Closed Geodesics and Isometries on Compact Lorentz Solvmanifolds

ABSTRACT The aim of this work is a deep study of existence of closed geodesics on Lorentz homogeneous spaces of the form , where is a solvable Lie group endowed with a bi‐invariant Lorentz metric and is a cocompact lattice. Conditions to assert closedness of light, time, or spacelike geodesics on the compact quotient spaces are given. This study implicitly requires additional information about the lattices in each case. We found conditions for which every lightlike geodesic on the quotient space is closed. And more importantly, this situation depends on the lattice. Even in dimension 4, there are examples of compact solvmanifolds for which every lightlike geodesic is closed and others for which at every point there is only one direction in which lightlike geodesics close. For time and spacelike geodesics, the conclusions are different: There are both open and closed geodesics of type timelike and spacelike. Finally, we improve results of isometry groups on compact quotients, for which one needs information on the normalizer of the given lattice. We show that this normalizer is included in a certain subset, if some extra condition holds. Moreover this condition is related to that one that determines if every lightlike geodesic on is closed. We show explicit computations in dimensions four and six.