Linearization of Isotropic Automorphisms of Non-quadratic Elliptic CR-Manifolds in ℂ4

Ežov, Vladimir V.; Schmalz, Gerd

doi:10.1007/978-3-642-55627-2_6

Cited by 2 publications

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“…We will show that the maximal dimension of the isotropy for nonquadratic elliptic CR-manifolds is 4 and is attained exactly for manifolds that correspond to the ODE's y ′′ = y k (y − xy ′ ) 3 y ′′ = y ℓ y ′ (y − xy ′ ) 2 + Cy 2ℓ+2 (y − xy ′ ) 3 where k, ℓ are non-negative integers and C is a complex constant. Thus, according to earlier results by the authors [3], the possible dimensions of the isotropy of elliptic CR-manifolds are 10, 4, 3, 2, 1, 0. This is somewhat unexpected, because the corresponding numbers for analogous hyperbolic manifolds are 10, 6, 5, 3, 2, 1, 0 (see [6]).…”

Section: Introductionmentioning

confidence: 93%

Elliptic CR-manifolds and shear invariant ordinary differential equations with additional symmetries

Ezhov

Schmalz

2007

Ark. Mat.

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We classify the ordinary differential equations that correspond to elliptic CRmanifolds with maximal isotropy. It follows that the dimension of the isotropy group of an elliptic CR-manifold can only be 10 (for the quadric), 4 (for the listed examples) or less. This is in contrast with the situation of hyperbolic CR-manifolds, where the dimension can be 10 (for the quadric), 6 or 5 (for semi-quadrics) or less than 4. We also prove that, for all elliptic CR-manifolds with non-linearizable isotropy group, except for two special manifolds, the points with non-linearizable isotropy form exactly some complex curve on the manifold.

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Section: Introductionmentioning

confidence: 93%

Elliptic CR-manifolds and shear invariant ordinary differential equations with additional symmetries

Ezhov

Schmalz

2007

Ark. Mat.

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“…It can be proved by the methods in [22], [23] that the isotropy group of an elliptic CRmanifold M is linearizable if M is not torsion-free. Moreover, we may assume that a nonlinearizable infinitesimal automorphism w of the torsion-free manifold V ¼ z 1 z 2 þ Nðz 1 ; z 2 ; hÞ coincides with w Q up to terms of higher order.…”

Section: Non-linearizable Automorphisms Of Elliptic Cr-manifoldsmentioning

confidence: 99%

“…Application of similar techniques to elliptic manifolds [22], [23] showed that manifolds with non-linearizable isotropy group must be torsion-free and its equations must satisfy extremely strong recursive conditions that seemed to be unsatisfiable.…”

Section: Introductionmentioning

confidence: 99%

“…The study of certain symmetries of ODE in the spirit of S. Lie [27], [28] and A. Tresse [46] leads us to a complete description of elliptic manifolds with non-trivial G 1 . It has been proved earlier by the authors [23] that only these manifolds may have nonlinearizable isotropy groups.…”

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confidence: 96%

“…half of the curvature components vanish). Thus, the phenomenon of non-linearizability is strongly related to quadrics.Application of similar techniques to elliptic manifolds [22], [23] showed that manifolds with non-linearizable isotropy group must be torsion-free and its equations must satisfy extremely strong recursive conditions that seemed to be unsatisfiable.In this article we adopt a di¤erent approach based on Cartan geometry [9], [37], [38] and on the relation between torsion-free elliptic CR-manifolds and complex second order Ezhov and Schmalz, Non-linearizable CR-automorphisms 216 Brought to you by | Freie Universität Berlin Authenticated Download Date | 7/8/15 6:36 AM ODE [43], [14], [34]. The study of certain symmetries of ODE in the spirit of S. Lie [27], [28] and A. Tresse [46] leads us to a complete description of elliptic manifolds with non-trivial G 1 .…”

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Non-linearizable CR-automorphisms, torsion-free elliptic CR-manifolds and second order ODE

Ezhov

Schmalz

2005

Journal für die reine und angewandte Mathematik (Crelles Journa

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The existence of non-linearizable isotropic CR-automorphisms of Levi non-degenerate hypersurfaces in complex space is characteristic for quadrics. In this paper we discover elliptic CR-manifolds of CR-dimension and codimension two that are not related to the quadric and that have non-linearizable isotropic automorphisms. This is a new and unexpected phenomenon. A complete description of such manifolds is given. IntroductionThe interest to invariants and automomorphisms of real submanifolds in complex space goes back to 1907, when Poincaré [35] realized that there are ''more'' hypersurfaces in C 2 than biholomorphic mappings. He asked for a classification of hypersurfaces with respect to holomorphic coordinate changes. This problem was solved by E. Cartan [12] (in 1932 for C 2 ) and by N. Tanaka [44] (in 1967 for C N ) and independently by S. Chern and J. Moser [13] (1974), who obtained a complete description of the invariants of real hypersurfaces. The celebrated papers by Cartan and by Chern and Moser initiated fruitful developments in di¤erent directions including theory of normal forms of CR-manifolds [48], [47], [31], [20], [21], CR-geometry [3], [6], [7], [5], [4], [30], [32], [22]. The theory of invariants of real hypersurfaces is a significant part of the parabolic theory of invariants, which was introduced in 1979 by C. Fe¤erman [24], [2], [10], [9], [37].A natural global CR-invariant with a clear geometric meaning is the automorphism group of the submanifold. Its punctual version is the isotropy group at a given point.Let M be a real submanifold in C N that contains the origin. The isotropy group Aut 0 M is defined as the group of germs of biholomorphic maps f with fð0Þ ¼ 0 and fðMÞ H M. The following theorem provides an upper bound for the dimension of this group.Theorem 0. Let M be a real-analytic, Levi non-degenerate, generic submanifold in C N passing through the origin. Then any isotropic automorphism f A Aut 0 M is uniquely determined by its first and second order derivatives at 0. The research of both authors was supported by Max-Planck-Institut fü r Mathematik Bonn. Brought to you by | Freie Universität Berlin Authenticated Download Date | 7/8/15 6:36 AM This theorem was proved by Chern and Moser for hypersurfaces [13] and by Beloshapka [5] for submanifolds of higher codimension.It turns out that the manifold with maximal isotropy group in the class of manifolds with given Levi-form is the quadricwhere ðz; wÞ are suitable coordinates in the ambient space C nþk and hz; zi is the nondegenerate, vector-valued Levi form. For non-quadratic submanifolds (i.e. submanifolds that are not locally equivalent to the quadric) the dimension may significantly drop.The isotropy group of a hyperquadric has dimension ðn þ 1Þ 2 þ 1. For non-quadratic hypersurfaces this dimension does not exceed n 2 -the dimension of the group of pseudounitary n Â n matrices. In this case any isotropic automorphism is uniquely determined by the restriction of its di¤erential to the maximal complex subspace T CR 0 M H T 0 M, i.e. the...

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Linearization of Isotropic Automorphisms of Non-quadratic Elliptic CR-Manifolds in ℂ4

Cited by 2 publications

References 9 publications

Elliptic CR-manifolds and shear invariant ordinary differential equations with additional symmetries

Elliptic CR-manifolds and shear invariant ordinary differential equations with additional symmetries

Non-linearizable CR-automorphisms, torsion-free elliptic CR-manifolds and second order ODE

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