View source for Projective determination of a metric

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).

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" />.

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

<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>

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

<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>

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

<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>

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>

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>

<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

<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>

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

<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>

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

<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>

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

<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

<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>

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" />.

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.

====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>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi,   "Transformation groups in differential geometry" , Springer  (1972)</TD></TR></table>