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Keywords:
two-dimensional manifolds with affine connection; locally homogeneous connections
Summary:
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).
References:
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[2] Kobayashi S., Nomizu K.: Foundations of Differential Geometry I. Interscience Publ., New York (1963). MR 0152974 | Zbl 0119.37502
[3] Kowalski O., Opozda B., Vlášek Z.: Curvature homogeneity of affine connections on two-dimensional manifolds. Colloq. Math., ISSN 0010-1354, 81 1 123-139 (1999). MR 1716190
[4] Kowalski O., Opozda B., Vlášek Z.: 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). MR 1767180
[5] Kowalski O., Opozda B., Vlášek Z.: A classification of locally homogeneous connections on 2-dimensional manifolds via group-theoretical approach. to appear. MR 2041671
[6] Nomizu K., Sasaki T.: Affine Differential Geometry. Cambridge University Press. MR 1311248 | Zbl 1140.53001
[7] Opozda B.: Classification of locally homogeneous connections on 2-dimensional manifolds. preprint, 2002. Zbl 1063.53024
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