2021 №52

This paper is dedicated to the me­mo­ry of Professor Kowalski who was one of the leading re­sear­chers in the field of dif­fe­ren­tial geometry and especially Rie­mannian and affine geometry. He signif­icantly contributed to rai­sing the level of teaching dif­fe­ren­tial geometry by careful and sys­tematic preparation of lectures for students. Prof. Kowalski is the author or co-author of more than 170 professional articles in internation­ally recognized jour­nals, two monographs, text books for students. Prof. Kowalski collaborated with many mathematicians from other countries, particularly from Belgium, Italy, Ja­pan, Romania, Russia, Morocco, Spain and others. With the death of Professor Oldřich Kowalský mathe­matical community are losing a significant personality and an exceptional colleague, a kind and dedicated teacher, a man of high moral qualities.

It is known that an almost contact metric structure is induced on an arbitrary hypersurface of an almost Hermitian manifold. The case when the almost Hermitian manifold is nearly Kählerian and the almost contact metric structure on its hypersurface is quasi-Sasakian is considered. It is proved that non-Kählerian nearly Kählerian manifolds (in particular, the six-dimensional sphere equipped with the canonical nearly Kählerian structure) do not satisfy to the quasi-Sasakian hypersurfaces axiom.

We consider 6-dimensional planar submanifolds of Cayley algebra. As it is known, the so-called Brown — Gray three-fold vector cross prod­ucts induce almost Hermitian structures on such submanifolds. We select the case when the almost Hermitian structures on 6-dimensional planar submanifolds of Cayley algebra are Hermitian, i. e. these structures are in­tegrable.

It is proved that the Hermitian structure on a 6-dimensional planar submanifold of Cayley algebra is stable if and only if such submanifold is totally geodesic.

We continue to study of the Grassmann-like manifold  of -centered planes. A special case is considered when the center  de­scribes an -dimensional surface . We will denote this mani­fold by . An analogue of the strong Norden normalization of the manifold  is realized. It is proved that this normalization induces a connection in the bundle associated with the manifold . A geometric characteristic of this connection is given with the help of parallel displacements.

In our research we use the Cartan method of external forms and the group-theoretical method of Laptev. These methods are used by many geometers and physicists.

The Grassmann-like manifold is closely related to such a well-known and popular manifold as the Grassmann manifold. The Grassmann mani­fold is an example of a homogeneous space and forms an important fun­damental class of projective manifolds, and the projective space itself can be represented as a Grassmann manifold.

А non-holonomic Kenmotsu manifold equipped with a connection analogous to the generalized Tanaka — Webster connection, is consid­ered. The studied connection is obtained from the generalized Tanaka — Webster connection by replacing the first structural endomorphism by the second structural endomorphism. The obtained connection is also called in the work the generalized Tanaka — Webster connection.

Unlike a Kenmotsu manifold, the structure form of a non-holonomic Kenmotsu manifold is not closed. The consequence of this single differ­ence is a significant discrepancy in the properties of such manifolds. For example, it is proved in the paper that the alternation of the Ricci-Schouten tensor of a non-holonomic Kenmotsu manifold, which is a transverse analogue of the Ricci tensor, is proportional to the external differential of the structural form. At the same time, in the classical case of a Kenmotsu manifold, the Ricci — Schouten tensor is a symmetric tensor.

It is proved that a Tanaka — Webster connection is a metric connec­tion. It is also proved that from the fact that the alternation of the Ricci-Schouten tensor is proportional to the external differential of the structur­al form, the following statement holds: if a non-holonomic Kenmotsu manifold is an Einstein manifold with respect to the generalized Tanaka — Webster connection, then it is Ricci-flat with respect to the same con­nection.

In n-dimensional projective space Pn a manifold , i. e., a family of pairs of planes one of which is a hyperplane in the other, is considered. A principal bundle  arises over it, . A typi­cal fiber is the stationarity subgroup of the generator of pair of planes: external plane and its multidimensional center — hyperplane. The princi­pal bundle contains four factor-bundles.

