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Maximal complexifications of certain homogeneous Riemannian manifolds. (English) Zbl 1039.32009

Let \(M\) be a complete real-analytic Riemannian manifold. Identify \(M\) with the zero-section of \(TM\), and let \((\Omega)\) be a domain around it. A real-analytic complex structure on \(\Omega \) is said to be adapted to the geometry of \(M\) if geodesics \(( \gamma : R \rightarrow TM )\) can be extended to a complex domain in a certain natural way with this extension being holomorphic. If \(( \Omega )\) is small enough such an adapted complex structure exists. One is naturally interested in finding the largest such \(( \Omega).\)
In this paper, the authors identify this maximal domain for \(( M = G/ K)\) a homogeneous space satisfying \(( \dim _{C} G^{C} = \dim _{R} G)\) and a condition on the extendibility of the geodesic flow. They then apply these results to the three-dimensional Heisenberg group and to generalized Heisenberg groups. In either case, the unique maximal adapted complexification \(( \Omega _{H} )\) is neither holomorphically separable nor holomorphically convex. They are not Stein either, and to the knowledge of the authors and this reviewer are the first such examples. For the three-dimensional Heisenberg group the description of \(( \Omega _{H} )\) is quite explicit, for the characterization given in their main theorem is something that one can hope to calculate.

MSC:

32C09 Embedding of real-analytic manifolds
53C30 Differential geometry of homogeneous manifolds
32M05 Complex Lie groups, group actions on complex spaces
32Q28 Stein manifolds
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