Namespaces
Variants
Actions

Difference between revisions of "Projective determination of a metric"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
 +
<!--
 +
p0752201.png
 +
$#A+1 = 91 n = 0
 +
$#C+1 = 91 : ~/encyclopedia/old_files/data/P075/P.0705220 Projective determination of a metric
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
An introduction in subsets of a [[Projective space|projective space]], by methods of projective geometry, of a metric such that these subsets become isomorphic to a Euclidean, hyperbolic or elliptic space. This is achieved by distinguishing in the class of all projective transformations (cf. [[Projective transformation|Projective transformation]]) those transformations that generate in these subsets a group of transformations isomorphic to the corresponding group of motions. The presence of motions allows one  "to lay off"  segments of a straight line from a given point in a given direction, thereby introducing the concept of the length of a segment.
 
An introduction in subsets of a [[Projective space|projective space]], by methods of projective geometry, of a metric such that these subsets become isomorphic to a Euclidean, hyperbolic or elliptic space. This is achieved by distinguishing in the class of all projective transformations (cf. [[Projective transformation|Projective transformation]]) those transformations that generate in these subsets a group of transformations isomorphic to the corresponding group of motions. The presence of motions allows one  "to lay off"  segments of a straight line from a given point in a given direction, thereby introducing the concept of the length of a segment.
  
To obtain the Euclidean determination of a metric in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p0752201.png" />-dimensional projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p0752202.png" />, one should distinguish in this space an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p0752203.png" />-dimensional hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p0752204.png" />, called the ideal hyperplane, and establish in this hyperplane an elliptic polar correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p0752205.png" /> of points and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p0752206.png" />-dimensional hyperplanes (that is, a polar correspondence under which no point belongs to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p0752207.png" />-dimensional plane corresponding to it).
+
To obtain the Euclidean determination of a metric in the $  n $-
 +
dimensional projective space $  P $,  
 +
one should distinguish in this space an $  ( n - 1 ) $-
 +
dimensional hyperplane $  \pi $,  
 +
called the ideal hyperplane, and establish in this hyperplane an elliptic polar correspondence $  \Pi $
 +
of points and $  ( n - 2 ) $-
 +
dimensional hyperplanes (that is, a polar correspondence under which no point belongs to the $  ( n - 2 ) $-
 +
dimensional plane corresponding to it).
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p0752208.png" /> is a subset of the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p0752209.png" /> obtained by removing from it an ideal hyperplane; and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522010.png" /> be points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522011.png" />. Two segments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522013.png" /> are said to be congruent if there exists a projective transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522014.png" /> taking the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522016.png" /> to the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522018.png" />, respectively, and preserving the [[Polarity|polarity]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522019.png" />.
+
Suppose that $  E _ {n} $
 +
is a subset of the projective space $  P $
 +
obtained by removing from it an ideal hyperplane; and let $  X, Y , X  ^  \prime  , Y  ^  \prime  $
 +
be points in $  E _ {n} $.  
 +
Two segments $  XY $
 +
and $  X  ^  \prime  Y  ^  \prime  $
 +
are said to be congruent if there exists a projective transformation $  \phi $
 +
taking the points $  X $
 +
and $  Y $
 +
to the points $  X  ^  \prime  $
 +
and $  Y  ^  \prime  $,  
 +
respectively, and preserving the [[Polarity|polarity]] $  \Pi $.
  
The concept of congruence of segments thus defined allows one to introduce a metric of a Euclidean space in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522020.png" />. For this, in the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522021.png" /> a system of [[Projective coordinates|projective coordinates]] is introduced with the basis simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522022.png" />, where the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522023.png" /> does not not belong to the ideal hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522024.png" /> while the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522025.png" /> do. Suppose that the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522026.png" /> in this coordinate system has the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522027.png" />, and that the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522029.png" />, have the coordinates
+
The concept of congruence of segments thus defined allows one to introduce a metric of a Euclidean space in $  E _ {n} $.  
 +
For this, in the projective space $  P $
 +
a system of [[Projective coordinates|projective coordinates]] is introduced with the basis simplex $  OA _ {1} {} \dots A _ {n} $,  
 +
where the point $  O $
 +
does not not belong to the ideal hyperplane $  \pi $
 +
while the points $  A _ {1} \dots A _ {n} $
 +
do. Suppose that the point $  O $
 +
in this coordinate system has the coordinates $  0 \dots 0 , 1 $,
 +
and that the points $  A _ {i} $,  
 +
$  i = 1 \dots n $,  
 +
have the coordinates
  
