The equivalence and the embedding problems for CR-structures of any codimension

Abstract

We give a solution to the equivalence and the embedding problems for smooth CR-submanifolds of complex spaces (and, more generally, for abstract CR-manifolds) in terms of complete differential systems in jet bundles satisfied by all CR-equivalences or CR-embeddings respectively (local and global). For the equivalence problem, the manifolds are assumed to be of finite type and finitely nondegenerate. These are higher order generalizations of the corresponding nondegeneracy conditions for the Levi form. It is shown by a simple example that these nondegeneracy conditions cannot be even slightly relaxed to more general known conditions. In particular, for essentially finite hypersurfaces in $\C^2$, such a complete system may not exist in general. For the embedding problem, the source manifold is assumed to be of finite type and the embeddings to be finitely nondegenerate. Situations are given, where the last condition is automatically satisfied by all CR-embeddings.

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• On the convergence of formal mappings
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Cited by in corpus (17)

• Finite jet determination of CR mappings
• Stationary discs and finite jet determination for non-degenerate generic real submanifolds
• A Counterexample to the $2$-jet determination Chern-Moser Theorem in higher codimension
• Sufficient and necessary conditions for local rigidity of CR mappings and higher order infinitesimal deformations
• Topological Aspects of Holomorphic Mappings of Hyperquadrics from ${\mathbb{C}}^2$ to ${\mathbb{C}}^3$
• Lie group structures on automorphism groups of real-analytic CR manifolds
• Construction of counterexamples to the $2-$jet determination Chern-Moser Theorem in higher codimension
• Rigidity of CR-immersions into spheres
• Finite jet determination of CR embeddings
• On different notions of homogeneity for CR-manifolds
• Local equivalence of symmetric hypersurfaces in ${\mathbb{C}}^2$
• The Catlin multitype and biholomorphic equivalence of models
• Images of real submanifolds under finite holomorphic mappings
• Stationary discs for smooth hypersurfaces of finite type and finite jet determination
• CR-structures of codimension 2 on tangent bundles in Riemann-Finsler geometry
• Projection on Segre varieties and determination of holomorphic mappings between real submanifolds
• Transversality of CR mappings