Spray simulation. III (Q1909835)

In [1]-[3]\ [the author, Sov. Math., Dokl. 44, No. 3, 474-478 (1992); translation from Dokl. Akad. Nauk SSSR 320, No. 3, 531-535 (1991; Zbl 0776.58029), Russ. Math. 36, No. 6, 60-66 (1992); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1992, No. 6(361), 63-70 (1992; Zbl 0778.58056), and Russ. Math. 38, No. 10, 22-28 (1994); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1994, No. 10(389), 26-32 (1994; see the paper above)] we have constructed a spray model for an arbitrary quasigeodesic flow (QF for short), i.e., for a second order ordinary differential equation. This model has made it possible to define exponential mappings and to obtain results in two-end problem for a QF (see [1], [2]), to introduce notions on connection and projective parameters of QF, to find geometric criteria of finite pointwise isomorphisms of QF, and to solve the triviality problem for QF (see [3]). In the present article we use the spray simulation to find geometric criteria for pointwise infinitesimal symmetries of arbitrary QFs, meanwhile the greater part of results still concerns one-dimensional QF. In particular, we define a QF dual to a given one, classify pairs of mutually dual one-dimensional QFs whose connections are projectively equivalent to affine connections, with respect to the degree of projective mobility.