On Sasakian hypersurfaces in 6-dimensional Hermitian

M. B. Banaru

Sbornik. Mathematics, Tome 194 (2003) no. 8, pp. 1125-1136

Résumé

A criterion for the minimality of a Sasakian hypersurface in a 6-dimensional Hermitian submanifold of the octave algebra is found. It is proved that the type number of a Sasakian hypersurface in a 6-dimensional Hermitian submanifold of the octave algebra is four or five. It is also proved that a Sasakian hypersurface in a 6-dimensional Hermitian submanifold of the Cayley algebra is minimal if and only if it is ruled.

Bibliographie

[1] Arnold V. I., Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1989 | MR

[2] Khart N., Geometricheskoe kvantovanie v deistvii, Mir, M., 1985 | MR

[3] Kiritchenko V. F., “Sur la géométrie des variétés approximativement cosymplectiques”, C.R. Acad. Sci. Paris Sér. I Math., 295:12 (1982), 673–676 | MR | Zbl

[4] Blair D. E., “The theory of quasi-Sasakian structures”, Differential Geom., 1 (1967), 331–345 | MR | Zbl

[5] Blair D. E., “Contact manifolds in Riemannian geometry”, Lecture Notes in Math., 509, 1976, 1–145 | MR

[6] Sasaki S., Hatakeyama Y., “On differential manifolds with certain structures which are closely related to almost contact structures. II”, Tôhoku Math. J. (2), 13:2 (1961), 281–294 | DOI | MR | Zbl

[7] Tashiro Y., “On contact structures of hypersurfaces in almost complex manifolds”, Tôhoku Math. J. (2), 15:1 (1963), 62–78 | DOI | MR | Zbl

[8] Kirichenko V. F., Rodina E. V., “O geometrii transsasakievykh i pochti transsasakievykh mnogoobrazii”, Fundament. i prikl. matem., 3:3 (1997), 837–846 | MR | Zbl

[9] Nakayama S., “On a classification of almost contact metric structures”, Tensor (N.S.), 9:1 (1968), 1–7 | MR

[10] Banaru M., “Six theorems on six-dimensional Hermitian submanifolds of Cayley algebra”, Izv. AN Respubliki Moldova, 34:3 (2000), 3–10 | MR | Zbl

[11] Banaru M., “On six-dimensional Hermitian submanifolds of Cayley algebra satisfying the $G$-cosymplectic hypersurfaces axiom”, Annuaire Univ. Sofia Fac. Math. Inform., 94 (2000), 91–96 | MR

[12] Stepanova L., Banaru M., “On hypersurfaces of quasi-Kählerian manifolds”, An. Ştiinţ. Univ. Al. I. Cuza Iaşi Math. (N.S.), XLVII:1 (2001), 165–170 | MR

[13] Banaru M. B., “Ermitova geometriya 6-mernykh podmnogoobrazii algebry Keli”, Matem. sb., 193:5 (2002), 3–16 | MR | Zbl

[14] Arseneva O. E., Kirichenko V. F., “Avtodualnaya geometriya obobschennykh ermitovykh poverkhnostei”, Matem. sb., 189:1 (1998), 21–44 | MR | Zbl

[15] Gray A., Hervella L. M., “The sixteen classes of almost Hermitian manifolds and their linear invariants”, Ann. Mat. Pura Appl. (4), 123:4 (1980), 35–58 | DOI | MR | Zbl

[16] Gray A., “Vector cross products on manifolds”, Trans. Amer. Math. Soc., 141 (1969), 465–504 | DOI | MR | Zbl

[17] Freidental G., “Oktavy, osobye gruppy i oktavnaya geometriya”, Matematika. Sb. per., 1:1 (1957), 117–153

[18] Postnikov M. M., Lektsii po geometrii. Semestr IV. Differentsialnaya geometriya, Nauka, M., 1988 | MR | Zbl

[19] Kirichenko V. F., “Klassifikatsiya kelerovykh struktur, indutsirovannykh 3-vektornymi proizvedeniyami na 6-mernykh podmnogoobraziyakh algebry Keli”, Izv. vuzov. Ser. matem., 1980, no. 8, 32–38 | MR | Zbl

[20] Likhnerovich A., Teoriya svyaznostei v tselom i gruppy golonomii, IL, M., 1960

[21] Gray A., “Some examples of almost Hermitian manifolds”, Illinois J. Math., 10:2 (1966), 353–366 | MR | Zbl

[22] Kobayasi Sh., Nomidzu K., Osnovy differentsialnoi geometrii, t. 2, Nauka, M., 1981

[23] Banaru M. B., “Klassy Greya–Khervelly pochti ermitovykh struktur na 6-mernykh podmnogoobraziyakh algebry Keli”, Nauchnye trudy MPGU im. V. I. Lenina, 1994, 36–38

[24] Stepanova L. V., “Kvazisasakieva struktura na giperpoverkhnostyakh ermitovykh mnogoobrazii”, Nauchnye trudy MPGU im. V. I. Lenina., 1995, 187–191

[25] Banaru M. B., “Tenzory Kirichenko”, Issledovaniya po kraevym zadacham kompleksnogo analiza i differentsialnym uravneniyam, no. 2, SGPU, Smolensk, 2000, 42–48 | MR

[26] Kirichenko V. F., “Metody obobschennoi ermitovoi geometrii v teorii pochti kontaktnykh mnogoobrazii”, Itogi nauki i tekhniki. Problemy geom., 18, VINITI, M., 1986, 25–71 | MR

[27] Norden A. P., Teoriya poverkhnostei, GITTL, M., 1956

[28] Kurihara H., “The type number on real hypersurfaces in a quaternionic space form”, Tsukuba J. Math., 24 (2000), 127–132 | MR | Zbl

[29] Banaru M., “On six-dimensional $Gl$-submanifolds of Cayley algebra”, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 40 (2001), 17–21 | MR | Zbl

[30] Banaru M., Banaru G., “About six-dimensional planar Hermitian submanifolds of Cayley algebra”, Bul. Ştiinţ. Univ. Politeh. Timiş. Ser. Mat. Fiz., 46 (60):1 (2001), 13–17 | MR | Zbl

[31] Banaru M. B., “O $W_3$-mnogoobraziyakh, udovletvoryayuschikh aksiome $G$-kosimplekticheskikh giperpoverkhnostei”, Trudy XXIV Konferentsii molodykh uchenykh mekhaniko-matematicheskogo fakulteta MGU im. M. V. Lomonosova, eds. D. V. Georgievskii, A. N. Yakivchik, Izd-vo MGU, M., 2002

[32] Calabi E., “Construction and properties of some 6-dimensional almost complex manifolds”, Trans. Amer. Math. Soc., 87:2 (1958), 407–438 | DOI | MR | Zbl

[33] Gray A., “Six-dimensional almost complex manifolds defined by means of three-fold vector cross products”, Tôhoku Math. J. (2), 21:4 (1969), 614–620 | DOI | MR | Zbl

[34] Kirichenko V. F., “Ermitova geometriya 6-mernykh simmetricheskikh podmnogoobrazii algebry Keli”, Vestn. MGU. Ser. matem., mekh., 1994, no. 3, 6–13 | MR | Zbl

[35] Banaru M. B., Kirichenko V. F., “Ermitova geometriya 6-mernykh podmnogoobrazii algebry Keli”, UMN, 49:1 (1994), 205–206 | MR | Zbl