MATHEMATICS

Pseudo-riemannian metrics on a variety of applied covectors

M. S. Bukhtyak

Tomsk State University, Tomsk, Russian Federation

Abstract: Based on the three-dimensional affine space $A_3$, a six-dimensional point-vector space $E_6$ is constructed, where its point is an ordered pair consisting of a point from $A_3$ and a covector, and its vector is an ordered pair consisting of a vector and a covector. There is a pseudo-Euclidean metrics of signature in $E_6$ $(3,3)$. The problem of finding all affine semi-invariant pseudo-Riemannian metrics in the tangent fibration of a given space is solved. It is shown that finding semi-invariant metrics leads to finding invariant metrics, and there is a one-parameter family of such metrics (including the pseudo-Euclidean metrics as the trivial case). For the given family of metrics, the Levi-Civita connection is constructed, and a description of geodesic lines of this connection in the general case is given.

Keywords: affine space, point-vector space, covector, pseudo-Euclidean metrics, pseudo-Riemannian metrics, Levi-Civita connection, geodesic lines.

UDC: 514.764.2

MSC: 53B05

Received: 24.10.2023
Accepted: June 3, 2024

DOI: 10.17223/19988621/89/2