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.

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

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.

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 paper is devoted to affine connection in the frame bundle associ­ated with a manifold which structure equations and derivation formulas are constructed using deformations of the exterior and ordinary differen­tials. Curvature and torsion objects of this connection are not tensors. A cha­racteristic of a curvature which is a convolution of a deformation tensor and a torsion, is considered. Torsion-free connections are not distingui­shed on the introduced manifold, even in the case of symmetric deforma­tion, a class of semi-symmetric connections is distinguished, which is an analogue of symmetric connection on an ordinary smooth manifold. It is proved that if the connection deformation tensor is symmetric or zero, then the connection is semi-symmetric. Analogues of torsion-free and cur­vature-free connections are constructed. The torsion and curvature of this connection are expressed in terms of the symmetric deformation tensor for the connection. Canonical connection is a special case of this connec­tion, it is semi-symmetric and curvature-free.

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.

The theory of motions in generalized spaces is one of the directions in modern differential geometry. Such scientists as E. Cartan, P. K. Rashev­sky, P. A. Shirokov, I. P. Egorov, A.Ya. Sultanov and other scientists were engaged in the study of movements in various spaces of affine connec­tions. The question of movements in direct products of two spaces of af­fine connection was considered in M. V. Morgun’s work. In the case of a direct product of more than two spaces of affine con­nection, the question of the dimension of Lie algebras of infinitesimal affine transformations of a given space remained open. In this article, an estimate of the upper bound of the dimension of the Lie algebra of infinitesimal affine transformations of affine connection spaces, representing a direct reproduction of at least three non-projective Euclidean spaces of a certain type, is obtained. To solve this problem, a system of linear homogeneous equations is obtained, which is satisfied by the components of an arbitrary infinitesi­mal affine transformation. This system is found using the properties of the Lie derivative applied to the tensor field of curvature of the spaces under consideration. The evaluation of the rank of this system allows us to ob­tain an estimate from below of the rank of the matrix of the system under consideration.