Vol 223 (2023)

We study the action of the multiplicative group of positive real numbers on various families of a metric space, which consists of multiplying all distances in the metric space by the same positive real number; such an action is called similarity. Stationary subgroups of such actions are studied.

In this paper, we obtain criteria for the Φ-invariance and the η-invariance for almost contact metric structures and a criterion for a characteristic vector ξ to be a Killing vector. We find all classes of almost contact metric structures from the Kirichenko classification that are Φ-invariant, η-invariant, and ξ is a Killing vector. Also, we prove that for any almost contact metric structure, ξ cannot be conformal Killing vector distinct from a Killing vector.

The isotropic Schouten–Weyl tensor was previously studied in the case of three-dimensional Lie groups with a left-invariant Lorentzian metric. In the case of locally homogeneous pseudo-Riemannian spaces with a nontrivial isotropy subgroup, manifolds with an isotropic Weyl tensor were classified. In this paper, we obtain a classification of four-dimensional, locally homogeneous pseudo-Riemannian manifolds with an isotropic Schouten—Weyl tensor. Some results on the curvature tensors of similar manifolds are obtained.

The shadow problem for horospheres is related to the problem of global isometric embedding of surfaces of revolution of constant negative curvature into the three-dimensional Euclidean space. Euclidean surfaces of revolution of constant negative curvature are globally isometric to parts of tangent cones of horospheres in the three-dimensional Lobachevsky space. In this work, meridians of Euclidean pseudospherical surfaces of revolution are expressed in terms of metric characteristics in the hyperbolic space, namely, in terms of the distance from the vertex of the tangent cone to the horosphere or through the distance from the polar of the vertex to the horosphere.

In this paper, we consider problems of stabilization of stationary motions (equilibrium positions and regular precessions) of a satellite near the center of mass in gravitational and magnetic fields under the assumption that the center of mass moves in a circular orbit. Solutions for a number of problems of stabilizing stationary motions of a satellite with the help of magnetic systems are proposed. We present the results of mathematical modeling of the algorithms, which confirm the effectiveness of the developed methodology. This paper is the fourth part of the work. The first part is: Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. — 2023. —220. —P. 71–85. The second part is: Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. — 2023. — 221. — P. 71–92. The third part is: Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory .— 2023. — 222. — P. 42–63.

In this paper, we consider the generalized Bochner technique, which is a natural development of the classical Bochner technique. As an illustration, we prove some vanishing theorems on Ricci solitons, conformal and projective mappings of complete Riemannian manifolds.

We consider a class of three-dimensional forms that admit a certain group of symmetries and are similar to tubular regions. Boundary-value problems in the invariant form in such domains can be reduced to problems with one spatial variable in a special coordinate system. A class of forms is proposed, their properties are proved, and examples are given.