On some tensors of six-dimensional Hermitian planar submanifolds of Cayley algebra

Abstract

In the present note, we consider six-dimensional Hermitian planar sub­manifolds of Cayley algebra. The almost Hermitian structure on such a six-dimensional submanifold is induced by means of so-called Brown — Gray three-fold vector cross products in Cayley algebra. The six-dimen­sio­nal Hermitian planar submanifolds of the octave algebra contain all six-dimensional Kählerian submanifolds of Cayley algebra. However, there exist non-Kählerian six-dimensional Hermitian planar submanifolds in the octave algebra. The components of the tensor of the Riemannian curvature for a six-dimensional almost Hermitian planar submanifold of Cayley algebra are computed. Remark that the tensor of Riemannian curvature plays a fun­damental role in geometry of almost Hermitian manifolds. Knowing all components of the tensor of the Riemannian curvature for a six-dimen­sio­nal almost Hermitian planar submanifold of the octave algebra, it is possible to study so-called Gray’s identities for this submanifold. The components of the Ricci tensor and of the tensor of conformal curvature (known also as Weyl tensor) for a six-dimensional almost Her­mitian planar submanifold of Cayley algebra are also computed.

Keywords

• almost hermitian structure
• tensor of riemannian curva­ture
• ricci tensor
• tensor of conformal curvature
• six-dimensional hermit­ian planar submanifold of cayley algebra