2025 №56

An example of an unstable Hermitian structure on a 6-dimensional submanifold of the octave algebra

Abstract

It is known that each of the two so-called Gray — Brown
3-fold vec­tor cross products in the Cayley algebra induces an almost Hermitian struc­ture on its 6-dimensional oriented submani­fold. As it is also known, the almost Hermitian structures induced by different 3-fold vector cross pro­ducts in the octave algebra on the same submanifold can differ sig­ni­ficantly from each other. For example, one of these almost Hermitian struc­tures can be Kähle­rian, while the other is not. Such an almost Her­mi­tian structure is called stable if its dual structure (that is, the structure in­du­ced by another 3-fold vector cross product in the Cayley algebra) be­longs to the same Gray — Hervella class of almost Hermitian structures.

In the present note, we give a specific example of an unstable Her­mitian structure on a 6-dimensional submanifold of the Cayley algebra, namely, on a locally symmetric submanifold of the octave algebra.

Reference

1.  Gray, A.: Vector cross products on manifolds. Trans. Amer. Math. Soc., 41, 465—504 (1969).

2.  Kirichenko, V. F.: The stability of almost Hermitian structures on sub­manifolds of a Cayley algebra. Russian Math. Surveys, 35:1, 215—216 (1980).

3.  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).

4.  Banaru, M. B., Banaru, G. A.: On stability of Hermitian structures on 6-dimensional planar submanifolds of Cayley algebra. DGMF, 52, 23—29 (2021).

5.  Kirichenko, V. F.: Differential-geometric structures on manifolds. Odessa, 2013.

6.  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).

7.  Banaru, M. B.: Hermitian geometry of the 6-dimensional subma­ni­folds of Cayley algebra. PhD thesis. Moscow State Pedagogical Univer­sity V. I. Lenin (1993).

8.  Banaru, M. B.: On skew-symplectic hypersurfaces of six-dimen­sio­nal Kählerian submanifolds of the Cayley algebra. Russian Math. (Iz­ves­tia Vuzov), 47:7, 60—63 (2003).

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

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

11.  Kirichenko, V. F.: Hermitian geometry of six-dimensional sym­met­ric submanifolds of the Cayley algebra. Mosk. Univ. Bull. Ser. 1. Mat. Mekh., 3, 6—13 (1994).

12.  Banaru, M. B.: On locally symmetric 6-dimensional Hermitian submanifolds of Cayley algebra. DGMF, 47, 11—17 (2016).

13.  Gray, A.: Some examples of almost Hermitian manifolds. Illinois J. Math., 10:2, 353—366 (1966).

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

15.  Banaru, M. B.: On the six-dimensional sphere with a nearly Käh­lerian structure. J. Math. Sci., New York, 245:5, 553—567 (2020).