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 mappings.
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Received: November 2, 2007.
Supported by grant No. 201/05/2707 of The Grant Agency of Czech Republic, by the Council of Czech Government MSM 6198959214, and by the Mathematisches Institut der Universität Erlangen-Nürnberg.
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Mikeš, J., Strambach, K. Differentiable Structures on Elementary Geometries. Result. Math. 53, 153–172 (2009). https://doi.org/10.1007/s00025-008-0296-2
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DOI: https://doi.org/10.1007/s00025-008-0296-2