Abstract
We consider compact Leviflat homogeneous CauchyâRiemann (CR) manifolds. In this setting, the Levi-foliation exists and we show that all its leaves are homogeneous and biholomorphic. We analyze separately the structure of orbits in complex projective spaces and parallelizable homogeneous CR-manifolds in our context and then combine the projective and parallelizable cases. In codimensions one and two, we also give a classification.
Highlights
We study the geometry of compact homogeneous Leviflat CR-manifolds
Because the CR-structure in the homogeneous setting is analytic, it turns out that there is a foliation of the CR-manifold whose leaves have tangent bundles corresponding to the distribution given by the zero spaces of the Levi form and we call this the Levi-foliation
It is well known that every homogeneous CR-manifold admits a homogeneous fibration, called the CR-normalizer fibration, whose fiber is parallelizable and whose base is an orbit in some projective space [23, 35], etc
Summary
3.1 that the leaves of this Levi-foliation on the homogeneous CR-manifold are homogeneous themselves under the action of a complex Lie subgroup of G. This setting is very special, since this implies that all the leaves are biholomorphic to one another. The quotient by the radical orbits is a compact homogeneous space of a maximal complex semisimple factor of the Lie group G, again with discrete isotropy. There is not much room left for transversal directions and the classification of the surfaces that occur as the corresponding leaf-spaces is well known, see [39] and [22] This yields the classification in the projective setting
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