On the geometry and topology of flows and foliations on surfaces and the Anosov problem

A study is made of flows with finitely many equilibrium states and of foliations with finitely many singularities of saddle type with integer and half-integer index on closed surfaces, and for a metric of constant curvature the role of geodesics is established in the asymptotic behaviour of semitrajectories of flows and semileaves of foliations upon lifting to the unbranched or branched universal covering.