Pseudo-riemannian metrics on a variety of applied covectors

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.