Classification of locally homogeneous affine connections in two dimensions is a nontrivial problem. (See [5] and [7] for two different versions of the solution.) Using a basic formula by B. Opozda, [7], we prove that all locally homogeneous torsion-less affine connections defined in open domains of a 2-dimensional manifold depend essentially on at most 4 parameters (see Theorem 2.4).
@article{119382,
author = {Old\v rich Kowalski and Zden\v ek Vl\'a\v sek},
title = {On the local moduli space of locally homogeneous affine connections in plane domains},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {44},
year = {2003},
pages = {229-234},
zbl = {1097.53009},
mrnumber = {2026160},
language = {en},
url = {http://dml.mathdoc.fr/item/119382}
}
Kowalski, Oldřich; Vlášek, Zdeněk. On the local moduli space of locally homogeneous affine connections in plane domains. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) pp. 229-234. http://gdmltest.u-ga.fr/item/119382/
Transformation Groups in Differential Geometry, Springer-Verlag, New York (1972). (1972) | MR 0355886 | Zbl 0246.53031
Foundations of Differential Geometry I, Interscience Publ., New York (1963). (1963) | MR 0152974 | Zbl 0119.37502
Curvature homogeneity of affine connections on two-dimensional manifolds, Colloq. Math., ISSN 0010-1354, 81 1 123-139 (1999). (1999) | MR 1716190
A classification of locally homogeneous affine connections with skew-symmetric Ricci tensor on 2-dimensional manifolds, Monatsh. Math., ISSN 0026-9255, 130 Springer-Verlag, Wien 109-125 (2000). (2000) | MR 1767180
A classification of locally homogeneous connections on 2-dimensional manifolds via group-theoretical approach, to appear. | MR 2041671
Affine Differential Geometry, Cambridge University Press. | MR 1311248 | Zbl 1140.53001
Classification of locally homogeneous connections on 2-dimensional manifolds, preprint, 2002. | Zbl 1063.53024