Beltrami’s Theorem (1865)
determines the Riemann spaces whose geodesics behave locally like affine lines. Its
global form (proved much later) states that a complete simply connected Riemann
space with this property is a euclidean, hyperbolic or spherical space. Here we
establish what we believe to be the most general meaningful version of this theorem.
We define a general class of complete metric spaces, called chord spaces, which
possess distinguished extremals. In our case these must behave locally like affine
lines, but they need not be the only extremals. This situation occurs in many
important spaces.
Our principal result is that such a space (of n dimensions, n > 1) can be mapped
topologically and geodesically either on the entire n-sphere Sn or on an arbitrary
open convex subset of an open hemisphere of Sn, considered as the affine space An.
Examples of such spaces with some unexpected phenomena and the significance of
chord spaces in general are discussed.
