On metrizability of locally homogeneous affine 2-dimensional manifolds

Alena Vanžurová

Faculty of Science, Department of Algebra and Geometry, Palacký University, Tř. 17. listopadu 12, 771 46 Olomouc, Czech Republic
Department of Mathematics, Faculty of Civil Engineering, Brno University of Technology, Veveří 331/95, 602 00 Brno, Czech Republic


Abstract: In [19] 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 [9], [1]) 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.

AMSclassification: primary 53B05; secondary 53B20.

Keywords: manifold, connection, metric.

DOI: 10.5817/AM2013-5-347