We provide examples of nonlocally, compact, geodesic Ptolemy metric spaces which are not uniquely geodesic. On the other hand, we show that locally, compact, geodesic Ptolemy metric spaces are uniquely geodesic. Moreover, we prove that a metric space is CAT(0) if and only if it is Busemann convex and Ptolemy.

Given, in an arbitrary spacetime (M, g), a 2-dimensional timelike submanifold Σ and an observer field n on Σ, we assign gravitational, centrifugal, Coriolis and Euler forces to every particle worldline λ in Σ with respect to n. We prove that centrifugal and Coriolis forces vanish, for all λ in Σ with respect to any n, if and only if Σ is a photon 2-surface, i.e., generated by two families of lightlike geodesics. We further demonstrate that a photon 2-surface can be characterized in terms of gyroscope transport and we give several mathematical criteria for the existence of photon 2-surfaces. Finally, examples of photon 2-surfaces in conformally flat spacetimes, in Schwarzschild and Reissner-Nordström spacetimes, and in Gödel spacetime are worked out.

It is shown that the Hilbert geometry (D, h D ) associated to a bounded convex domain D ⊂ E n is isometric to a normed vector space (V, || · ||) if and only if D is an open n-simplex. One further result on the asymptotic geometry of Hilbert's metric is obtained with corollaries for the behavior of geodesics. Finally we prove that every geodesic ray in a Hilbert geometry converges to a point of the boundary.

We generalize the classical de Rham decomposition theorem for Riemannian manifolds to the setting of geodesic metric spaces of finite dimension.

Using a four points inequality for the boundary of CAT(−1)-spaces we study the relation between Gromov hyperbolic spaces and CAT(−1)-spaces.