On affine symmetries of quasigeodesic flows (Q2387046)

A quasi-geodesic flow (QF, spray) on an \((m-1)\)-dimensional manifold \(M\) is considered in local description as a 2nd order ordinary differential equation \[ \frac{d^2x^i}{d t^2}=f^i\left( x^j,t, \frac{d x^i}{d t} \right), \] denoted \(f=(M,f)\). \((M,f)\) is called quadratic, if the dependence of the \(f^i\) on the derivatives \(d x^i/dt\) is quadratic, i.e., \[ f^i=-\Gamma^i_{jk}\frac{d x^i}{d t} \frac{d x^j}{d t}-2\, B^i_j(x^s,t)\frac{d x^j}{d t}-A^i(x^s,t), \] where \(\Gamma^i_{jk}\) are Chistoffel symbols of a torsionfree affine connection and \(B^i_j, A^i\) are tensor fields on \(M\times{\mathbb R}\). In chapter I, infinitesimal point symmetries of quasi-geodesic flows are studied. This leads to an investigation of the Lie algebra of infinitesimal affine motions. Classical results of \textit{I. P. Egorov} [Motions in spaces of affine connection, in: Motions in Spaces of Affine Connections, Izdat. Kazan. Univ., Kazan, pp. 5--179 (1965)] are applied, errors and gaps in Egorov's proofs are revised. In chapter II, nonprojective-Euclidean and equiaffine symmetries of quadratic quasi-geodesic flows are studied.