<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/p/p075/p075220/p07522030.png" /></td> </tr></table>
+
$$
 +
x _ {1}  = 0 \dots x _ {i-} 1  = 0 ,\
 +
x _ {i}  = 1 , x _ {i+} 1  = 0 \dots x _ {n+} 1  = 0 .
 +
$$
  
Then the elliptic polar correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522031.png" /> defined in the hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522032.png" /> can be written in the form
+
Then the elliptic polar correspondence $  \Pi $
 +
defined in the hyperplane $  \pi $
 +
can be written in 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/p/p075/p075220/p07522033.png" /></td> </tr></table>
+
$$
 +
u _ {i}  = \sum _ { j= } 1 ^ { n }  a _ {ij} x _ {j} ,\ \
 +
i = 1 \dots n .
 +
$$
  
The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522034.png" /> of this correspondence is symmetric, and the quadratic form
+
The matrix $  ( a _ {ij} ) $
 +
of this correspondence is symmetric, and the quadratic 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/p/p075/p075220/p07522035.png" /></td> </tr></table>
+
$$
 +
Q ( x _ {1} \dots x _ {n} )  = \sum a _ {ij} x _ {i} x _ {j}  $$
  
 
corresponding to it is positive definite. Let
 
corresponding to it is positive definite. Let
  
<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/p/p075/p075220/p07522036.png" /></td> </tr></table>
+
$$
 +
= ( a _ {1} : \dots : a _ {n+} 1 ) \ \
 +
\textrm{ and } \  Y  = ( b _ {1} : \dots : b _ {n+} 1 )
 +
$$
 +
 
 +
be two points in  $  E _ {n} $(
 +
that is,  $  a _ {n+} 1 \neq 0 $,
 +
$  b _ {n+} 1 \neq 0 $).  
 +
One may set
  
be two points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522037.png" /> (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522039.png" />). One may set
+
$$
  
<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/p/p075/p075220/p07522040.png" /></td> </tr></table>
+
 +
\frac{a _ 1}{a _ n+}
 +
  1  = x _ {1} \dots
 +
 +
\frac{a _ n}{a _ n+}
 +
  1  = x _ {n} ;
 +
$$
  
<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/p/p075/p075220/p07522041.png" /></td> </tr></table>
+
$$
  
Then the distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522042.png" /> between the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522044.png" /> is defined by
+
 +
\frac{b _ 1}{b _ n+}
 +
  1  = y _ {1} \dots
 +
 +
\frac{b _ n}{b _ n+}
 +
  1  = y _ {n} .
 +
$$
  
<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/p/p075/p075220/p07522045.png" /></td> </tr></table>
+
Then the distance  $  \rho $
 +
between the points  $  X $
 +
and  $  Y $
 +
is defined by
  
For a projective determination of the metric of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522046.png" />-dimensional hyperbolic space, in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522047.png" />-dimensional projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522048.png" /> a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522049.png" /> of interior points of a real oval hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522050.png" /> of order two is considered. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522051.png" /> be points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522052.png" />; then the segments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522054.png" /> are assumed to be congruent if there is a projective transformation of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522055.png" /> under which the hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522056.png" /> is mapped onto itself and the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522058.png" /> are taken to the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522060.png" />, respectively. The concept of congruence of segments thus introduced establishes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522061.png" /> the metric of the hyperbolic space. The length of a segment in this metric is defined by
+
$$
 +
\rho ( X , Y )  = \sqrt {Q ( x _ {1} - y _ {1} \dots x _ {n} - y _ {n} ) } .
 +
$$
  
