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Differentiable structures on elementary geometries. (English) Zbl 1179.51009
The classical examples of \(2\)-dimensional stable planes, i.e. the real projective, affine and hyperbolic planes, can be endowed with a Riemannian metric such that the lines are geodesics. By a result of G. Gerlich [Arch. Math. 79, No. 4, 317–320 (2002; Zbl 1022.51012)], this attribute distinguishes the real projective plane among the vast number of \(2\)-dimensional compact projective planes. Therefore, it seems to be natural to ask for examples of \(2\)-dimensional stable planes whose lines are geodesics w.r.t. a Riemannian metric or at least w.r.t. an affine connection.
The paper under review looks at two classes of \(\mathbb R^2\)-planes whose lines are \(C^2\)-curves, namely the generalized shift \(\mathbb R^2\)-planes and the generalized Moulton planes. Among these, precisely the (ordinary) Moulton planes admit an affine connection \(\nabla\) such that the lines are geodesics w.r.t. \(\nabla\). Moreover, the authors classify the possible affine connections in question and determine the corresponding groups of affine mappings.

51H25 Geometries with differentiable structure
51H10 Topological linear incidence structures
53B05 Linear and affine connections
53C22 Geodesics in global differential geometry
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