2023 №54(2)

The detailed description of the construction of the canonical form on the higher order frame bundle over an n-dimensional smooth manifold is given. In particular, it is shown that some vector space isomorphism play­ing the key role in this construction is defined correctly, i. e. it depends only on the frame of order p + 1 and does not depend on the choice of its representative, i. e. a local diffeomorphism which (p + 1)-jet is exactly this frame. This isomorphism acts from the direct sum of n-dimensional arithmetic space and the Lie algebra of the p-th order differential group to the tangent space to the p-th order frame bundle over the manifold at the p-th order frame lying “below”. The action of this isomorphism can be splitted into two its restrictions. The first one acts from the first direct sum­mand, and the second one acts from the second direct summand. It is shown that the first restriction depends only on the choice of the (p + 1)-fra­me, while the second one is closely related to fundamental vector fields and therefore does not depend of this frame at all.

In this paper, we study infinitesimal transformations of the tangent bundle of a common path space. The general path space is a genera­liza­tion space of the affine connectivity. By affine connectivity of the com­mon path space, we construct an affine connection on the tangent bundle. For the infinitesimal transformation of the tangent bundle, a system of in­va­riance equations for the constructed affine connectivity is compiled. This system is a system of second-order differential equations with res­pect to the components of the infinitesimal transformation. The main re­sults of the article are obtained by analyzing this system taking into account the properties of homogeneous functions. It is shown that the comp­lete lift of an infinitesimal transformation of base is an infinitesimal af­fine motion of a tangent bundle if and only if the infinitesimal transfor­ma­tion of base is an affine motion in the general path space. Necessary and sufficient conditions are found that the infinitesimal transformation of a tangent bundle generated by a vertical vector field leaves the affine connec­ti­vity of the tangent bundle invariant. Conditions are given that are neces­sa­ry and sufficient so that the infinitesimal transformation of a tangent bundle with affine connectivity that preserves layers is an affine motion.

This paper relates to differential geometry, and the research technique is based on G. F. Laptev’s method of extensions and envelopments, which generalizes E. Cartan's method of moving frame and exterior forms.

A manifold is studied, the structure equations and derivational for­mu­las of which are built using the deformations of the exterior and ordinary differentials. The manifold in question is a deformation of an ordinary smooth manifold. The bundles of non-symmetrical coframes and frames of the second order on this manifold are examined and an affine con­nec­tion is given. It is proved that the curvature and torsion of this connection are not tensors. A canonical connection is built. It is shown that the cano­ni­cal connectionis flat and non-symmetrical.

In the present paper we consider pointwise orthogonal split­ting of the space of well-known TT-tensors on Rieman­nian manifolds. Tensors of the first subspace belong to the ker­nel of the Bourguignon Laplacian, and the tensors of the se­cond subspace belong to the kernel of the Sampson Lap­la­cian. We give examples and prove Liouville-type non-exis­tence theorems of these tensors.

Linear algebras over a given field arise when studying various problems of algebra, analysis and geometry.  The operation of differentiation, which originated in mathematical analysis, was transferred to the theory of linear algebras over a field, as well as to the theory of rings.

The set of all differentiations of a linear algebra themselves form a linear algebra. This algebra is called the algebra of differentiations. At the same time, this algebra admits the structure of a Lie algebra. If the algebra whose differentiations are considered is finite-dimensional, then its Lie algebra of differentiations will also be finite-dimensional. Therefore, the­re is a natural problem of determining the dimension of the Lie algebras of the differentiations of the linear algebra under consideration or to obtain an estimate from above of the dimension of the algebra of differentiations.

To solve these problems, a system of linear homogeneous equations is obtained, which is satisfied by the components of arbitrary differen­tia­tion. Evaluation of the rank of this system allows us to obtain an estimate from below of the rank of the matrix under consideration.

The algebra of dual numbers was first introduced by V. K. Clifford in 1873. The algebras of plural and dual numbers are analogous to the algebra of complex numbers. Dual numbers form an algebra, but not a field, because only dual numbers with a real part not equal to zero have an inverse element.

In this work, automorphisms of algebras of plural numbers, which are a generalization of the algebra of dual numbers, are studied. Algebras of plural numbers were in the center of attention of the professor of Kazan University A. P. Shirokov. Studying the geometry of higher-order tangent bundles, he established that higher-order tangent bundles over smooth manifolds have the structure of a smooth manifold over algebras of plural numbers. This allowed him in the 70s of the twentieth century to construct a theory of lifts of tensor fields and linear connections from a smooth manifold to its tangent bundles of arbitrary order.

In this paper, we study automorphisms of the algebra of plural numbers. It is proved that the set of all automorphisms of the algebra of plural numbers forms a group. The structure of this group is described. The groups of automorphisms of the algebra of plural numbers with small dimension are indicated as examples.

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 Minding 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 nonlinear 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 pseudosphere is constructed. Using a mathematical package, the pseu­dos­phere and its Bianchi transform are constructed.

The principal bundles of the first order coframes and the second order coframes, as well as factor bundle of centroprojective (coaffine) coframes are considered. In the bundle of linear coframes a connection is given with the help of the field of connection object. The torsion and curvature tensors of this linear connection are determined. Special connections are singled out: torsion-free, curvature-free. The space of a linear connection devoid of torsion and curvature is an affine group, that served as the basis for classical name «affine connection».

Under the specializations of a manifold, strong and weak projectivity conditions was introduced, which make it possible to single out the cof­ra­me bundles. The connections in these principal bundles are called strong and weak projective connections.

In the case of symmetric linear connection, when the torsion is absent, the object of classic projective connection is considered. Connec­tion forms are introduced and their structure equations are found. Hence it follows that classic projective connection is neither fundamental-group nor linear differential-geometric. It is proved, that the curvature object of this connection forms a quasitensor together with the connection object only. It is shown, that classic projective connection degenerates into diffe­rent from the original linear connection on the image of a section of some homogeneous bundle.