JOURNAL BROWSE
Search
Advanced SearchSearch Tips
NORMAL SYSTEMS OF COORDINATES ON MANIFOLDS OF CHERN-MOSER TYPE
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
NORMAL SYSTEMS OF COORDINATES ON MANIFOLDS OF CHERN-MOSER TYPE
Schmalz, Gerd; Spiro, Andrea;
  PDF(new window)
 Abstract
It is known that the CR geometries of Levi non-degen-erate hypersurfaces in and of the elliptic or hyperbolic CR submanifolds of codimension two in share many common features. In this paper, a special class of normalized coordinates is introduced for any CR manifold M which is one of the above three kinds and it is shown that the explicit expression in these coordinates of an isotropy automorphism , is equal to the expression of a corresponding element of the automorphism group of the homogeneous model. As an application of this property, an extension theorem for CR maps is obtained.
 Keywords
CR structures;Chern-Moser bundle;normal coordinates;
 Language
English
 Cited by
1.
The Equivalence Problem for Five-dimensional Levi Degenerate CR Manifolds, International Mathematics Research Notices, 2013  crossref(new windwow)
 References
1.
V. K. Belosapka, A uniqueness theorem for automorphisms of a non-degenerate surface in the complex space (in Russian), Mat. Zametki 47 (1990), no. 3, 17–22. crossref(new window)

2.
S. S. Chern and J. Moser, Real Hypersurfaces in Complex Manifolds, Acta Math. 133 (1974), 219–271. crossref(new window)

3.
A. Cap and H. Schichl, Parabolic Geometries and Canonical Cartan Connections, Hokkaido Math. J. 29 (2000), no. 3, 453–505.

4.
A. Cap and G. Schmalz, Partially integrable almost CR manifolds of CR dimension and codimension two in “Lie Groups, Geometric Structures and Differential Equations–One Hundred Years After Sophus Lie” T.Morimoto, H. Sato, and K. Yamaguchi (eds.) Adv. Stud. in Pure Math., vol. 37, 2002

5.
V. V. Ezhov and G. Schmalz, Normal forms and two-dimensional chains of an elliptic CR surface in $\mathbb{C}^4$, J. Geom. Anal. 6 (1996), no. 4, 495–529. crossref(new window)

6.
A. V. Loboda, On local automorphisms of real analytic hypersurfaces (in Russian), Izv. Akad. Nauk SSSR (Ser. Mat.) 45 (1981), no. 3, 620–645. crossref(new window)

7.
A. V. Loboda, Generic real analytic manifolds of codimension 2 in $\mathbb{C}^4$ and their biholo-morphic mappings, Izv. Akad. Nauk SSSR (Ser. Mat.) 52, no. 5, 970–990; Engl. transl. in Math. USSR Izv. 33 (1989), no.2, 295–315.

8.
G. Schmalz, Über die Automorphismen einer streng pseudokonvexen CR-Manningfaltigkeit der Kodimension 2 im $\mathbb{C}^4$, Math. Nachr. 196 (1998), 189–229. crossref(new window)

9.
G. Schmalz, Remarks on CR-manifolds of Codimension 2 in $\mathbb{C}^4$, Proceeding Winter School Geometry and Physics, Srni 1998, Supp. Rend. Circ. Matem. Palermo, Ser. II 59 (1999), 171–180

10.
G. Schmalz and J. Slovak, The Geometry of Hyperbolic and Elliptic CR manifolds of codimension two, Asian J. Math. 4 (2000), no. 3, 565–598.

11.
G. Schmalz and A. Spiro, Explicit construction of a Chern-Moser connection for CR manifolds of codimension two, preprint (2002)

12.
J. Slovak, Parabolic geometries, part of the DrSc Dissertation, preprint IGA 11/97

13.
A. Spiro, Smooth real hypersurfaces in $C^n$ with non compact isotropy groups of CR transformations, Geom. Dedicata 67 (1997), 199–221. crossref(new window)

14.
N. Tanaka, On the equivalence problem associated with simple graded Lie algebras, Hokkaido Math. J. 8 (1979), 131-190.

15.
A. G. Vitushkin, Holomorphic Mappings and the Geometry of Hypersurfaces, in Encyclopaedia of Mathematical Sciences vol. 7 (Several Complex Variables I), VINITI-Springer-Verlag, (1985-1990)