On metrizability of locally homogeneous affine 2-dimensional manifolds. (English) Zbl 1313.53021

Summary: In [A. Vanžurová and P. Žáčková, Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 48, 157–170 (2009; Zbl 1195.53023)] we proved a theorem which shows how to find, under particular assumptions guaranteeing metrizability (among others, recurrency of the curvature is necessary), all (at least local) pseudo-Riemannian metrics compatible with a given torsion-less linear connection without flat points on a two-dimensional affine manifold. The result has the form of an implication only; if there are flat points, or if curvature is not recurrent, we have no good answer in general, which can be also demonstrated by examples. Note that in higher dimension, the problem is not easy to solve.
Here we try to apply this apparatus to the two main types (A and B from [O. Kowalski, B. Opozda and Z. Vlášek, Cent. Eur. J. Math. 2, No. 1, 87–102, electronic only (2004; Zbl 1060.53013)], [T. Arias-Marco and O. Kowalski, Monatsh. Math. 153, No. 1, 1–18 (2008; Zbl 1155.53009)]) of torsion-less locally homogeneous connections defined in open domains of 2-manifolds. We prove that in dimension two a symmetric linear connection with constant Christoffels is metrizable if and only if it is locally flat. On the other hand, in the class of connections of type B there are even non-flat metrizable connections.


53B05 Linear and affine connections
53B20 Local Riemannian geometry
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