Lorentzian geodesic flows (Q1915931)

The paper studies the class of \(C_Q\) Lorentzian manifolds \((M, g)\), \(Q\geq 0\), that is \((M, g)\) is time-oriented, future 1-connected, nonspacelike complete, and the sectional curvature of any timelike hyperplane is bounded from above by \(Q^2\). It is proved that the set of timelike closed geodesics is dense in the quotient of the future timelike unit tangent bundle modulo a suitable group of isometries called vicious. Moreover, it is proved a rigidity theorem for two-dimensional \(C_Q\) manifolds. More precisely, it is shown that any orientable, two-dimensional \(C_Q\) manifold, \(Q> 0\), admitting a vicious isometry group, must have constant curvature.