Desargues geometry

geometry of a Desargues space, Desarguesian geometry

A geodesic geometry in which the role of geodesics is played by ordinary straight lines. More exactly, a Desargues space $R$ is a $G$-space which may be topologically mapped into a projective space $P^n$ so that each geodesic of $R$ is mapped into a straight line of $P^n$.

For $R$ to be a Desargues space, the following conditions are necessary and sufficient:

1) the geodesic passing through two different points must be unique;

2) if $\dim R=2$, both the direct and the converse Desargues assumption must be valid, provided the intersections occurring in these assumptions exist;

3) if $\dim R>2$, any three points in $R$ must lie in one plane.

Any $R$ mapped into $P^n$ must either cover all of $P^n$, and in such a case the geodesics of $R$ are all circles of the same length, or else $R$ must not contain any points of some hyperplane, and may be considered as an open convex domain of an affine space.

In the Riemannian case the only Desargues geometries are the Euclidean, hyperbolic and elliptic geometries, i.e. very strong mobility properties follow from the Desargues-nature of the space (Beltrami's theorem). This is an example of a noteworthy theorem of Riemannian geometry which has no analogues in more general spaces. If the differentiability conditions are sufficiently strong, a method of construction of a Desargues geometry has been proposed, but it was only A.V. Pogorelov [2] who offered the ultimate and general solution of this so-called Hilbert's 4th problem on the metrization of the projective space or its convex subspaces without any regularity assumptions. Another example of a Desargues geometry which is useful in the study of spaces with a non-positive curvature is provided by the Hilbert geometry.

An important example of a non-Riemann Desargues geometry is the Minkowski geometry, which may be regarded as the prototype of all non-Riemann geometries (including the Finsler geometry).

References

[1]	H. Busemann, "The geometry of geodesics" , Acad. Press (1955)
[2]	A.V. Pogorelov, "Hilbert's fourth problem" , Winston & Wiley (1979) (In Russian)

Comments

The notion treated in the article above is a generalization (to more general spaces) of a Desarguesian plane: An affine or projective plane in which the Desargues assumption holds, cf. [a1].

For the notion of a $G$-space, cf. Finsler space, generalized; Space with an indefinite metric; and especially Geodesic geometry. This meaning of $G$-space should not be confused with the notion of $G$-space used in differential and algebraic topology, which is simply a topological space equipped with an action of a group $G$.

References

[a1]	D.R. Hughes, F.C. Piper, "Projective planes" , Springer (1973)