Spray modeling. I (Q2366390)

This paper is contributed in memory of late Professor Ya. L. Shapiro, by investigating the quasi-geodesic flow homomorphisms. It is defined as the flow of a second order ordinary differential equation \(D\) on a manifold \(M\), having the expression in a local coordinate system \[ d^ 2x^ i/dt^ 2=D^ i(X^ j,t,dx^ j/dt),\quad 1\leq i,j\leq n=\dim M. \] A submersion \(\varphi:\overline M\to M\) is called a homomorphism of such a flow \(\overline D\) on \(\overline M\) onto \(D\) on \(M\) if \(\varphi\) carries trajectories of the former one onto those of the latter one. \(\varphi\) is also said to be a spray model. The set \(((\overline M,\overline D),\varphi,M)\) with \(\overline M=M\times R\), \(\varphi:\overline M\to M\) is then a natural projection, and it can be proven that \(D\) has the expression \[ d^ 2x^ i/dt^ 2=(dt/d\tau)^ 2D^ i\left(x^ j,t,{dx^ j/d\tau\over dt/d\tau}\right), d^ 2t/d\tau^ 2=0. \] Let a vector field along a quasi-geodesic flow, which satisfies the variation equation, be called a Jacobi field. Assuming that \(\overline x(\tau)=(x(t),\;t(\tau))\), \(\tilde x(t)=x(\tau=(t-t_ 0)/\lambda^ 0)\) and denoting by \(\overline\Lambda\) and \(\Lambda\) the Jacobi vector fields along \(\overline x(\tau)\) and \(\tilde x(t)\), one can prove that the restriction of the corresponding \(\overline X(\tau)\to X(t)\) to \(\overline\Lambda\) is an isomorphism of \(\overline\Lambda\) to \(\Lambda\). Let \(\overline O\) be the domain of a global flow \(\overline\beta\) i.e. \(\overline O\to T\overline M:(\overline v,t)\equiv(\overline p,\overline\lambda,\tau)\to\overline\beta(\overline p,\overline\lambda,\tau)\). Then, for any \(\overline p=(p,t)\in\overline M\) and \(\overline\lambda=(\lambda,\lambda^ 0)\in\overline M\), there exists in the exponential mapping the remarkable relation: \(\lim_{\tau\to 0}(\overline{\text{Exp}}_{\overline p})_{*\tau\lambda}=\text{id}:\overline M_{\overline p}\to\overline M_{\overline p}\), and with the aid of this fact, the author could define the Shapiro's so-called fine exponential mappings for \(((\overline M,\overline D),\varphi,M)\).