Current issue

We consider obtained in 2024 by M. Y. Abass a minimality criterion for hypersurfaces with classical generalized Kenmotsu structure in a Vaisman — Gray manifold. This condition is known to have arisen in the study of hypersurfaces equipped with certain kinds of almost contact metric structures in almost Hermitian manifolds belonging to various Gray — Hervella classes. The con­di­tion was sometimes a minimality criterion for an almost con­tact met­ric hypersurface of an almost Hermitian manifold; in other ca­ses, it turned out to be only necessary or only sufficient.

In the present note, we formulate two problems:

1) to analyze in detail the above-mentioned minimality con­di­tion for an almost contact metric hypersurface of an almost Her­mi­tian manifold;

2) to find out how the Abass minimality condition for a hy­per­sur­face with the classical generalized Kenmotsu structure in a Vais­man — Gray manifold is related to the minimality of a hyper­sur­face with the Kirichenko — Uskorev structure, which is also a ge­ne­ralization of the Kenmotsu structure.

In the present paper, we formulate conditions for the cons­tan­cy of the scalar curvature of an -dimensional  compact Rie­mannian manifold . In particular, conditions for the cons­tan­cy of the scalar curvature of  in the case of the quasi-ne­ga­tive Ricci tensor are found. Conditions are also obtained for a com­pact Riemannian manifold  to be an Einstein manifold.

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.

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.