On geodesic points of six-dimensional planar Hermitian submanifolds of Cayley algebra
- DOI
- 10.5922/0321-4796-2026-57-1-2
- Pages
- 28-38
Abstract
In this short note, we study six-dimensional planar submanifolds of Cayley algebra on which the so-called Brown — Gray three-fold vector cross products in the octave algebra induce a Hermitian structure. Such submanifolds of the octave algebra were introduced into consideration by the Russian geometer V. F. Kirichenko in the 90s of the last century. In such six-dimensional planar Hermitian submanifolds, we consider geodesic points, that is, points at which the configuration tensor of the submanifold vanishes. We establish that at geodesic points of six-dimensional planar Hermitian submanifolds of the octave algebra, both the Weyl tensor of conformal curvature and the conharmonic curvature tensor vanish. We also establish that the vanishing of the scalar curvature of a six-dimensional planar submanifold of Cayley algebra implies the vanishing of both the Weyl tensor of conformal curvature and the conharmonic curvature tensor.
Two problems related to geodesic points of six-dimensional planar submanifolds of the octave algebra are formulated. Solutions, or significant advances in solving of these problems, appear entirely possible.
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