Differentiable Structures on Elementary Geometries

Abstract.

Using Beltrami’s differential equation we show that the real affine plane is the only generalized shift ${\mathbb{R}}^2$-plane such that their lines are geodesics with respect to an affine connection. Among the generalized Moulton planes only the Moulton planes admit affine connections $\nabla$ such that their lines are geodesics with respect to $\nabla$. For Moulton planes we classify to large extent all such connections $\nabla$ and determine the corresponding groups of affine map**s.