## On six-dimensional $$G_1$$-submanifolds of Cayley algebra.(English)Zbl 1059.53045

The classification of the almost Hermitian structures has been done by A. Gray and L. M. Hervella [Ann. Mat. Pura Appl., IV. Ser. 123, 35–58 (1980; Zbl 0444.53032)]. With the terminology of the above paper, the author proves that 6-dimensional $$G_1$$-submanifolds of the octave algebra are $$W+W$$-manifolds. Some interesting corollaries follow the main result. This article has been also published in Taiwanese J. Math. 6, No. 3, 383–388 (2002; Zbl 1030.53061).

### MSC:

 53C40 Global submanifolds

### Keywords:

Hermitian structures; octave algebra

### Citations:

Zbl 0444.53032; Zbl 1030.53061
Full Text:

### References:

 [1] Gray A., Hervella L. M.: The sixteen classes of almost Hermitian manifolds and their linear invariant. Ann. Mat., Pure ed Appl. 123, 4 (1980), 35-58. · Zbl 0444.53032 [2] Hervella L. M., Vidal E.: Novelles géométries pseudo-Kählériennes G1 et G2. C. R. Acad. Sci. Paris 283 (1976), 115-118. [3] Gray A.: Vector cross products on manifolds. Trans. Amer. Math. Soc. 141 (1969), 465-504. · Zbl 0182.24603 [4] Gray A.: Some examples of almost Hermitian manifolds. Ill. J. Math. 10, 2 (1966), 353-366. · Zbl 0183.50803 [5] Vaisman I.: On locally conformal almost Kähler manifolds. Israel J. Math. 24 (1976), 338-351. · Zbl 0335.53055 [6] Kirichenko V. F.: On nearly-Kählerian structures induced by means of 3-vector cross products on six-dimensional submanifolds of Cayley algebra. Vestnik MGU 3 (1973), 70-75. [7] Kirichenko V. F.: Classification of Kählerian structures induced by means of 3-vector cross products on six-dimensional submanifolds of Cayley algebra. Izvestia Vuzov, Kazan, 8 (1980), 32-38. · Zbl 0449.53049 [8] Kirichenko V. F.: Rigidity of almost Hermitian structures induced by means of 3-vector cross products on six-dimensional submanifolds of Cayley algebra. Ukrain Geom. Coll. 25 (1982), 60-68. · Zbl 0508.53045 [9] Freudenthal H.: Octaves, singular groups and octaves geometry. Coll. Math., Moscow, 1957, 117-153. [10] Lichnerovicz A.: Théorie globale des connexions et des groupes d’holonomie. Cremonese, Roma, 1955. · Zbl 0116.39101 [11] Cartan E.: Riemannian geometry in an orthogonal frame. MGU, Moscow, 1960. [12] Banaru M.: Hermitian geometry of six-dimensional submanifolds of Cayley algebra. MSPU, Moscow, 1993. · Zbl 1031.53087 [13] Banaru M.: On almost Hermitian structures induced by 3-vector cross products on six-dimensional submanifolds of Cayley algebra. Polyanalytical Functions, Smolensk, 1997, 113-117.
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