Projective metric

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)

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)