Abstract
We investigate the nondegeneracy of higher order Levi forms on weakly nondegenerate homogeneous CR manifolds. Improving previous results, we prove that general orbits of real forms in complex flag manifolds have order less or equal than 3 and the compact ones less or equal 2. Finally we construct by Lie extensions weakly nondegenerate CR vector bundles with arbitrary orders of nondegeneracy.
Highlights
The Levi form is a basic invariant of C R geometry
Sufficient more general conditions preventing a C R manifold M from being foliated by complex leaves of positive dimension or from having an infinite dimensional group of local C R automorphisms can be expressed by the nondegeneracy of higher order Levi forms
If GR is a real form of a complex Lie algebra G and q the Lie algebra of its closed subgroup Q, M is locally C R diffeomorphic to the orbit of GR in the complex homogeneous space G /Q and its C R structure is induced by the complex structure of G /Q
Summary
The Levi form is a basic invariant of C R geometry (see e.g. [9]). It is a hermitian symmetric form on the space of tangent holomorphic vector fields, which, when the C R codimension is larger than one, is vector valued. Sufficient more general conditions preventing a C R manifold M from being foliated by complex leaves of positive dimension or from having an infinite dimensional group of local C R automorphisms can be expressed by the nondegeneracy of higher order Levi forms Iterations of the Levi forms can be described by building descending chains of algebras of vector fields, whose lengths can be taken as a measure of nondegeneracy In [10] Fels posed the question of the existence of weakly nondegenerate homogeneous C R manifolds with Levi order larger than 3. 3 we exhibit, by constructing some C R vector bundles over CP1, weakly nondegenerate homogeneous C R manifolds having Levi order q, for every positive integer q
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