Geodesic geometry

geometry of geodesics

The geometry of a metric space (a $G$-space) characterized by the fact that extensions of geodesic lines (cf. Geodesic line), defined as locally shortest lines, are unique.

A $G$-space is defined by the following system of axioms:

1) $G$ is a metric space; $\rho(x,y)$ is the distance in the space;

2) $G$ is finitely compact, i.e. bounded infinite sets in $G$ have limit points;

3) $G$ is convex in the sense of Menger, i.e. for two points $x\neq y$ there exists a third point $z$ distinct from $x$ and $y$ and such that

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

4) for each point $a$ there exists an $r>0$ such that in the ball $\rho(a,x)<r$ there exists, for the points $x\neq y$, a third point $z$ distinct from $x$ and $y$ with $\rho(x,y)+\rho(y,z)=\rho(x,z)$ (the axiom of local extension);

5) if, in axiom 4), two points $z_1$ and $z_2$ such that $\rho(y,z_1)=\rho(y,z_2)$ have been found, then $z_1=z_2$ (the axiom of unique extension).

The class of $G$-spaces includes, in particular, Riemannian spaces and Finsler spaces (cf. Finsler space; Riemannian space).

$G$-spaces in which extension of a geodesic is possible in general and any segment of the geodesic remains a shortest line are called straight spaces. They include, for example, Euclidean, Minkowski and Lobachevskii spaces, and all simply-connected Riemannian spaces of non-positive curvature. In straight spaces and in some $G$-spaces of special type (elliptic), a geodesic is determined by two points.

In general $G$-spaces, unlike in Minkowski spaces, a sphere is not always convex. Perpendicularity, defined in terms of the shortest line to the geodesic, is not necessarily symmetric, unlike in Euclidean spaces. Criteria have been formulated in terms of $G$-spaces to distinguish Euclidean spaces, spheric spaces, and Minkowski spaces.

The theory of $G$-spaces showed that many results of differential geometry are not connected with conditions of differentiability. This theory extended the studies on Finsler spaces; made it possible to study those metrizations of affine and projective spaces which convert straight lines to geodesic lines; and to study the freedom of choice of geodesic nets under metrization. A number of hitherto unsolved problems are connected with the possible topological structure of $G$-spaces [1].

References

[1]	H. Busemann, "The geometry of geodesics" , Acad. Press (1955)

Comments

For further generalizations and results see [a1]. The phrase "G-space" , in analogy with $G$-set, is also used to denote the totally different notion of a (topological) space with an action of a (topological) group $G$ on it.

References

[a1]	M. Gromov, "Structures métriques pour les variétés Riemanniennes" , F. Nathan (1981) (Translated from Russian)