Maximal complexifications of certain homogeneous Riemannian manifolds

Abstract:

Let $M=G/K$ be a homogeneous Riemannian manifold with $\dim _{\mathbb {C}} G^{\mathbb {C}} = \dim _{\mathbb {R}} G$, where $G^{\mathbb {C}}$ denotes the universal complexification of $G$. Under certain extensibility assumptions on the geodesic flow of $M$, we give a characterization of the maximal domain of definition in $TM$ for the adapted complex structure and show that it is unique. For instance, this can be done for generalized Heisenberg groups and naturally reductive homogeneous Riemannian spaces. As an application it is shown that the case of generalized Heisenberg groups yields examples of maximal domains of definition for the adapted complex structure which are neither holomorphically separable nor holomorphically convex.