Curvatures of curves in $\mathbb{R}^{2}$ endowed with a canonical linear connection

Abstract

We investigate the geometry of plane curves in $\mathbb{R}^2$ endowed with a canonical linear connection. By introducing the notions of the tangential and geodesic curvatures, we prove the existence and uniqueness theorems for curves with prescribed curvatures. We find all curves with constant curvature and we generalize the curves whose curvatures are linear functions of the arc length parameter.