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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 .

We introduce the concept of radially symmetric pseudo-Finsler spaces, which generalize the notion of symmetric Finsler spaces, and prove that this concept is equivalent to the preservation of flag curvature by parallel transport together with reversibility. As a consequence, reversible pseudo-Finsler manifolds with constant flag curvature are radially symmetric.

Tokamak edge (scrape-off layer (SOL)) plasmas can exhibit non-local transport in the direction parallel to the magnetic field due to steep temperature gradients. This effect along with its consequences has been explored at equilibrium for a range of conditions, from sheath-limited to detached, using the 1D kinetic electron code SOL-KiT, where the electrons are treated kinetically and compared to a self-consistent fluid model. Line-averaged suppression of the kinetic heat flux (compared to Spitzer-Härm) of up to 50% is observed, contrasting with up to 98% enhancement of the sheath heat transmission coefficient, γ _e . Simple scaling laws in terms of basic SOL parameters for both effects are presented. By implementing these scalings as corrections to the fluid model, we find good agreement with the kinetic model for target electron temperatures. It is found that the strongest kinetic effects in γ _e are observed at low-intermediate collisionalities, and tend to increase (keeping upstream collisionality fixed) at increasing upstream densities and temperatures. On the other hand, the heat flux suppression is found to increase monotonically as upstream collisionality decreases. The conditions simulated encompass collisionalities relevant to current and future tokamaks.

It is known that the Yang-Mills equations for a connection are equivalent to the Levy-Laplace equation for the parallel transport associated with this connection (see the paper [4] by L. Accardi, P. Gibilisco and I. V. Volovich). The Levy-Laplace equation is the Laplace equation for Levy Laplacian, which can be defined as the Cesaro mean of the second-order directional derivatives along the vectors from the orthonormal basis of some Hilbert space. The author’s work [11] has proved for the case of a four-dimensional Euclidean space, that with a certain choice of the orthonormal basis, the Laplace-Levy equation for a parallel transport becomes equivalent to the self-duality equations for a connection.A connection that is a solution to the self-duality equations is called an instanton and is a solution to the Yang-Mills equations. In the paper we define the Levy Laplacian for the case of a four-dimensional Riemannian manifold. This operator is a generalization of both the Levy Laplacian, introduced by the author in [11], and the Levy Laplacian, introduced in [3] by L. Accardi and О. G. Smolyanov for a Riemannian manifold.In the paper we consider the case of a line bundle over a four-dimensional Riemannian manifold and Maxwell's equations (the commutative case of the Yang-Mills equations). We find the conditions, which imply that from the fact that the parallel transport is a harmonic functional for the introduced Levy Laplacian follows that the appropriate connection is a solution to the self-duality equations. In addition, the paper considers the relationship of the introduced Levy Laplacian and the Laplace-Beltrami operator on a manifold. It can be expected that the results of the paper can be generalized to the non-commutative case of Yang-Mills fields.

A procedure to obtain a closed form of the shifting operators along a known geodesic line on a surface as a solution of a system of linear algebraic equations is proposed. Its correctness is numerically demonstrated in the case of a helicoid surface and a spherical one. The future use of these operators in finite element approximations of tensor fields in non-Euclidean spaces is announced.

A proposal is made for what could well be the most natural symmetrical Riemannian spaces which are homogeneous but not isotropic, i.e. of what could well be the most natural class of symmetrical spaces beyond the spaces of constant Riemannian curvature, that is, beyond the spaces which are homogeneous and isotropic, or, still, the spaces which satisfy the axiom of free mobility.