Spray simulation and point trajectory morphisms of quasigeodesic flows (Q1603060)

One considers two manifolds \(M\) and \(N\) each one endowed with a spray \(f\) and \(h\), respectively. A mapping \(\Phi : M\times R\longrightarrow N\times R\) is called a point quasiisomorphism if it maps the integral curves of \(f\) to the integral curves of \(h\). Various characterisations of quasiisomorphisms are given and particular cases are discussed. A trajectory automorphism of a spray is a diffeomorphism of \(M\) onto itself which maps trajectories onto trajectories. Infinitesimal trajectory symmetries of \(f\) are defined and characterised. In the 2-dimensional case a classification of sprays with respect to the point trajectory mobility is given. This is based on results by \textit{J. Levine} [Ann. Math. (2) 52, 456-477 (1950; Zbl 0038.34603)].