Almost conformal transformation in a four dimensional Riemannian manifold with an additional structure

TL;DR

This paper studies a special four-dimensional Riemannian manifold with an additional structure, exploring conditions for a related metric to be positive and for the structure to be parallel, while analyzing angle transformations and convergence properties.

Contribution

It introduces a new metric construction on a four-dimensional manifold with circulant tensor structures and analyzes conditions for conformality and parallelism of the additional structure.

Findings

Conditions for the new metric to be positive definite.

Criteria for the structure q to be parallel with respect to the Levi-Civita connection.

Convergence of angle series related to the structure q.

Abstract

We consider a four dimensional Riemannian manifold M with a metric g and affinor structure q. The local coordinates of these tensors are circulant matrices. Their first orders are (A, B, C, B), A, B, C\in FM and (0, 1, 0, 0), respectively. We construct another metric \tilde{g} on M. We find the conditions for \tilde{g} to be a positively defined metric, and for q to be a parallel structure with respect to the Riemannian connection of g. Further, let x be an arbitrary vector in T_{p}M, where p is a point on M. Let \phi and \phi be the angles between x and qx, x and q^{2}x with respect to g. We express the angles between x and qx, x and q^{2}x with respect to $\tilde{g}$ with the help of the angles $\phi$ and \phi. Also,we construct two series {\phi_{n}}and {\phi_{n}}. We prove that every of it is an increasing one and it is converge.