Geodesics and Almost Geodesics Curves

We determine in $R^n$ the form of curves $C$ for which also any image under an $(n-1)$-dimensional algebraic torus is a geodesic or an almost geodesic with respect to an affine connections $\nabla$ with constant coefficients and calculate explicitly the components of $\nabla$. In this paper we consider the special case for the connection $\nabla$ when any curve from a set of images of $C$ is almost geodesic with respect to $\nabla$.