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Differential Geometry of Manifolds

2025 №56

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Banaru G.A.

On conharmonic curvature tensor of six-dimensional Kählerian submanifolds of Cayley algebra

Abstract

Conharmonic transformations are conformal transformations that preserve the property of harmonicity of smooth functions. This type of transformation was introduced into consideration in the 50s of the last century by the Japanese mathematician Y. Ishii. It is known that such transformations have a tensor invariant — the so-called conhar­mo­nic curvature tensor. Note that complementing the Riemannian struc­ture to an almost Hermitian structure allows us to single out so­me additional conharmonic invariants.

In this paper, we consider the conharmonic curvature tensor of
6-di­men­sional Kählerian submanifolds of the octave algebra. The Käh­le­rian (and in the general case, almost Hermitian) structure on such sub­manifolds is induced by the so-called Gray — Brown
3-vector cross products in the Cayley algebra.

The main result of the work is the calculation of the so-called spect­rum of the conharmonic curvature tensor for an arbitrary 6-di­men­sional Kählerian submanifold of the octave algebra. By the con­cept of the spectrum of a tensor, we mean the minimal set of the com­po­nents in the space of the associated G-structure that completely de­termines this tensor.

Reference


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