The class of second order equations which Riemannian geometry can be applied to
Let \(M\) be a \(C^ \infty\) manifold without boundary and \(TM\) the tangent bundle of \(M\). A second order differential system on \(M\) is considered to be a vector field \(V\) on \(TM\). Let \(f^ t\) be the flow on \(TM\) generated by \(V\). In this note is studied the case that \(f^ t\) has an invariant hypersurface \(S\) in \(TM\) such that all fibres \(S_ x\) are star-shaped hypersurfaces around the origin in \(T_ x M\), \(x \in M\). Using the flows whose trajectories are geodesics and certain positive continuous functions, the author obtains several nontrivial conditions equivalent to the existence of a complementary \(f^ t\)-invariant distribution.
scientific article; zbMATH DE number 469053