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A metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075270/p0752701.png" /> in a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075270/p0752702.png" /> of a projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075270/p0752703.png" /> such that shortest paths with respect to this metric are parts of or entire projective straight lines. It is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075270/p0752704.png" /> does not belong to a hypersurface and that: 1) for any three non-collinear points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075270/p0752705.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075270/p0752706.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075270/p0752707.png" /> the triangle inequality holds in the strict sense:
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A metric $\rho(x,y)$ in a subset $R$ of a projective space $P^n$ such that shortest paths with respect to this metric are parts of or entire projective straight lines. It is assumed that $R$ does not belong to a hypersurface and that: 1) for any three non-collinear points $x$, $y$ and $z$ the triangle inequality holds in the strict sense:
  
<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/p075270/p0752708.png" /></td> </tr></table>
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$$\rho(x,y)+\rho(y,z)>\rho(x,z);$$
  
and 2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075270/p0752709.png" /> are different points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075270/p07527010.png" />, then the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075270/p07527011.png" /> of the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075270/p07527012.png" /> through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075270/p07527013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075270/p07527014.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075270/p07527015.png" /> is either all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075270/p07527016.png" /> (a large circle), or is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075270/p07527017.png" /> by discarding some segment (which may reduce to a point) (a metric straight line).
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and 2) if $x,y$ are different points in $R$, then the intersection $l(x,y)$ of the straight line $l$ through $x$ and $y$ with $R$ is either all of $l$ (a large circle), or is obtained from $l$ by discarding some segment (which may reduce to a point) (a metric straight line).
  
The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075270/p07527018.png" />, provided with a projective metric, is called a projective-metric space.
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The set $R$, provided with a projective metric, is called a projective-metric space.
  
In one and the same projective-metric space there cannot exist simultaneously both types of straight lines: They are either all metric straight lines (i.e. isometric to an interval in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075270/p07527019.png" />), or they are all large circles of the same length (Hamel's theorem). Spaces of the first kind are called open (they coincide with subspaces of an affine space, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075270/p07527020.png" /> from which a hypersurface has been deleted); the geometry of open projective-metric spaces is also called [[Hilbert geometry|Hilbert geometry]]. Spaces of the second kind are called closed (they coincide with the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075270/p07527021.png" />).
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In one and the same projective-metric space there cannot exist simultaneously both types of straight lines: They are either all metric straight lines (i.e. isometric to an interval in $\mathbf R$), or they are all large circles of the same length (Hamel's theorem). Spaces of the first kind are called open (they coincide with subspaces of an affine space, i.e. $P^n$ from which a hypersurface has been deleted); the geometry of open projective-metric spaces is also called [[Hilbert geometry|Hilbert geometry]]. Spaces of the second kind are called closed (they coincide with the whole of $P^n$).
  
 
The problem of determining all projective metrics is the so-called fourth problem of Hilbert (cf. [[#References|[2]]]), and a complete solution of it was given by A.V. Pogorelov (1974).
 
The problem of determining all projective metrics is the so-called fourth problem of Hilbert (cf. [[#References|[2]]]), and a complete solution of it was given by A.V. Pogorelov (1974).
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The so-called [[Projective determination of a metric|projective determination of a metric]] is related to projective metrics, as a particular case. It consists of introducing in a subset of a projective space, by methods of projective geometry, a metric such that this subset becomes isomorphic to a Euclidean, elliptic or hyperbolic space. E.g., the geometry of open projective-metric spaces, whose subsets coincide with all of affine space, is called Minkowski geometry. Euclidean geometry is a Hilbert geometry and a Minkowski geometry simultaneously.
 
The so-called [[Projective determination of a metric|projective determination of a metric]] is related to projective metrics, as a particular case. It consists of introducing in a subset of a projective space, by methods of projective geometry, a metric such that this subset becomes isomorphic to a Euclidean, elliptic or hyperbolic space. E.g., the geometry of open projective-metric spaces, whose subsets coincide with all of affine space, is called Minkowski geometry. Euclidean geometry is a Hilbert geometry and a Minkowski geometry simultaneously.
  
Hyperbolic geometry is a Hilbert geometry in which there exist reflections at all straight lines. The subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075270/p07527022.png" /> has a hyperbolic geometry if and only if it is the interior of an ellipsoid.
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Hyperbolic geometry is a Hilbert geometry in which there exist reflections at all straight lines. The subset $R$ has a hyperbolic geometry if and only if it is the interior of an ellipsoid.
  
 
Elliptic geometry (or [[Riemann geometry|Riemann geometry]]) is the geometry of a projective-metric space of the second kind.
 
Elliptic geometry (or [[Riemann geometry|Riemann geometry]]) is the geometry of a projective-metric space of the second kind.

Latest revision as of 16:16, 1 May 2014

A metric $\rho(x,y)$ in a subset $R$ of a projective space $P^n$ such that shortest paths with respect to this metric are parts of or entire projective straight lines. It is assumed that $R$ does not belong to a hypersurface and that: 1) for any three non-collinear points $x$, $y$ and $z$ the triangle inequality holds in the strict sense:

$$\rho(x,y)+\rho(y,z)>\rho(x,z);$$

and 2) if $x,y$ are different points in $R$, then the intersection $l(x,y)$ of the straight line $l$ through $x$ and $y$ with $R$ is either all of $l$ (a large circle), or is obtained from $l$ by discarding some segment (which may reduce to a point) (a metric straight line).

The set $R$, provided with a projective metric, is called a projective-metric space.

In one and the same projective-metric space there cannot exist simultaneously both types of straight lines: They are either all metric straight lines (i.e. isometric to an interval in $\mathbf R$), or they are all large circles of the same length (Hamel's theorem). Spaces of the first kind are called open (they coincide with subspaces of an affine space, i.e. $P^n$ from which a hypersurface has been deleted); the geometry of open projective-metric spaces is also called Hilbert geometry. Spaces of the second kind are called closed (they coincide with the whole of $P^n$).

The problem of determining all projective metrics is the so-called fourth problem of Hilbert (cf. [2]), and a complete solution of it was given by A.V. Pogorelov (1974).

The so-called projective determination of a metric is related to projective metrics, as a particular case. It consists of introducing in a subset of a projective space, by methods of projective geometry, a metric such that this subset becomes isomorphic to a Euclidean, elliptic or hyperbolic space. E.g., the geometry of open projective-metric spaces, whose subsets coincide with all of affine space, is called Minkowski geometry. Euclidean geometry is a Hilbert geometry and a Minkowski geometry simultaneously.

Hyperbolic geometry is a Hilbert geometry in which there exist reflections at all straight lines. The subset $R$ has a hyperbolic geometry if and only if it is the interior of an ellipsoid.

Elliptic geometry (or Riemann geometry) is the geometry of a projective-metric space of the second kind.

References

[1] P.J. Kelley, "Projective geometry and projective metrics" , Acad. Press (1953)
[2] "Hilbert's problems" Bull. Amer. Math. Soc. , 8 (1902) pp. 437–479 (Translated from German)


Comments

References

[a1] H. Busemann, "The geometry of geodesics" , Acad. Press (1955)
[a2] H. Busemann, "Metric methods in Finsler spaces and in the foundations of geometry" , Princeton Univ. Press (1942)
How to Cite This Entry:
Projective metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Projective_metric&oldid=32094
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article