Vector fields with distributions and invariants of ODEs (Q1951367)

On a manifold \(M\) of dimension \((k+1)m+1\), a pair \((X, \mathcal{V})\) consisting of a vector field and a distribution is given. The authors call it a dynamical pair if the following conditions hold. (R1) \(\operatorname{rk}(\mathcal{V}^0=\mathcal{V})=m\), \(\operatorname{rk}(\mathcal{V}^{i+1}=\mathcal{V}^i+[X, \mathcal{V}^i])=(i+2)m\) for \(i=0,\dotsc ,m\); (R2) \(\mathcal{V}^k\oplus \operatorname{span}\{X\}=TM\). They associate to this framework a class of curvature operators and a canonical connection with respect to a canonical splitting of \(TM\). The equivalence problem, namely the (local) equivalence of two dynamical pairs, is solved by using the curvature and torsion of a principal connection corresponding to a \(\operatorname{GL}(m)\)-structure on \(M\). With this formalism there are obtained invariants of ODEs under time scale preserving equivalence, with a special emphasis on mechanical control systems and Veronese webs.