then it permits a maximal intransitive group of motions of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101054.png" />, and, conversely, every Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101055.png" /> that permits a maximal intransitive group of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101056.png" /> is a subprojective space. Riemannian subprojective spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101057.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101058.png" /> with an affine connection is called a generalized subprojective space if its geodesic lines lie in Euclidean planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101060.png" />, that pass through a fixed plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101061.png" /> (at a finite or infinite distance).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.F. Kagan,   "Subprojective spaces" , Moscow  (1961)  (In Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.A. Schouten,   "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer  (1954)  (Translated from German)</TD></TR></table>