Projective metric

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

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

The set , 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 ), 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. 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 ).

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 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)