Infinitesimal projective transformations in the tangent bundle of general space of paths (Q1387276)

Let \(M(G)\) be a general space of paths \((n>2)\) defined on the basis of the functions \(G^i(x, \dot x)\), homogeneous of degree 2 in the variables \(\dot x^i=dx^i/dt\). Its connection is given by \(G^i_{km}= \partial^2G/ \partial \dot x^k\partial \dot x^m\), and if a vector field \(v\) satisfies \({\mathcal L}_v G^i= \dot x^iP\) with \(P=q_k \dot x^k\) (or, equivalently, \({\mathcal L}_v G^i_{km}= \delta^i_k P_m+ \delta^i_mP_k +\dot x^iP_{km})\), it is said to define an infinitesimal projective transformation of \(M(G)\). In this paper, the validity of the following theorem is proved: in order for a vector field \(\widetilde V\) on the tangent bundle \(T(M(G))\) over \(M(G)\) to determine the infinitesimal projective transformation of the connection \(\Gamma \) which is the natural lift of \(G\), it is necessary and sufficient that it be of the form \[ \widetilde v={^Nu} +{^Vv}+ {^{HX}A} +{^{VX}B} +p(\dot x){^{VX} \text{id}}, \] where \(u\) and \(v\) are vector fields on \(M\) defining an infinitesimal projective and affine transformation, respectively, the left indices \(N,V\) indicate the normal and vertical lifts, respectively, the left indices \(VX\) and \(HX\) indicate the vertical and horizontal vector lifts, respectively, \(p(x)\) is a parallel vector field, and \(A(x)\) and \(B(x)\) are tensor fields of type (1,1) closely related to the curvature tensor of \(G\).