A fundamental-group connection is set by the Laptev — Lumiste method in the associated fibering. It is shown that the connection object contains four subobjects that define connections in the corresponding fac­tor-bundles. It is proved that the curvature object of fundamental-group connection forms pseudotensor. It contains four subpseudotensors, which are curvature objects of the corresponding subconnections.

The composite equipment of the family of hypercentered planes set by means of a point lying in the plane and not belonging to its hypercent­er and an (n – m – 1)-dimensional plane, which does not have common points with the hypercentered plane. It is proved, that composite equip­ment induces the fundamental-group connections of two types in the as­sociated fibering.

The notion of an almost quasi-Sasakian manifold is introduced. A ma­nifold with an almost quasi-Sasakian structure is a generalization of a quasi-Sasakian manifold; the difference is that an almost quasi-Sasakian manifold is almost normal. A characteristic criterion for an almost quasi-Sasakian manifold is formulated. Conditions are found under which al­most quasi-Sasakian manifolds are quasi-Sasakian manifolds. In particu­lar, an almost quasi-Sasakian manifold is a quasi-Sasakian manifold if and only if the first and second structure endomorphisms commute. An extended almost contact metric structure is defined on the distribution of an almost contact metric manifold. It follows from the definition of an extended structure that it is a quasi-Sasakian structure only if the original structure is cosymplectic with zero Schouten curvature tensor. It is proved that the constructed extended almost contact metric structure is the struc­ture of an almost quasi-Sasakian manifold if and only if the Schouten ten­sor of the original manifold is equal to zero. Relationships are found be­tween the second structure endomorphisms of the original and extended structures.

The study continues in a three-dimensional affine space of complexes of three-parameter families of ellipsoids, considered earlier in a number of works by the author. A variety of ellipsoids is studied when the ends of the coordinate vectors coincide with the focal points, and the first coordi­nate straight line describes a cylindrical surface, while on the generating element there are at least three focal points that do not lie on one straight line and on one plane passing through center, and defining three conju­gate directions. A complex of ellipsoids is distinguished from the indicat­ed manifold provided that the indicatrices of the second and third coordi­nate vectors describe surfaces with tangent planes parallel to the third coordinate plane, and the end of the second coordinate vector describes a line with a tangent parallel to the first coordinate vector. An existence theorem for the variety under study is proved. The geometric properties of the complex under consideration are found. It is proved that the end of the first coordinate vector, points of the first coordinate line, and also the first coordinate plane describe a two-parameter family of planes, the end of the third coordinate vector describes a two-parameter family of cylindrical planes, a point of the third coordinate plane describes a one-parameter family of lines with tangents parallel to the first coordinate vector. The characteristic manifold of a generating element consists of six points: the vertex of the frame, three ends of the coordinate vectors, and two ends: the sum of the first and second coordinate vectors, as well as the sum of the first and third coordinate vectors. The focal manifold of the ellipsoid, the complex under study, consists of only three points, which are the ends of the coordinate vectors.

The basis for this study of affine connections in linear frame bundle over a smooth manifold is the structure equations of the bundle. An affine connection is given in this bundle by the Laptev — Lumiste method. The differential equations are written for components of the deformation ten­sor from an affine connection to the symmetrical canonical one. The ex­pressions for the components of the torsion tensor for two-dimensional and three-dimensional manifolds were found.

For a two-dimensional manifold, the affine torsion is a fraction, in the numerator there is a linear combination of two fiber coordinates which coefficients are two functions depending on the base coordinates (the co­ordinates on the base), and in the denominator there is the determinant composed of the fiber coordinates (the coordinates in a fiber). For a three-dimensional manifold, the arbitrariness of the numerator is determined by nine functions depending on the base coordinates.

