Namespaces
Variants
Actions

Difference between revisions of "Subprojective space"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
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).
+
<!--
 +
s0910101.png
 +
$#A+1 = 61 n = 0
 +
$#C+1 = 61 : ~/encyclopedia/old_files/data/S091/S.0901010 Subprojective space
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
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
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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>
+
One of the generalizations of spaces of constant curvature (of projective space). One defines a  $  k $-
 +
fold [[Projective space|projective space]] with an [[Affine connection|affine connection]] and expresses its geodesic lines in some coordinate system by a system of  $  ( n- 1) $
 +
equations of which exactly  $  k $
 +
are linear. When  $  k= n- 2 $,
 +
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).
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101010.png" /> are the Kronecker symbols and
+
Let  $  A _ {n} $
 +
be an  $  n $-
 +
dimensional subprojective space with a torsion-free affine connection. With respect to a projective coordinate system  $  x  ^ {i} $
 +
of  $  A _ {n} $,
 +
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/s09101011.png" /></td> </tr></table>
+
$$
 +
\Gamma _ {jk}  ^ {i}  = x  ^ {i} f _ {jk} + \delta _ {k}  ^ {i} p _ {j} + \delta _ {j}  ^ {i} p _ {k} ,\ \
 +
f _ {jk}  = f _ {kj} ,
 +
$$
  
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.
+
where  $  \delta _ {j}  ^ {i} $
 +
are the Kronecker symbols and
  
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
+
$$
 +
p _ {k}  =
 +
\frac{1}{2}
 +
( \Gamma _ {kk}  ^ {k} - f _ {kk} x  ^ {k} ).
 +
$$
  
<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>
+
In this coordinate system, all two-dimensional Euclidean planes on which the geodesic lines of  $  A _ {n} $
 +
are situated pass through the coordinate origin.
  
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:
+
In general, in a subprojective space $  A _ {n} $
 +
there exists a canonical coordinate system  $  x  ^ {i} $
 +
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/s09101017.png" /></td> </tr></table>
+
$$
 +
\Gamma _ {jk}  ^ {i}  = x  ^ {i} f _ {jk} .
 +
$$
 +
 
 +
A Riemannian subprojective space  $  V _ {n} $
 +
is defined in the same way; its metric reduces to one of three possible forms:
 +
 
 +
$$
 +
ds  ^ {2}  = g _ {\alpha \gamma }  dx  ^  \alpha  dx  ^  \gamma  ,
 +
$$
  
 
where
 
where
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101018.png" />,
+
1) $  g _ {ik} = \partial  _ {i} \tau \partial  _ {k} \tau + ( \lambda / \theta ) \partial  _ {ik} ( v/ \lambda ) $,
  
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.
+
2) $  g _ {ik} = \partial  _ {i} \tau \partial  _ {k} \tau + c( \lambda / \theta ) \partial  _ {ik} \lambda $,  
 +
$  c = \textrm{ const } \neq 0 $;  
 +
here $  \theta $
 +
is an arbitrary function in the coordinates $  x  ^ {i} $,  
 +
$  \theta $
 +
is a function of the variable $  \lambda / \theta $,  
 +
$  v $
 +
is a quadratic form in the $  x  ^ {i} $,  
 +
$  \lambda $
 +
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
 
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>
+
$$
 +
\theta  ds  ^ {2}  = a _ {\alpha \beta }  dx  ^  \alpha  dx  ^  \beta  + 2 \
 +
d \lambda  dx  ^ {n-} 1 + 2  dv  dx  ^ {n} ,
 +
$$
 +
 
 +
where  $  a _ {\alpha \beta }  = \textrm{ const } $,
 +
$  \mathop{\rm det}  | a _ {\alpha \beta }  | \neq 0 $,
 +
$  \theta $
 +
is a homogeneous function of degree one in  $  x  ^ {n-} 1 $
 +
and  $  x  ^ {n} $,
 +
and  $  \lambda $
 +
and  $  v $
 +
are functions related by
  
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>
+
\frac{1}{2}
 +
a _ {\alpha \beta }  x  ^  \alpha  x  ^  \beta  + \lambda x  ^ {n-} 1 + vx
 +
^ {n}  = 0,\ \
 +
\alpha , \beta = 1 \dots n- 2 .
 +
$$
  
