Stable projective planes with Riemannian metrics (Q1849611)

The classical \(2\)-dimensional stable planes, i.e.\ the projective, the affine and the hyperbolic planes, can be endowed with a Riemannian metric such that the lines are precisely the geodesics. In the paper under review, the following results are proved: Let \((M,g)\) be a complete \(2\)-dimensional Riemannian manifold such that every two distinct points are joined by a unique geodesic. Then taking the set of all geodesics as line system turns \(M\) into a stable plane. If such a stable plane is a projective plane, then it is isomorphic to the real projective plane and, moreover, \(g\) equals the ususal Riemannian metric (up to a scalar).