In the first-order frame a tangentially r-framed hyperband is given in the projective space. For simplicity of presentation, we adapt the frame by the field of the 1st kind normals. The tensor of nonholonomicity of cloth­ing L-planes field is introduced. The vanishing the nonholonomic tensor leads to three different interpretations of the hyperband. With the help of ТL-virtual normals of the 1st and 2nd kind of framed L-planes, we come to the following conclusion: in a third order differential neighborhood the bundle of the hyperband second kind normals generates a one-parameter bundle of ТL-virtual first and second kind normals, which correspond to each other in bijection. We consider focal images associated with the hy­perband, with the help of which the Norden — Timofeev plane of the indicated hyperband is constructed. The geometric interpretation of the object defining the Norden — Timofeev surface was found by R. F. Dom­brovsky for tangentially r-framed surfaces in the projective space. We note that the field of ТL-virtual first kind normals induces the field of the Norden — Timofeev planes, this is the field of the 2nd kind regular hyper­band normals. It is proved that with each the 1st kind ТL-virtual normal is induced a bundle of Cartan planes in the 1st kind normal at a fixed point of the hyperband.

In conclusion, we consider the p-structures of the tangent planes field at the base surface of the hyperband.

A 2n-dimensional differentiable manifold M with -structure is a Riemannian almost para­complex manifold. In the present paper, we consider con­formal transformations of metrics of Riemannian para­complex manifolds. In particular, a number of vanishing theorems for such transformations are proved using the Bochner technique.

In this paper, we study a system of linear equations that define the Lie algebra of differentiations DerA of an arbitrary finite-dimensional linear algebra over a field. A system of equations is obtained, which is satisfied by the components of an arbitrary differentiation with respect to a fixed basis of algebra A. This system is a system of linear homogeneous equa­tions. The law of transformation of the matrix of this system is proved. The invariance of the rank of the matrix of this system in the transition to a new basis in algebra is proved. Next, we consider the possibility of ap­plying the obtained results in differential geometry when estimating the dimensions of groups of affine transformations from above. As an exam­ple, the method of I. P. Egorov is given for studying the dimensions of Lie algebras of affine vector fields on smooth manifolds equipped with linear connections having non-zero torsion tensor fields.

The theory of tangent bundles over a differentiable manifold M be­longs to the geometry and topology of manifolds and is an intensively developing area of the theory of fiber spaces. The foundations of the theo­ry of fibered spaces were laid in the works of S. Eresman, A. Weil, A. Mo­ri­moto, S. Sasaki, K. Yano, S. Ishihara. Among Russian scientists, tangent bund­les were investigated by A. P. Shirokov, V. V. Vishnevsky, V. V. Shu­rygin, B. N. Shapukov and their students.

In the study of automorphisms of generalized spaces, the question of infinitesimal transformations of connections in these spaces is of great importance. K. Yano, G. Vrancianu, P. A. Shirokov, I. P. Egorov, A. Z. Pet­rov, A. V. Aminova and others have studied movements in different spac­es. The works of K. Sato and S. Tanno are devoted to the motions and au­tomorphisms of tangent bundles. Infinitesimal affine collineations in tan­gent bundles with a synectic connection were considered by H. Shadyev.

At present, the question of the motions of fibered spaces is considered in the works of A. Ya. Sultanov, in which infinitesimal transformations of a bundle of linear frames with a complete lift connection, the Lie algebra of holomorphic affine vector fields in arbitrary Weyl bundles are investi­gated. In this paper we obtain exact upper bounds for the dimensions of Lie algebras of infinitesimal affine transformations in tangent bundles with a synectic connection A. P. Shyrokov.

The work is devoted to the study of the deformation of one-sided sur­faces. Let a normal vector be drawn along a closed curve on the surface. If, when returning to the original point, the direction of the normal coin­cides with the original direction of the normal, then the surface is called two-sided. Otherwise, we have a one-sided surface. Unilateral surfaces include: crossed cap, Roman surface, Boya surface, Klein bottle. Roman surface, Boya surface and crossed hood are a model of the projective plane.

It is proved that if the surface is a model of a Moebius strip, of a Klein bottle, of projective plane, then the surface deformation is a Moebius strip model, a Klein bottle model, projective plane model respectively.

Using a mathematical package, graphs are built the surfaces under consideration.