C(7) semi-groups are finitely presented semi-groups were each relation decomposes into at least seven pieces. We give an example showing that, unlike for C(7) groups, the set of geodesics under some generating set is not an automatic structure (furthermore, even just a subset of the geodesics is not an automatic structure). We then show that under further restrictions, which imply embeddability...

We generalize the optical geometry to include, not only conformally static spacetimes, but any spacetime that admits a hypersurface orthogonal shearless congruence of worldlines. In the generalized optical geometry, which in general is time dependent, photons move with constant speed along spatial geodesics. Gyroscopes moving along spatial geodesics do not precess. We also classify inertial for...

The set of spatial rigid body motions forms a Lie group known as the special Euclidean group in three dimensions, (3). Chasles’s theorem states that there exists a screw motion between two arbitrary elements of (3). In this paper we investigate whether there exist a Riemannian metric whose geodesics are screw motions. We prove that no Riemannian metric with such geodesics exists and we show tha...

We construct geodesics in the Wasserstein space of probability measures along which all the measures have an upper bound on their density that is determined by the densities of the endpoints of the geodesic. Using these geodesics we show that a local Poincaré inequality and the measure contraction property follow from the Ricci curvature bounds defined by Sturm. We also show for a large class o...

We interpret magnetic billiards as Finsler ones and describe an analog of the string construction for magnetic billiards. Finsler billiards for which the law “angle of incidence equals angle of reflection” are described. We characterize the Finsler metrics in the plane whose geodesics are circles of a fixed radius. This is a magnetic analog of Hilbert’s fourth problem asking to describe the Fin...