2023 №54(1)

The article presents a very brief biography of a corresponding mem­ber of the Russian Academy of Natural Sciences, Honored Scientist of the Russian Federation, Honorary Doctor of Sciences of the I. Kant Baltic Federal University, professor-consultant of the Institute of Physical and Mathematical Sciences and Information Technologies of I. Kant BFU. The information about the scientific and pedagogical work of the scientist for 68 years is given. The article analyzes the active life position of Vladislav Stepanovich Malakhovsky during his years of study at school, at Tomsk University, as well as during his work at Tomsk University and the I. Kant Baltic Federal University up to December 14, 2022. Links are given to individual articles in which Vladislav Stepanovich's activities in all areas are described in more detail, including the content of recent pub­lications on number theory.

The notion of type constancy was introduced by Alfred Gray for nearly Kählerian manifolds and later generalized by Vadim F. Kirichenko and Irina V. Tret’yakova for all Gray — Hervella classes of almost Her­mitian manifolds. In the present note, we consider the notion of type con­stancy for some six-dimensional almost Hermitian planar submanifolds of Cayley algebra. The almost Hermitian structure on such six-dimensional submanifolds is induced by means of so-called Brown — Gray three-fold vector cross products in Cayley algebra. We select the case when six-dimensional submanifolds of Cayley algebra are locally symmetric.

It is proved that six-dimensional locally symmetric submanifolds of Ricci type of Cayley algebra are almost Hermitian manifolds of zero con­stant type. This result means that six-dimensional locally symmetric sub­manifolds of Ricci type of Cayley algebra possess a property of six-dimensional Kählerian submanifolds of Cayley algebra. However, there exist non-Kählerian six-dimensional locally symmetric submanifolds of Ricci type in Cayley algebra.

In the present note, we consider the introduced by Lidia Vasil’evna Stepanova notion of an -quasi-umbilical hypersurface in an almost Her­mitian manifold. We show that the notion of an -quasi-umbilical hyper­surface in an almost Hermitian manifold is connected with the notion of a minimal hypersurface in this manifold.

Using the classical theory of minimal hypersurfaces in Riemannian manifolds and Kirichenko — Stepanova general theory of almost contact metric hypersurfaces in almost Hermitian manifolds, we establish that an -quasi-umbilical hypersurface of a nearly Kählerian manifold is minimal if and only if this hypersurface is totally umbilical.

Taking into account the connection between the notions of a minimal hypersurface and of an -quasi-umbilical hypersurface in an almost Her­mitian manifold, we conclude that some well-known results in the theory of almost contact metric hypersurfaces in almost Hermitian manifolds can be reformulated.

The problem of the existence of a non-umbilical minimal -quasi-umbilical hypersurface of a quasi-Kählerian manifold is posed.

We mark out the most important results obtained by outstanding Rus­sian geometer Vadim Feodorovich Kirichenko in the theory of almost Hermitian and almost contact metric manifolds.

The Grassmann manifold  is the set of all -dimensional planes of an -dimensional projective space, with

dim.

One of the submanifolds of the Grassmann manifold is a complex of -planes if the dimension of the complex exceeds the difference .

We continue to study the cocongruence of -dimensional planes us­ing the Cartan — Laptev method. In an -dimensional projective space, the cocongruence of -dimensional planes can be given by the following equations .

Compositional equipment of a given cocongruence by fields of

()-planes :

and points

allows one to define connections of three types in the associated bundle, and one of the three connections is average with respect to the other two. The deformation of these connections is considered and it is shown that the object of deformation is a pseudotensor.

We introduce the deformation object  of the connection of the sec­ond type with respect to the connection of the first type. The deformation of the connection of the third type with respect to the connection of the first type is , and the deformation of the connection of the third type with respect to the connection of the second type is .

In the present paper, we use the method of continuations and cover­ages of G. F. Laptev with assignment of connections in the principal bun­dle.

The concept of a nonholonomic para-Kenmotsu manifold is intro­duced. A nonholonomic para-Kenmotsu manifold is a natural generaliza­tion of a para-Kenmotsu manifold; the distribution of a nonholonomic para-Kenmotsu manifold does not need to be involutive. Properly nonho­lonomic para-Kenmotsu manifolds are singled out, these are nonho­lono­mic para-Kenmotsu manifolds with non-involutive distribution. On an al­most (para-)contact metric manifold, we introduce a metric connec­tion with torsion, which is called a connection of Levi-Civita type in this pa­per. In the case of a nonholonomic para-Kenmotsu manifold, such a con­nection has a simpler structure than the Levi-Civita connection, and in so­me cases it turns out to be preferable from an applied point of view. A Le­vi-Civita type connection coincides with a Levi-Civita connec­tion if and only if a nonholonomic para-Kenmotsu manifold reduc­es to a para-Ken­motsu manifold. It is proved that a proper nonholonomic para-Ken­motsu manifold cannot carry the structure of an Einstein mani­fold with respect to a connection of the Levi-Civita type.

In this article, a sub-Riemannian manifold of contact type is under­stood as a Riemannian manifold equipped with a regular distribution of codimension-one and by a unit structure vector field orthogonal to this distribution. This vector field is called a structural. On a sub-Riemannian manifold of contact type, a quarter-symmetric connection is defined, which is associated with an endomorphism that preserves the distribution of the sub-Riemannian manifold. It is proved that if the connection under study is metric, then the endomorphism associated to it is uniquely de­fined. The structure of the associated endomorphism is found. In the case when the structure vector field is a field of infinitesimal isometries, the quarter-symmetric connection is called the canonical N-connection. An expression is found for the curvature tensor of the canonical N-connection in terms of the Riemann curvature tensor. The properties of the Schouten curvature tensor are investigated, which provide, in particular, the neces­sary symmetries of the curvature tensor of an N-connection for its sec­tional curvature to be well-defined. A relation between the sectional cur­vature of the canonical N-connection and the sectional curvature of the Levi-Civita connection is found. Necessary and sufficient conditions are found under which the sectional curvature of the N-connection and the sectional curvature of the Levi-Civita connection coincide.

In this paper, we study a special class of hyperbands, i. e., a framed hyperband distribution. The study of hyperbands and their generalizations in spaces with different fundamental groups is of great interest in connec­tion with numerous applications in mathematics and physics. A special place is occupied by regular hyperstrips, for which the characteristic planes of families of principal tangent hyperplanes do not contain direc­tions tangent to the basal surface of the hyperstrip. In this paper, we use the method of external differential forms of E. Cartan and the group-theoretic method of G. F. Laptev.

We consider a regular hyperband distribution of an affine space equipped with a field of conjugate planes with respect to an asymptotic bundle of tensors of the basic surface. The definition of the studied hy­perband distribution in the affine space with respect to the 1st order frame is given and the existence theorem is proved. A sequence of fundamental geometric objects of the 1st and 2nd order of a hyperband distribution equip­ped with a field of conjugate planes is constructed. Fields of qua­sitensors are constructed that define the fields of normals of the first kind of the distribution of the characteristics of the hyperband distribution. In a differential neighborhood of the 2nd order, the fields of Transon normals of the 1st and 2nd kind are constructed. The conditions for the coincidence of the Transon normal and the Blaschke normal are found.