The aim of this paper is to classify (lócally) all torsion-less locally homogeneous affine connections on two-dimensional manifolds from a group-theoretical point of view. For this purpose, we are using the classification of all non-equivalent transitive Lie algebras of vector fields in ℝ2 according to P.J. Olver [7].
@article{bwmeta1.element.doi-10_2478_BF02475953,
author = {Old\v rich Kowalski and Barbara Opozda and Zden\v ek Vl\'a\v sek},
title = {A classification of locally homogeneous connections on 2-dimensional manifolds via group-theoretical approach},
journal = {Open Mathematics},
volume = {2},
year = {2004},
pages = {87-102},
zbl = {1060.53013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_BF02475953}
}
Oldřich Kowalski; Barbara Opozda; Zdeněk Vlášek. A classification of locally homogeneous connections on 2-dimensional manifolds via group-theoretical approach. Open Mathematics, Tome 2 (2004) pp. 87-102. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_BF02475953/
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[2] S. Kobayashi, K. Nomizu: Foundations of Differential Geometry I, Interscience Publ., New York, 1963.
[3] O. Kowalski, B. Opozda, Z. Vlášek: “Curvature homogeneity of affine connections on two-dimensional manifolds”, Colloquium Mathematicum, Vol. 81, (1999), pp. 123–139. | Zbl 0942.53019
[4] O. Kowalski, B. Opozda, Z. Vlášek: “A classification of locally homogeneous affine connections with skew-symmetric Ricci tensor on 2-dimensional manifolds”, Monatshefte für Mathematik, (2000), pp. 109–125. | Zbl 0993.53008
[5] O. Kowalski, Z. Vlášek: “On the local moduli space of locally homogeneous affine connections in plane domains”, Comment. Math. Univ. Carolinae, (2003), pp. 229–234. | Zbl 1097.53009
[6] K. Nomizu, T. Sasaki: Affine Differential Geometry, Cambridge University Press, Cambridge, 1994.
[7] P.J. Olver: Equivalence, Invariants and Symmetry Cambridge University Press, Cambridge, 1995.
[8] B. Opozda: “Classification of locally homogeneous connections on 2-dimensional manifolds”, to appear in Diff. Geom. Appl..