<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/p/p075/p075220/p07522062.png" /></td> </tr></table>
+
For a projective determination of the metric of the  $  n $-
 +
dimensional hyperbolic space, in the  $  n $-
 +
dimensional projective space  $  P $
 +
a set  $  U $
 +
of interior points of a real oval hypersurface  $  S $
 +
of order two is considered. Let  $  X , Y , X  ^  \prime  , Y  ^  \prime  $
 +
be points in  $  U $;  
 +
then the segments  $  XY $
 +
and  $  X  ^  \prime  Y  ^  \prime  $
 +
are assumed to be congruent if there is a projective transformation of the space  $  P $
 +
under which the hypersurface  $  S $
 +
is mapped onto itself and the points  $  X $
 +
and  $  Y $
 +
are taken to the points  $  X  ^  \prime  $
 +
and  $  Y  ^  \prime  $,
 +
respectively. The concept of congruence of segments thus introduced establishes in  $  U $
 +
the metric of the hyperbolic space. The length of a segment in this metric is defined by
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522064.png" /> are the points of intersection of the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522065.png" /> with the hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522067.png" /> is a positive number related to the curvature of the Lobachevskii space. To introduce an elliptic metric in the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522068.png" />, one considers an elliptic polar correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522069.png" /> in this space. Two segments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522071.png" /> are said to be congruent if there exists a projective transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522072.png" /> taking the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522074.png" /> to the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522076.png" />, respectively, and preserving the polar mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522077.png" /> (that is, for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522078.png" /> and its polar <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522079.png" />, the polar of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522080.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522081.png" />). If the elliptic polar correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522082.png" /> is given by the relations
+
$$
 +
\rho ( X , Y )  = c  |  \mathop{\rm ln}  ( XYPQ ) | ,
 +
$$
  
<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/p/p075/p075220/p07522083.png" /></td> </tr></table>
+
where  $  P $
 +
and  $  Q $
 +
are the points of intersection of the straight line  $  XY $
 +
with the hypersurface  $  S $
 +
and  $  c $
 +
is a positive number related to the curvature of the Lobachevskii space. To introduce an elliptic metric in the projective space  $  P $,
 +
one considers an elliptic polar correspondence  $  \Pi $
 +
in this space. Two segments  $  XY $
 +
and  $  X  ^  \prime  Y  ^  \prime  $
 +
are said to be congruent if there exists a projective transformation  $  \phi $
 +
taking the points  $  X $
 +
and  $  Y $
 +
to the points  $  X  ^  \prime  $
 +
and  $  Y  ^  \prime  $,
 +
respectively, and preserving the polar mapping  $  \Pi $(
 +
that is, for any point  $  M $
 +
and its polar  $  m $,
 +
the polar of the point  $  \phi ( M) $
 +
is  $  \phi ( m) $).  
 +
If the elliptic polar correspondence  $  \Pi $
 +
is given by the relations
  
then the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522084.png" /> is symmetric and the quadratic form corresponding to it is positive definite. Now, if
+
$$
 +
u _ {i}  = \sum _ { j= } 1 ^ { n+ }  1 a _ {ij} x _ {j} ,\ \
 +
i = 1 \dots n + 1 ,
 +
$$
  
<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/p/p075/p075220/p07522085.png" /></td> </tr></table>
+
then the matrix  $  ( a _ {ij} ) $
 +
is symmetric and the quadratic form corresponding to it is positive definite. Now, if
 +
 
 +
$$
 +
= ( x _ {1} : \dots : x _ {n+} 1 ) ,\ \
 +
= ( y _ {1} : \dots : y _ {n+} 1 ) ,
 +
$$
  
 
then
 
then
  
<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/p/p075/p075220/p07522086.png" /></td> </tr></table>
+
$$
 +
\rho ( X , Y )  =   \mathop{\rm arccos} 
 +
\frac{| B ( X , Y ) | }{\sqrt {B ( X , X ) } \sqrt {B ( Y , Y ) } }
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522087.png" /> is the bilinear form given by the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522088.png" />.
+
where $  B $
 +
is the bilinear form given by the matrix $  ( a _ {ij} ) $.
  
