On metrizability of a class of 2-manifolds with linear connection. (English) Zbl 1299.53034

Summary: In [the author 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 that guarantee metrizability (among others, recurrency of the curvature is necessary), all (at least local) pseudo-Riemannian metrics compatible with a given torsion-free linear connection without flat points in a domain of two-dimensional manifold. The result has the form of an implication only. In general, if there are flat points, or if curvature is not recurrent, we cannot give any good answer as it can also be demonstrated by examples. Note that in higher dimension, the problem of metrizability is not easy to solve, [O. Kowalski, Math. Z. 125, 129–138 (1972; Zbl 0234.53024); Note Mat. 8, 1–11 (1988; Zbl 0699.53038)]. Here, we try to apply this apparatus to the class of (torsion-free, locally homogeneous) connections with constant Christoffels in open domains of 2-manifolds (called connections of Type A in [T. Arias-Marco and O. Kowalski, Monatsh. Math. 153, 1–18 (2008; Zbl 1155.53009); O. Kowalski et al., Cent. Eur. J. Math. 2, 87–102 (2004; Zbl 1060.53013)]).


53B05 Linear and affine connections
53B20 Local Riemannian geometry