Complexifications of geodesic flows and adapted complex structures

The existence of adapted complex structures for real-analytic Riemannian manifolds is examined under the point view of complexifications of geodesic flows. If the geodesic flow can be complexified to a complete holomorphic flow a sufficient criterion is given as to when a domain in the tangent bundle of the Riemannian manifold is a maximal domain of definition of an adapted complex structure. This criterion allows to determine maximal domains of definition for adapted complex structures of Berger spheres and of the Heisenberg group.