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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.

The article is dedicated to the memory of the scientist-geo­me­ter Yuri Ivanovich Popov, one of the representatives of the Kali­nin­grad geometric school, professor of the Immanuel Kant Baltic Federal University. The scientific and pedagogical work of the scien­tist during 55 years is described. The area of scientific inte­rests of Yu. I. Popov was the differential geometry of hyperstrips and composite distributions. He is the author of 161 scientific pa­pers and 54 textbooks, the list is presented in this paper.

The theory of connections of manifolds has wide application in ma­the­matics and physics.

Flat connection affects the type of parallel translations.

In the present paper, we study the curvature and torsion of connec­tions on the Grassmann manifold of points.

In projective space, the region described by a point is considered. A prin­cipal bundle arises over the domain, the typical fiber of which is the sta­tio­narity subgroup of a point.

A fundamental group connection according to G. F. Laptev is given in this bundle. It is shown that the Bortolotti’s clothing of the domain in­duces centroprojective connections of three types in the associated bund­le, and these connections have zero curvature and the connections are tor­sion-free.

A geometric characteristics of the resulting connections are given:

1) when the covariant differentials vanish in all three connec­tions, the clothing hyperplane is immovable;

2) a simple subobject  of connection objects , ,  is characterized by the central projection of the plane  adja­cent to the hyperplane  to the original plane  from the center — the point .

In modern differential geometry, one of the main problems of the geo­metry of a space with a differential-geometric structure is the study of the group of affine transformations (automorphisms) of this space. The stu­dies of automorphisms in various spaces of affine connections are de­vo­ted to the works of E. Cartan, P. K. Rashevsky, P. A. Shirokov, I. P. Ego­rov, A.Ya. Sultanov and other scientists.

Affine conversions in direct products of two spaces with affine con­nec­tion were considered in the works of M. V. Morgun. In the case of di­rect products of more than two spaces with affine connection, the ques­tion of affine envelopes, these spaces are stable.

In the article Glebova M. V. and Sultanov A.Ya an estimate was ob­tai­ned for the dimension of the Lie algebra of infinitesimal affine trans­for­mations of spaces with affine connection that represent a direct pro­duct of at least three non-projective Euclidean spaces of the special condition. Such spaces are called spaces of the first type.

In this paper, the accuracy of this estimate is proven. To solve the prob­lem, a system of linear homogeneous equations is investi­ga­ted, which is satisfied by the components of an arbitrary infinite­si­mal affine trans­for­mation.

This system is obtained using the properties of the Lie deriva­tive app­lied to the tensor field of curvature of the spaces under con­sideration. An es­timate of the rank of this system made it pos­sible to obtain a lower es­ti­mate for the rank of the matrix of the ori­ginal system.

The free boundary problem of water waves in three space di­men­sions without surface tension is considered. The problem con­sists of the Laplace equation on the velocity potential, and of the ki­nematic and dynamic boundary conditions. T. Brooke Benjamin and P. Olver have showed that the methods of group analysis of dif­ferential equations can be applied to such problems. The group analysis is based on finding infinitesimal symmetries inherent to the problem. The key point is that each infinitesimal symmetry ge­ne­rates a one-parameter group of symmetries, and that transfor­ming a given solution of the problem by any of the symmetries pro­duces a continuous family of other solutions. The aim of the pre­sent paper is demonstration of application of the group analysis to the problem. The base infinitesimal symmetries of the problem are deduced, their physical meanings are revealed, and their com­mu­tators are computed. It is shown that all the infinitesimal sym­met­ries of the problem form a 13-dimensional non-solvable Lie al­geb­ra, and the corresponding Lie group of symmetries is generated by horizontal and vertical translations, time translation, variation of base-level for potential, horizontal rotations, horizontal and ve­ti­cal Galilean boosts, vertical acceleration, gravity-compensated ro­ta­tions, and scaling. All the results agree with the ones obtained by T. Brooke Benjamin and P. Olver.

A manifold is considered whose structure equations are const­ructed using a deformation of the exterior differential. The torsion and curvature objects of the affine connection on this manifold are not tensors. The curvature object is a tensor, and it is vanishing, only for a canonical connection. The torsion object coincides with the antisymmetry of fiber coordinates and it is non-vanishing even for the canonical connection. Unlike the torsion-free Levi-Civita con­nection, the canonical connection has vanishing curvature and non-vanishing torsion.

Generalized Ricci and Bianchi identities are constructed for cur­vature and torsion of the affine connection on this manifold. Ho­wever, the repeated deformed differential for the basis forms and connection forms vanishes only along a line on the manifold. For the canonical connection, these identities take on a classical form. Moreover, in this case, the repeated deformed differential for the connection forms is identically equal to zero, and the repeated deformed differential for the basis forms vanishes only along the line on the manifold.

The York decomposition of the space of symmetric two-ten­sors originated in theoretical physics and has found applications in Riemannian geometry, as illustrated by its use in Besse’s fa­mous monograph on Einstein manifolds. In this paper, we derive York decompositions for Codazzi, Killing and Rucci tensors on a clo­sed Riemannian manifold. In particular, we derive the York de­com­positions for the Codazzi, Killing and Ricci tensors with cons­tant trace.