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From 1950s, it is known that an almost contact metric structure is in­duced on an arbitrary oriented hypersurface in an almost Hermitian mani­fold. In accordance with the definition, an almost Hermitian manifold satisfies the axiom of almost contact hypersurfaces endowed with a some property, if an almost contact hypersurface with this property passes through every point of considered almost Hermitian manifold.

In the present note, we discuss some problems related to almost con­tact metric hypersurfaces axioms for almost Hermitian manifolds. In par­ticular, we select some special types of almost contact metric hypersur­faces axioms for almost Hermitian manifolds. We mark out the axioms consisting of the conditions for the almost contact metric structure on the hypersurface of an almost Hermitian manifold to belong to a special class (for example, to the class of Sasakian or quasi-Sasakian structures). We also mark out the axioms that are related to the second fundamental form of the immersion of the almost contact metric hypersurface into an almost Hermitian manifold.

Almost contact metric structures on odd-dimensional manifolds are considered. The first group of the Cartan structural equations of an arbi­trary almost contact metric structure written in an A-frame (i. e., in a fra­me adapted to this almost contact metric structure) is studied. It is proved that the fifth and sixth Kirichenko structural tensors of the almost contact metric structure vanish if and only if the structural contact form is closed.

A rigged hyperstrip distribution is a special class of hyperstrips. The study of hyperstrips and their generalizations in spaces with various fun­damental groups is of great interest due to 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 base surface of the hyperstrip. In this work, we use E. Cartan’s method of external differen­tial forms and the group-theoretic method of G. F. Laptev.

In affine space, a hyperstrip distribution is considered, which at each point of the base surface is equipped with a tangent plane and a conjugate tangent line. The specification of the studied hyperstrip distribution in an affine space with respect to a 1st order reference and an existence theorem are given. The fields of affine normals of the 1st kind for Blaschke and Transon are constructed and the conditions for their coincidence are found. The definition of normal affine connection and normal centroaffine connection on the studied framed hyperstrip distribution is given.

A detailed obtaining of the expressions for the scalar components of the canonical form on higher order frame bundles over a smooth manifold has been done. The canonical form on the frame bundle of order p + 1 over an n-dimensional smooth manifold is a vector-valued differential 1-form with values in the tangent space to the p-th order frame bundle over the n-di­mensional arithmetical space at the unit of the p-th order differential group. The scalar components of the canonical form are its coefficients with respect to natural basis of the tangent space. For every frame, there exists a polynomial mapping representing the frame in a given local chart on the manifold. Therefore, for any tangent vector to the frame bundle there is a first order Taylor expansion of one-parametric family of poly­nomial mappings representing the tangent vector. We obtain the formulas of the scalar components from the equations for coefficients of the two expansions for some tangent vector.

The concept of a common path space was introduced by J. Duqlas. M. S. Knebelman was the first to consider affine and projective move­ments in these spaces. The general path space is a generalization of the space of affine connectivity. In this paper, we study spaces of paths that admit groups of affine motions with one-dimensional orbits. For each representation in the form of algebra of vector fields of the abelian Lie algebra and the Lr algebra containing the abelian ideal Lr-1, a system of equations of infinitesimal affine motions is compiled. The vector fields of each of these representations are operators of a group of transformations with one-dimensional orbits. Integrating this system, general spaces of paths are defined that admit a group of affine motions with one-dimensional orbits, the operators of which are the vector fields of these representations. The maximum order of these groups is set. It is shown that the spaces of paths admitting a group of affine motions with one-dimensional orbits of maximum order are projectively flat. The conditions that are necessary and sufficient for the space of paths to admit a group of affine motions with one-dimensional orbits of maximum order are given.

Among Thurston's famous list of eight three-dimensional geometries is the geometry of the manifold Sol. The variety Sol is a connected simply connected Lie group of real matrices of a special form. The manifold Sol has a left-invariant pseudo-Riemannian metric for which the group of left shifts is the maximal simply transitive isometry group. In this paper, we prove that on the manifold Sol there exists a left-invariant differential 1-form, which, together with the left-invariant pseudo-Riemannian metric, defines a paracontact metric structure on Sol. A three-parameter family of left-invariant paracontact metric connections is found, that is, linear con­nec­tions invariant under left shifts, in which the structure tensors of the par­acontact structure are covariantly constant. Among these connections, a flat connection is distinguished. It has been established that some geo­desics of a flat connection are geodesics of a truncated connection, which is an orthogonal projection of the original connection onto a 2n-dimen­sional con­tact distribution. This means that this connection is con­sistent with the con­tact distribution. Thus, the manifold Sol has a pseudo-sub-Riemannian structure determined by a completely non-holonomic contact distribution and the restriction of the original pseudo-Riemannian metric to it.

In the present paper, we prove that if  is an -dimensional  compact Riemannian manifold and if  where ,  and  are the sectional and Ricci curvatures of  respectively, then  is diffeomorphic to a spherical space form  where  is a finite group of isometries acting freely. In particular, if  is simply connected, then it is diffeo­mor­phic to the Euclidian sphere

In this work, Lie algebras of differentiation of linear algebra, the op­eration of multiplication in which is defined using a linear form and two fixed elements of the main field are studied. In the first part of the work, a definition of differentiation of linear algebra is given, a system of linear homogeneous equations is obtained, which is satisfied by the components of arbitrary differentiation. An embedding of the Lie algebra of differenti­ations into the Lie algebra of square matrices of order n over the field P is constructed. This made it possible to give an upper bound for the dimen­sion of the Lie algebra of derivations. It has been proven that the dimen­sion of the algebra of differentiation of the algebras under study is equal to n2 – n, where n is the dimension of the algebra. Next we give a result on the maximum dimension of the Lie algebra of derivations of a linear alge­bra with identity. Based on the above facts, it is proven that the algebras under study cannot have a unit.

The work is devoted to the study of the Bianchi transform for surfac­es of constant negative Gaussian curvature. The surfaces of rotation of constant negative Gaussian curvature are the Mining top, the Minding coil, the pseudosphere (Beltrami surface). Surfaces of constant negative Gaussian curvature also include Kuens surface and the Dinis surface. The study of surfaces of constant negative Gaussian curvature (pseudospheri­cal surfaces) is of great importance for the interpretation of Lobachevsky planimetry. The connection of the geometric characteristics of pseudo­spherical surfaces with the theory of networks, with the theory of solitons, with non-linear differential equations and sin-Gordon equations is estab­lished. The sin-Gordon equation plays an important role in modern phys­ics. Bianchi transformations make it possible to obtain new pseudospheri­cal surfaces from a given pseudospherical surface. The Bianchi transform for the Minding coil is constructed. Using a mathematical package, the Minding coil  and its Bianchi transform are constructed.