In all the cases considered (if a real projective space is completed to a complex projective space), under the projective transformations defining the congruence of segments, that is, under motions, some hypersurfaces of the second order remain invariant; these are called absolutes. In the case of a Euclidean determination of a metric, the absolute is an imaginary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522089.png" />-dimensional oval surface of order two. In the case of a hyperbolic determination of a metric, the absolute is an oval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522090.png" />-dimensional real hypersurface of order two. In the case of an elliptic determination of a metric, the absolute is an imaginary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075220/p07522091.png" />-dimensional oval hypersurface of order two.
+
In all the cases considered (if a real projective space is completed to a complex projective space), under the projective transformations defining the congruence of segments, that is, under motions, some hypersurfaces of the second order remain invariant; these are called absolutes. In the case of a Euclidean determination of a metric, the absolute is an imaginary $  ( n - 2 ) $-
 +
dimensional oval surface of order two. In the case of a hyperbolic determination of a metric, the absolute is an oval $  ( n - 1 ) $-
 +
dimensional real hypersurface of order two. In the case of an elliptic determination of a metric, the absolute is an imaginary $  ( n - 1 ) $-
 +
dimensional oval hypersurface of order two.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.V. Efimov,  "Higher geometry" , MIR  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.A. Glagolev,  "Projective geometry" , Moscow  (1963)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Busemann,  P.J. Kelly,  "Projective geometry and projective metrics" , Acad. Press  (1953)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.V. Efimov,  "Higher geometry" , MIR  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.A. Glagolev,  "Projective geometry" , Moscow  (1963)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Busemann,  P.J. Kelly,  "Projective geometry and projective metrics" , Acad. Press  (1953)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi,  "Transformation groups in differential geometry" , Springer  (1972)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi,  "Transformation groups in differential geometry" , Springer  (1972)</TD></TR></table>

Revision as of 08:08, 6 June 2020


An introduction in subsets of a projective space, by methods of projective geometry, of a metric such that these subsets become isomorphic to a Euclidean, hyperbolic or elliptic space. This is achieved by distinguishing in the class of all projective transformations (cf. Projective transformation) those transformations that generate in these subsets a group of transformations isomorphic to the corresponding group of motions. The presence of motions allows one "to lay off" segments of a straight line from a given point in a given direction, thereby introducing the concept of the length of a segment.

To obtain the Euclidean determination of a metric in the $ n $- dimensional projective space $ P $, one should distinguish in this space an $ ( n - 1 ) $- dimensional hyperplane $ \pi $, called the ideal hyperplane, and establish in this hyperplane an elliptic polar correspondence $ \Pi $ of points and $ ( n - 2 ) $- dimensional hyperplanes (that is, a polar correspondence under which no point belongs to the $ ( n - 2 ) $- dimensional plane corresponding to it).

Suppose that $ E _ {n} $ is a subset of the projective space $ P $ obtained by removing from it an ideal hyperplane; and let $ X, Y , X ^ \prime , Y ^ \prime $ be points in $ E _ {n} $. Two segments $ XY $ and $ X ^ \prime Y ^ \prime $ are said to be congruent if there exists a projective transformation $ \phi $ taking the points $ X $ and $ Y $ to the points $ X ^ \prime $ and $ Y ^ \prime $, respectively, and preserving the polarity $ \Pi $.

The concept of congruence of segments thus defined allows one to introduce a metric of a Euclidean space in $ E _ {n} $. For this, in the projective space $ P $ a system of projective coordinates is introduced with the basis simplex $ OA _ {1} {} \dots A _ {n} $, where the point $ O $ does not not belong to the ideal hyperplane $ \pi $ while the points $ A _ {1} \dots A _ {n} $ do. Suppose that the point $ O $ in this coordinate system has the coordinates $ 0 \dots 0 , 1 $, and that the points $ A _ {i} $, $ i = 1 \dots n $, have the coordinates

$$ x _ {1} = 0 \dots x _ {i-} 1 = 0 ,\ x _ {i} = 1 , x _ {i+} 1 = 0 \dots x _ {n+} 1 = 0 . $$

Then the elliptic polar correspondence $ \Pi $ defined in the hyperplane $ \pi $ can be written in the form

$$ u _ {i} = \sum _ { j= } 1 ^ { n } a _ {ij} x _ {j} ,\ \ i = 1 \dots n . $$

The matrix $ ( a _ {ij} ) $ of this correspondence is symmetric, and the quadratic form

$$ Q ( x _ {1} \dots x _ {n} ) = \sum a _ {ij} x _ {i} x _ {j} $$

corresponding to it is positive definite. Let

$$ X = ( a _ {1} : \dots : a _ {n+} 1 ) \ \ \textrm{ and } \ Y = ( b _ {1} : \dots : b _ {n+} 1 ) $$

