Geodesic
On the affine homogeneity of tubular type surfaces in $\mathbb C^3$
Informatics and Automation, Analytic and geometric issues of complex analysis, Tome 279 (2012), pp. 102-119.

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We present a systematic approach to solving the problem of affine homogeneity of real hypersurfaces in the three-dimensional complex space. This question is an important part of the general problem of holomorphic classification of homogeneous real hypersurfaces in three-dimensional complex spaces. In contrast to the two-dimensional case, the whole problem (just as its affine part) has not yet been fully studied, although there exist a large number of examples of homogeneous manifolds. We study only the class of tubular type surfaces, which is defined by conditions imposed on the $2$-jet of their canonical equations and generalizes the class of tube manifolds. We discuss the procedure of describing all matrix Lie algebras corresponding to the homogeneous manifolds under consideration. In the class that we study, we distinguish four cases depending on the third-order Taylor coefficients of the canonical equations; in three of these cases, the Lie algebras and the corresponding affine homogeneous surfaces are completely described. The key point of the proposed approach is the solution of a large system of quadratic equations that corresponds to each of the homogeneous surfaces. 
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A. V. Loboda; T. T. D. Nguyẽn. On the affine homogeneity of tubular type surfaces in $\mathbb C^3$. Informatics and Automation, Analytic and geometric issues of complex analysis, Tome 279 (2012), pp. 102-119. http://geodesic.mathdoc.fr/item/TRSPY_2012_279_a8/

[1] Cartan É., “Sur la géométrie pseudo-conforme des hypersurfaces de l'espace de deux variables complexes”, Ann. Mat. Pura ed Appl. Ser. 4, 11 (1932), 17–90 ; ØE uvres complètes, Part 2, v. 2, Gauthier-Villars, Paris, 1953, 1231–1304 | DOI | MR | Zbl

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

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

[4] Fels G., Kaup W., “Classification of Levi degenerate homogeneous CR-manifolds in dimension 5”, Acta math., 201 (2008), 1–82 | DOI | MR | Zbl

[5] Demin A. M., Loboda A. V., “Primer 2-parametricheskogo semeistva affinno odnorodnykh veschestvennykh giperpoverkhnostei v $\mathbb C^3$”, Mat. zametki, 84:5 (2008), 791–794 | DOI | MR | Zbl

[6] Danilov M. S., Loboda A. V., “Ob affinnoi odnorodnosti indefinitnykh veschestvennykh giperpoverkhnostei prostranstva $\mathbb C^3$”, Mat. zametki, 88:6 (2010), 867–884 | DOI | MR | Zbl

[7] Beloshapka V. K., Kossovskiy I. G., “Homogeneous hypersurfaces in $\mathbb C^3$, associated with a model CR-cubic”, J. Geom. Anal., 20:3 (2010), 538–564 | DOI | MR | Zbl

[8] Loboda A. V., “Affinno odnorodnye veschestvennye giperpoverkhnosti 3-mernogo kompleksnogo prostranstva”, Vestn. Voronezh. gos. un-ta. Fizika. Matematika, 2009, no. 2, 71–91

[9] Loboda A. V., “Klassifikatsiya affinno odnorodnykh nevyrozhdennykh po Levi veschestvennykh giperpoverkhnostei prostranstva $\mathbb C^2$”, Sovremennye problemy matematiki i mekhaniki, Vyp. 3: K 100-letiyu so dnya rozhdeniya N. V. Efimova, v. 6, Matematika, Izd-vo MGU, M., 2011, 56–68

[10] Shabat B. V., Vvedenie v kompleksnyi analiz. Ch. 2: Funktsii neskolkikh peremennykh, Nauka, M., 1976 | MR

[11] Nguen T. T. Z., “Affinno odnorodnye poverkhnosti trubchatogo tipa v $\mathbb C^3$”, Sovremennye metody teorii funktsii i smezhnye problemy, Tez. dokl. Voronezh. zimn. mat. shk., Voronezh, 2011, 236–237

[12] Eastwood M., Ezhov V., “On affine normal forms and a classification of homogeneous surfaces in affine three-space”, Geom. dedicata, 77 (1999), 11–69 | DOI | MR | Zbl

[13] Loboda A. V., “Ob odnom semeistve algebr Li, svyazannykh s odnorodnymi poverkhnostyami”, Tr. MIAN, 253, 2006, 111–126 | MR

[14] Fels G., Kaup W., Nilpotent algebras and affinely homogeneous surfaces, E-print, 2011, arXiv: 1101.3088v2[math.AC] | MR

[15] Dadok J., Yang P., “Automorphisms of tube domains and spherical hypersurfaces”, Amer. J. Math., 107:4 (1985), 999–1013 | DOI | MR | Zbl

[16] Chern S. S., Moser J. K., “Real hypersurfaces in complex manifolds”, Acta math., 133 (1974), 219–271 | DOI | MR