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.
+
The functions $  \lambda $
 +
and $  v $
 +
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" />:
+
All three cases can be reduced to a uniform expression by the choice of coordinates $  z  ^ {i} $:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091010/s09101040.png" />,
+
1) $  ds  ^ {2} = e ^ {- 2 \mu ( z  ^ {1} ) } ( e _ {1}  dz ^ {1  ^ {2} } + \dots + e _ {n}  dz ^ {n  ^ {2} } ) $,
  
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" />,
+
2) $  ds  ^ {2} = e ^ {- 2 \mu ( z) } \sum _ {i=} 1  ^ {n} ( e _ {i}  dz ^ {i  ^ {2} } ) $,  
 +
$  z = ( \sum _ {i=} 1  ^ {n} e _ {i}  dz ^ {i  ^ {2} } )  ^ {1/2} $,
  
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.
+
3) $  ds  ^ {2} = e ^ {- 2 \mu ( z  ^ {1} ) } ( 2  dz  ^ {1}  dz  ^ {2} + \sum _ {i=} 3  ^ {n} e _ {i}  dz ^ {i  ^ {2} } ) $,  
 +
$  i = 3 \dots n $
 +
$  ( e _ {i} = \textrm{ const } ) $.  
 +
All Riemannian subprojective spaces $  V _ {n} $
 +
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).
+
Tensor criteria for conformal Euclidean subprojective spaces exist, distinguishing them from the class of all conformal Euclidean spaces. Every subprojective space $  V _ {n} $(
 +
apart from the case 3)) can be realized as a hypersurface in a Euclidean space $  E _ {n+} 1 $
 +
in the case 1), or as a hypersurface of rotation in $  E _ {n+} 1 $
 +
in the case 2). The converse is also true: Every hypersurface of rotation around a non-isotropic axis in a Euclidean space $  E _ {n+} 1 $,  
 +
$  n > 2 $,  
 +
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).
+
Motions in Riemannian subprojective spaces are defined in the usual way. The subprojective spaces $  V _ {n} $
 +
are characterized by the fact that if $  V _ {n} $
 +
is not a space of constant curvature, then it permits a maximal intransitive group of motions of order $  n( n- 1)/2 $,  
 +
and, conversely, every Riemannian space $  V _ {n} $
 +
that permits a maximal intransitive group of order $  n( n- 1)/2 $
 +
is a subprojective space. Riemannian subprojective spaces $  V _ {n} $
 +
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).
+
The concept of a subprojective space permits the following generalizations: A space $  A _ {n} $
 +
with an affine connection is called a generalized subprojective space if its geodesic lines lie in Euclidean planes $  E _ {r+} 1 $,  
 +
$  1 \leq  r \leq  n- 2 $,  
 +
that pass through a fixed plane $  E _ {r-} 1 $(
 +
at a finite or infinite distance).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.F. Kagan,  "Subprojective spaces" , Moscow  (1961)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.F. Kagan,  "Subprojective spaces" , Moscow  (1961)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Revision as of 08:24, 6 June 2020


One of the generalizations of spaces of constant curvature (of projective space). One defines a $ k $- fold projective space with an affine connection and expresses its geodesic lines in some coordinate system by a system of $ ( n- 1) $ equations of which exactly $ k $ are linear. When $ k= n- 2 $, 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 $ A _ {n} $ be an $ n $- dimensional subprojective space with a torsion-free affine connection. With respect to a projective coordinate system $ x ^ {i} $ of $ A _ {n} $, the coefficients of the connection take the form

$$ \Gamma _ {jk} ^ {i} = x ^ {i} f _ {jk} + \delta _ {k} ^ {i} p _ {j} + \delta _ {j} ^ {i} p _ {k} ,\ \ f _ {jk} = f _ {kj} , $$

where $ \delta _ {j} ^ {i} $ are the Kronecker symbols and

$$ p _ {k} = \frac{1}{2} ( \Gamma _ {kk} ^ {k} - f _ {kk} x ^ {k} ). $$

In this coordinate system, all two-dimensional Euclidean planes on which the geodesic lines of $ A _ {n} $ are situated pass through the coordinate origin.

