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 $a\in M$ as a tube submanifold of some ${\u2102}^{r}$. 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.

Classification: 32V05, 32V40, 32M25, 17B66

@article{ASNSP_2011_5_10_1_99_0, author = {Fels, Gregor and kaup, Wilhelm}, title = {Local tube realizations of CR-manifolds and maximal Abelian subalgebras}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 10}, number = {1}, year = {2011}, pages = {99-128}, zbl = {1229.32020}, mrnumber = {2829317}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2011_5_10_1_99_0} }

Fels, Gregor; kaup, Wilhelm. Local tube realizations of CR-manifolds and maximal Abelian subalgebras. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 1, pp. 99-128. http://www.numdam.org/item/ASNSP_2011_5_10_1_99_0/

[1] M. S. Baouendi, L. P. Rothschild and F. Treves, CR structures with group action and extendibility of CR functions, Invent. Math. **82** (1985), 359–396.
| MR 809720
| Zbl 0598.32019

[2] M. S. Baouendi - H. Jacobowitz and F. Treves, On the analyticity of CR mappings, Ann. of Math. **122** (1985), 365–400.
| MR 808223
| Zbl 0583.32021

[3] M. S. Baouendi - P. Ebenfelt and L. P. Rothschild, “Real Submanifolds in Complex Spaces and Their Mappings”, Princeton Math. Series, Vol. 47, Princeton Univ. Press, 1998. | MR 1668103 | Zbl 0944.32040

[4] M. S. Baouendi - L. Rothschild and D. Zaitsev, Equivalences of real submanifolds in complex space, J. Differential Geom. **59** (2001), 301–351.
| MR 1908985
| Zbl 1037.32030

[5] A. Boggess, “CR manifolds and the tangential Cauchy Riemann complex”, Studies in Advanced Mathematics, Boca Raton, FL: CRC Press, 1991. | MR 1211412 | Zbl 0760.32001

[6] S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta. Math. **133** (1974), 219–271.
| MR 425155
| Zbl 0302.32015

[7] J. Dadok and P. Yang, Automorphisms of tube domains and spherical hypersurfaces, Amer. J. Math. **107** (1985), 999–1013.
| MR 796910
| Zbl 0586.32035

[8] G. Fels, Locally homogeneous finitely nondegenerate CR-manifolds, Math. Res. Lett. **14** (2007), 693–922.
| MR 2357464
| Zbl 1155.32027

[9] G. Fels and W. Kaup, CR-manifolds of dimension 5: A Lie algebra approach, J. Reine Angew. Math. **604** (2007), 47–71.
| MR 2320313
| Zbl 1128.32023

[10] G. Fels and W. Kaup, Classification of Levi degenerate homogeneous CR-manifolds of dimension 5, Acta Math. **201** (2008), 1–82.
| MR 2448066
| Zbl 1171.32023

[11] G. Fels and W. Kaup, Classification of commutative algebras and tube realizations of hyperquadrics, arXiv:0906.5549.

[12] A. V. Isaev and M. A. Mishchenko, Classification of spherical tube hypersurfaces that have one minus in the Levi signature form, Math. USSR-Izv. **33** (1989), 441–472.
| MR 984213
| Zbl 0677.32003

[13] A. V. Isaev, Classification of spherical tube hypersurfaces that have two minuses in the Levi signature form, Math. Notes **46** (1989), 517–523.
| MR 1019254
| Zbl 0725.32001

[14] A. V. Isaev, Global properties of spherical tube hypersurfaces, Indiana Univ. Math. J. **42** (1993), 179–213.
| MR 1218712
| Zbl 0797.32014

[15] A. V. Isaev and W. Kaup, Regularization of local CR-automorphisms of real-analytic CR-manifolds, J. Geom. Anal. (2011), to appear. | MR 2868965 | Zbl 1261.32005

[16] W. Kaup, On the holomorphic structure of $G$-orbits in compact hermitian symmetric spaces, Math. Z. **249** (2005), 797–816.
| MR 2126217
| Zbl 1136.32007

[17] W. Kaup, CR-quadrics with a symmetry property, Manuscripta Math. **133** (2010), 505–517.
| MR 2729265
| Zbl 1202.32032

[18] W. Kaup and D. Zaitsev, On local CR-transformations of Levi degenerate group orbits in compact Hermitian symmetric spaces, J. Eur. Math. Soc. **8** (2006), 465–490.
| MR 2250168
| Zbl 1118.32019

[19] R. S. Palais, “A Global Formulation of the Lie Theory of Transformation Groups”, Mem. Amer. Math. Soc. 1957. | MR 121424 | Zbl 0178.26502

[20] N. Tanaka, On the pseudo-conformal geometry of hypersurfaces of the space of $n$ complex variables, J. Math. Soc. Japan **14** (1962), 397–429.
| MR 145555
| Zbl 0113.06303

[21] D. Zaitsev, On different notions of homogeneity for CR-manifolds, Asian J. Math. **11** (2007), 331–340.
| MR 2328898
| Zbl 1138.32018