“…On the other hand it is well known that every hypersurface S p,q := S 1 p,q has many local tube realizations, compare e.g. with [11,12]. whose sizes are determined by the requirement u 11 , v 11 , w 11 ∈ C m×m .…”
Section: Classification Of Involutions For Certain Cr-manifoldsmentioning
confidence: 99%
“…In the particular case of spherical hypersurfaces the following result has been obtained in [7] by solving a certain partial differential equation coming from the Chern-Moser theory [6]: For every r ≥ 2 there exist, up to affine equivalence, precisely r + 2 closed smooth tube submanifolds of C r that are locally CR-isomorphic to the Euclidean sphere S 2r−1 ⊂ C r . In [12,13] the same method has been used for a certain more general class of CR-flat manifolds. All the above results rely on Chern-Moser theory and therefore only apply to CR-manifolds that are Levi nondegenerate and of hypersurface type.…”
For every real-analytic CR-manifold M we give necessary and sufficient conditions that M can be realized in a suitable neighbourhood of a given point a ∈ M as a tube submanifold of some C r . We clarify the question of the 'right' equivalence between two local tube realizations of the CR-manifold germ (M, a) by introducing two different notions of affine equivalence. One of our key results is a procedure that reduces the classification of equivalence classes to a purely algebraic manipulation in terms of Lie theory.
“…On the other hand it is well known that every hypersurface S p,q := S 1 p,q has many local tube realizations, compare e.g. with [11,12]. whose sizes are determined by the requirement u 11 , v 11 , w 11 ∈ C m×m .…”
Section: Classification Of Involutions For Certain Cr-manifoldsmentioning
confidence: 99%
“…In the particular case of spherical hypersurfaces the following result has been obtained in [7] by solving a certain partial differential equation coming from the Chern-Moser theory [6]: For every r ≥ 2 there exist, up to affine equivalence, precisely r + 2 closed smooth tube submanifolds of C r that are locally CR-isomorphic to the Euclidean sphere S 2r−1 ⊂ C r . In [12,13] the same method has been used for a certain more general class of CR-flat manifolds. All the above results rely on Chern-Moser theory and therefore only apply to CR-manifolds that are Levi nondegenerate and of hypersurface type.…”
For every real-analytic CR-manifold M we give necessary and sufficient conditions that M can be realized in a suitable neighbourhood of a given point a ∈ M as a tube submanifold of some C r . We clarify the question of the 'right' equivalence between two local tube realizations of the CR-manifold germ (M, a) by introducing two different notions of affine equivalence. One of our key results is a procedure that reduces the classification of equivalence classes to a purely algebraic manipulation in terms of Lie theory.
“…For Levi nondegenerate tube manifolds (which necessarily cannot be conical) such examples can be found in [13]. In [25] even two affinely homogeneous examples are contained which are locally affinely non-equivalent but whose associated tube manifolds are locally CR-equivalent.…”
“…with non-degenerate Levi form of type (p−1, q−1). Therefore the classification problem for local tube realizations for both classes is the same, compare also [5], [10], [11], [12], [13], [14], [20] for partial results in this context.…”
Section: Introductionmentioning
confidence: 99%
“…The paper is organized as follows: In Section 2 we relate our results to existing results in the literature, in particular to those in [5], [11], [12]. In Section 3 we recall the necessary tools from [10] and give a short outline of the classification procedure.…”
In this paper we classify up to affine equivalence all local tube realizations of real hyperquadrics in C n . We show that this problem can be reduced to the classification, up to isomorphism, of commutative nilpotent real and complex algebras. We also develop some structure theory for commutative nilpotent algebras over arbitrary fields of characteristic zero.
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