In general, in a subprojective space $ A _ {n} $ there exists a canonical coordinate system $ x ^ {i} $ in which the coefficients of the connection take the simplest form

$$ \Gamma _ {jk} ^ {i} = x ^ {i} f _ {jk} . $$

A Riemannian subprojective space $ V _ {n} $ is defined in the same way; its metric reduces to one of three possible forms:

$$ ds ^ {2} = g _ {\alpha \gamma } dx ^ \alpha dx ^ \gamma , $$

where

1) $ g _ {ik} = \partial _ {i} \tau \partial _ {k} \tau + ( \lambda / \theta ) \partial _ {ik} ( v/ \lambda ) $,

2) $ g _ {ik} = \partial _ {i} \tau \partial _ {k} \tau + c( \lambda / \theta ) \partial _ {ik} \lambda $, $ c = \textrm{ const } \neq 0 $; here $ \theta $ is an arbitrary function in the coordinates $ x ^ {i} $, $ \theta $ is a function of the variable $ \lambda / \theta $, $ v $ is a quadratic form in the $ x ^ {i} $, $ \lambda $ 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

$$ \theta ds ^ {2} = a _ {\alpha \beta } dx ^ \alpha dx ^ \beta + 2 \ d \lambda dx ^ {n-} 1 + 2 dv dx ^ {n} , $$

where $ a _ {\alpha \beta } = \textrm{ const } $, $ \mathop{\rm det} | a _ {\alpha \beta } | \neq 0 $, $ \theta $ is a homogeneous function of degree one in $ x ^ {n-} 1 $ and $ x ^ {n} $, and $ \lambda $ and $ v $ are functions related by

$$ \frac{1}{2} a _ {\alpha \beta } x ^ \alpha x ^ \beta + \lambda x ^ {n-} 1 + vx ^ {n} = 0,\ \ \alpha , \beta = 1 \dots n- 2 . $$

The functions $ \lambda $ and $ v $ are not homogeneous of the first degree.

All three cases can be reduced to a uniform expression by the choice of coordinates $ z ^ {i} $:

1) $ ds ^ {2} = e ^ {- 2 \mu ( z ^ {1} ) } ( e _ {1} dz ^ {1 ^ {2} } + \dots + e _ {n} dz ^ {n ^ {2} } ) $,

2) $ ds ^ {2} = e ^ {- 2 \mu ( z) } \sum _ {i=} 1 ^ {n} ( e _ {i} dz ^ {i ^ {2} } ) $, $ z = ( \sum _ {i=} 1 ^ {n} e _ {i} dz ^ {i ^ {2} } ) ^ {1/2} $,

3) $ ds ^ {2} = e ^ {- 2 \mu ( z ^ {1} ) } ( 2 dz ^ {1} dz ^ {2} + \sum _ {i=} 3 ^ {n} e _ {i} dz ^ {i ^ {2} } ) $, $ i = 3 \dots n $ $ ( e _ {i} = \textrm{ const } ) $. All Riemannian subprojective spaces $ V _ {n} $ 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 $ V _ {n} $( apart from the case 3)) can be realized as a hypersurface in a Euclidean space $ E _ {n+} 1 $ in the case 1), or as a hypersurface of rotation in $ E _ {n+} 1 $ in the case 2). The converse is also true: Every hypersurface of rotation around a non-isotropic axis in a Euclidean space $ E _ {n+} 1 $, $ n > 2 $, 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 $ V _ {n} $ are characterized by the fact that if $ V _ {n} $ is not a space of constant curvature, then it permits a maximal intransitive group of motions of order $ n( n- 1)/2 $, and, conversely, every Riemannian space $ V _ {n} $ that permits a maximal intransitive group of order $ n( n- 1)/2 $ is a subprojective space. Riemannian subprojective spaces $ V _ {n} $ 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 $ A _ {n} $ with an affine connection is called a generalized subprojective space if its geodesic lines lie in Euclidean planes $ E _ {r+} 1 $, $ 1 \leq r \leq n- 2 $, that pass through a fixed plane $ E _ {r-} 1 $( at a finite or infinite distance).

References

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

Comments

References

[a1] J.A. Schouten, "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer (1954) (Translated from German)
How to Cite This Entry:
Subprojective space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subprojective_space&oldid=48899
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article