Geodesic
On complete list of affinely homogeneous surfaces of ($\varepsilon,0$)-types in the space~$\mathbb C^3$
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2015), pp. 75-81.

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The problem of describing the affine homogeneous real hypersurfaces in complex spaces is an important part of a difficult problem of holomorphic classification for homogeneous manifolds, that have not a complete solution till now, even in the $3$-dimensional case. The scheme developed by the authors which uses canonical affine equations and techniques of matrix Lie algebras, previously allowed to obtain a full description of two natural classes of affine-homogeneous real hypersurfaces in $3$-dimensional complex space. In this paper we present the complete description of one another class. It comprises the known partial examples, and (obtained with the use of symbolic computations) Lie algebras corresponding to the remaining homogeneous manifolds of discussed types. The main result is obtained by means of the integration of these algebras.
Keywords: complex space, homogeneous manifold, vector field, Lie algebra, canonical equation.
Mots-clés : affine transformation 
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A. V. Loboda; A. V. Shipovskaya. On complete list of affinely homogeneous surfaces of ($\varepsilon,0$)-types in the space~$\mathbb C^3$. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2015), pp. 75-81. http://geodesic.mathdoc.fr/item/IVM_2015_6_a8/

[1] Cartan E., “Sur la géometrie pseudoconforme des hypersurfaces de deux variables complèxes”, Ann. Math. Pura Appl. (4), 11 (1932), 17–90 | MR | Zbl

[2] Loboda A. V., “Odnorodnye strogo psevdovypuklye giperpoverkhnosti v $\mathbb C^3$ s dvumernymi gruppami izotropii”, Matem. sb., 192:12 (2001), 3–24 | DOI | MR | Zbl

[3] Fels G., Kaup W., “Classification of Levi degenerate homogeneous CR-manifolds in dimension 5”, Acta Math., 210 (2008), 1–82 | DOI | MR

[4] Beloshapka V. K., Kossovskiy I. G., “Classification of homogeneous CR-manifolds in dimension 4”, J. Math. Anal.Appl., 374:2 (2011), 655–672 | DOI | MR | Zbl

[5] Loboda A. V., Khodarev A. V., “Ob odnom semeistve affinno-odnorodnykh veschestvennykh giperpoverkhnostei $3$-mernogo kompleksnogo prostranstva”, Izv. vuzov. Matem., 2003, no. 10, 38–50 | MR | Zbl

[6] Loboda A. V., Nguen T. T. Z., “Ob affinnoi odnorodnosti poverkhnostei trubchatogo tipa v $\mathbb C^3$”, Tr. MIAN, 279, 2012, 102–119 | MR | Zbl

[7] Loboda A. V., “O polnom opisanii affinno-odnorodnykh veschestvennykh giperpoverkhnostei trubchatogo tipa prostranstva $\mathbb C^3$”, Voronezhsk. zimnyaya matem. shkola (VZMSh-2013), Tez. dokl., Voronezh, 2013, 144–145

[8] Atanov A. V., Loboda A. V., Shipovskaya A. V., Affine homogeneous strictly pseudoconvex hypersurfaces of the type $(1/2,0)$ in $\mathbb C^3$, arXiv: 1401.2252v1

[9] Nguen T. T. Z., “Affinno-odnorodnye veschestvennye giperpoverkhnosti trubchatogo tipa v $\mathbb C^3$”, Matem. zametki, 94:2 (2013), 246–265 | DOI | MR | Zbl

[10] Bishop R., Krittenden R., Geometriya mnogoobrazii, Mir, M., 1963