2024 №55(2)

In this paper, we formulate two theorems on the disap­pearance of the kernel of the Hodge — de Rham Laplacian and refine the estimate for its smallest eigenvalue on closed Riemannian manifolds.

In the present note, we consider six-dimensional Hermitian planar sub­manifolds of Cayley algebra. The almost Hermitian structure on such a six-dimensional submanifold is induced by means of so-called Brown — Gray three-fold vector cross products in Cayley algebra. The six-dimen­sio­nal Hermitian planar submanifolds of the octave algebra contain all six-dimensional Kählerian submanifolds of Cayley algebra. However, there exist non-Kählerian six-dimensional Hermitian planar submanifolds in the octave algebra.

The components of the tensor of the Riemannian curvature for a six-dimensional almost Hermitian planar submanifold of Cayley algebra are computed. Remark that the tensor of Riemannian curvature plays a fun­damental role in geometry of almost Hermitian manifolds. Knowing all components of the tensor of the Riemannian curvature for a six-dimen­sio­nal almost Hermitian planar submanifold of the octave algebra, it is possible to study so-called Gray’s identities for this submanifold.

The components of the Ricci tensor and of the tensor of conformal curvature (known also as Weyl tensor) for a six-dimensional almost Her­mitian planar submanifold of Cayley algebra are also computed.

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 clothing of a given cocongruence by fields of

()-planes :  

and points

allows one to define connections of three types in the associated bundle.

In the present paper, parallel transports of an analogue of Cartan plane are studied in the connections of three types. It is proved 4 theo­rems:

1. Parallel transport of the analogue of the Cartan plane  in an arbitrary connection is freely degenerate, i. e., in general, there are no spe­cial transports of this clothing plane.
2. In the group connection of the first type, the parallel transport of an analog of the Cartan plane is connected degenerate, i. e., the plane  will be fixed under parallel transport in this connection.
3. In the group connections of the second and third types, the parallel transport of the analogue of the Cartan plane is freely degenerate.
4. The analogue of the Cartan plane is transferred in parallel in a line­ar combination of the first type connection if and only if it is displaced in the plane .

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.

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.