# Triviality of scalar linear type isotropy subgroup by passing to an alternative canonical form of a hypersurface

Annales Polonici Mathematici (1998)

- Volume: 70, Issue: 1, page 85-97
- ISSN: 0066-2216

## Access Full Article

top## Abstract

top## How to cite

topVladimir V. Ežov. "Triviality of scalar linear type isotropy subgroup by passing to an alternative canonical form of a hypersurface." Annales Polonici Mathematici 70.1 (1998): 85-97. <http://eudml.org/doc/262799>.

@article{VladimirV1998,

abstract = {The Chern-Moser (CM) normal form of a real hypersurface in $ℂ^N$ can be obtained by considering automorphisms whose derivative acts as the identity on the complex tangent space. However, the CM normal form is also invariant under a larger group (pseudo-unitary linear transformations) and it is this property that makes the CM normal form special. Without this additional restriction, various types of normal forms occur. One of them helps to give a simple proof of a (previously complicated) theorem about triviality of the scalar linear type isotropy subgroup of a nonquadratic hypersurface. An example of an analogous nontrivial subgroup for a 2-codimensional CR surface in $ℂ^4$ is constructed. We also consider the question whether the group structure that is induced on the family of normalisations to the CM normal form via the parametrisation of the isotropy automorphism group of the underlining hyperquadric coincides with the natural composition operation on the biholomorphisms.},

author = {Vladimir V. Ežov},

journal = {Annales Polonici Mathematici},

keywords = {CR automorphisms; canonical form; scalar linear type isotropy; nonquadratic hypersurface},

language = {eng},

number = {1},

pages = {85-97},

title = {Triviality of scalar linear type isotropy subgroup by passing to an alternative canonical form of a hypersurface},

url = {http://eudml.org/doc/262799},

volume = {70},

year = {1998},

}

TY - JOUR

AU - Vladimir V. Ežov

TI - Triviality of scalar linear type isotropy subgroup by passing to an alternative canonical form of a hypersurface

JO - Annales Polonici Mathematici

PY - 1998

VL - 70

IS - 1

SP - 85

EP - 97

AB - The Chern-Moser (CM) normal form of a real hypersurface in $ℂ^N$ can be obtained by considering automorphisms whose derivative acts as the identity on the complex tangent space. However, the CM normal form is also invariant under a larger group (pseudo-unitary linear transformations) and it is this property that makes the CM normal form special. Without this additional restriction, various types of normal forms occur. One of them helps to give a simple proof of a (previously complicated) theorem about triviality of the scalar linear type isotropy subgroup of a nonquadratic hypersurface. An example of an analogous nontrivial subgroup for a 2-codimensional CR surface in $ℂ^4$ is constructed. We also consider the question whether the group structure that is induced on the family of normalisations to the CM normal form via the parametrisation of the isotropy automorphism group of the underlining hyperquadric coincides with the natural composition operation on the biholomorphisms.

LA - eng

KW - CR automorphisms; canonical form; scalar linear type isotropy; nonquadratic hypersurface

UR - http://eudml.org/doc/262799

ER -

## References

top- [Be] V. K. Beloshapka, On the dimension of the automorphism group of an analytic hypersurface, Izv. Akad. Nauk SSSR Ser. Mat. 93 (1979), 243-266 (in Russian).
- [CM] S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219-271. Zbl0302.32015
- [ES] V. V. Ežov and G. Schmalz, Holomorphic automorphisms of quadrics, Math. Z. 216 (1994), 453-470. Zbl0806.32007
- [Lo1] A. V. Loboda, On local automorphisms of real-analytic hypersurfaces, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), 620-645 (in Russian). Zbl0473.32016
- [Lo2] A. V. Loboda, Generic real-analytic manifolds of codimension 2 in ${\u2102}^{4}$ and their biholomorphic mappings, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), 970-990 (in Russian). Zbl0662.32018
- [Sch] G. Schmalz, Über die Automorphismen einer streng pseudokonvexen CR-Mannigfaltigkeit der Kodimension 2 im ${\u2102}^{4}$, Math. Nachr., to appear.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.