be two points in $ E _ {n} $( that is, $ a _ {n+} 1 \neq 0 $, $ b _ {n+} 1 \neq 0 $). One may set

$$ \frac{a _ 1}{a _ n+} 1 = x _ {1} \dots \frac{a _ n}{a _ n+} 1 = x _ {n} ; $$

$$ \frac{b _ 1}{b _ n+} 1 = y _ {1} \dots \frac{b _ n}{b _ n+} 1 = y _ {n} . $$

Then the distance $ \rho $ between the points $ X $ and $ Y $ is defined by

$$ \rho ( X , Y ) = \sqrt {Q ( x _ {1} - y _ {1} \dots x _ {n} - y _ {n} ) } . $$

For a projective determination of the metric of the $ n $- dimensional hyperbolic space, in the $ n $- dimensional projective space $ P $ a set $ U $ of interior points of a real oval hypersurface $ S $ of order two is considered. Let $ X , Y , X ^ \prime , Y ^ \prime $ be points in $ U $; then the segments $ XY $ and $ X ^ \prime Y ^ \prime $ are assumed to be congruent if there is a projective transformation of the space $ P $ under which the hypersurface $ S $ is mapped onto itself and the points $ X $ and $ Y $ are taken to the points $ X ^ \prime $ and $ Y ^ \prime $, respectively. The concept of congruence of segments thus introduced establishes in $ U $ the metric of the hyperbolic space. The length of a segment in this metric is defined by

$$ \rho ( X , Y ) = c | \mathop{\rm ln} ( XYPQ ) | , $$

where $ P $ and $ Q $ are the points of intersection of the straight line $ XY $ with the hypersurface $ S $ and $ c $ is a positive number related to the curvature of the Lobachevskii space. To introduce an elliptic metric in the projective space $ P $, one considers an elliptic polar correspondence $ \Pi $ in this space. Two segments $ XY $ and $ X ^ \prime Y ^ \prime $ are said to be congruent if there exists a projective transformation $ \phi $ taking the points $ X $ and $ Y $ to the points $ X ^ \prime $ and $ Y ^ \prime $, respectively, and preserving the polar mapping $ \Pi $( that is, for any point $ M $ and its polar $ m $, the polar of the point $ \phi ( M) $ is $ \phi ( m) $). If the elliptic polar correspondence $ \Pi $ is given by the relations

$$ u _ {i} = \sum _ { j= } 1 ^ { n+ } 1 a _ {ij} x _ {j} ,\ \ i = 1 \dots n + 1 , $$

then the matrix $ ( a _ {ij} ) $ is symmetric and the quadratic form corresponding to it is positive definite. Now, if

$$ X = ( x _ {1} : \dots : x _ {n+} 1 ) ,\ \ Y = ( y _ {1} : \dots : y _ {n+} 1 ) , $$

then

$$ \rho ( X , Y ) = \mathop{\rm arccos} \frac{| B ( X , Y ) | }{\sqrt {B ( X , X ) } \sqrt {B ( Y , Y ) } } , $$

where $ B $ is the bilinear form given by the matrix $ ( a _ {ij} ) $.

In all the cases considered (if a real projective space is completed to a complex projective space), under the projective transformations defining the congruence of segments, that is, under motions, some hypersurfaces of the second order remain invariant; these are called absolutes. In the case of a Euclidean determination of a metric, the absolute is an imaginary $ ( n - 2 ) $- dimensional oval surface of order two. In the case of a hyperbolic determination of a metric, the absolute is an oval $ ( n - 1 ) $- dimensional real hypersurface of order two. In the case of an elliptic determination of a metric, the absolute is an imaginary $ ( n - 1 ) $- dimensional oval hypersurface of order two.

References

[1] N.V. Efimov, "Higher geometry" , MIR (1980) (Translated from Russian)
[2] N.A. Glagolev, "Projective geometry" , Moscow (1963) (In Russian)
[3] H. Busemann, P.J. Kelly, "Projective geometry and projective metrics" , Acad. Press (1953)

Comments

References

[a1] S. Kobayashi, "Transformation groups in differential geometry" , Springer (1972)
How to Cite This Entry:
Projective determination of a metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Projective_determination_of_a_metric&oldid=48319
This article was adapted from an original article by P.S. ModenovA.S. Parkhomenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article