Search
Submit an article
Differential Geometry of Manifolds

2021 №52

Back to the list Download the article
Banaru M.B. Banaru G.A

On stability of Hermitian structures on 6-dimensional planar submanifolds of Cayley algebra

DOI
10.5922/0321-4796-2021-52-3
Pages
23-29
Cayley algebra 6-dimensional planar submanifold of Cay­ley algebra stability of the almost Hermitian structure

Abstract

We consider 6-dimensional planar submanifolds of Cayley algebra. As it is known, the so-called Brown — Gray three-fold vector cross prod­ucts induce almost Hermitian structures on such submanifolds. We select the case when the almost Hermitian structures on 6-dimensional planar submanifolds of Cayley algebra are Hermitian, i. e. these structures are in­tegrable.

It is proved that the Hermitian structure on a 6-dimensional planar submanifold of Cayley algebra is stable if and only if such submanifold is totally geodesic.

Reference

1. Gray, A.: Vector cross products on manifolds. Trans. Amer. Math. Soc., 141, 465—504 (1969).

2. Kirichenko, V. F.: Stability of the almost Hermitian structures, de­fined by means of three-fold vector cross products on six-dimensional sub­manifolds of Cayley algebra. Ukrain. Geom. Sbornik, 25, 60—68 (1982).

3. Banaru, M. B., Kirichenko, V. F.: The Hermitian geometry of the 6-dimensional submanifolds of Cayley algebra. Russian Mathematical Sur­veys, 49:1, 223—225 (1994).

4. Banaru, M. B.: Hermitian geometry of the 6-dimensional submani­folds of the Cayley algebra. Sb. Math., 193:5—6, 635—648 (2002).

5. Banaru, M. B., Banaru G. A.: A note on six-dimensional planar Hermitian submanifolds of Cayley algebra. Bul. Acad. Ştiinţe Repub. Moldova. Matem., 1:74, 23—32 (2014).

6. Banaru, M. B., Banaru, G. A.: 1-cosymplectic hypersurfaces axiom and six-dimensional planar Hermitian submanifolds of the Octonian. SUT J. Math., 51:1, 1—9 (2015).

7. Kirichenko, V. F.: Classification of Kählerian structures, defined by means of three-fold vector cross products on six-dimensional submani­folds of Cayley algebra. Izvestia Vuzov. Math., 8, 32—38 (1980).

8. Banaru, M. B.: Geometry of 6-dimensional Hermitian manifolds of the octave algebra. J. Math. Sci., 207:3, 354—388 (2015).

9. Banaru, M. B., Banaru, G. A.: On planar 6-dimensional Hermitian sub­manifolds of Cayley algebra. DGMF. Kaliningrad. 48, 21—25 (2017).