Kowalski, Oldřich; Opozda, Barbara; Vlášek, Zdeněk A classification of locally homogeneous connections on 2-dimensional manifolds via group-theoretical approach. (English) Zbl 1060.53013 Cent. Eur. J. Math. 2, No. 1, 87-102 (2004). P. J. Olver [Equivalences, Invariants and Symmetry (Cambridge University Press, Cambridge) (1995; Zbl 0837.58001)] classified all non-equivalent transitive Lie algebras of vector fields in \(\mathbb R^2\). In the present paper, the authors locally classify all torsion-less locally homogeneous affine connections on two-dimensional manifolds from a group-theoretical approach. Namely, for each specific transitive algebra \(A\) of vector fields from Olver’s classification, the authors look for all affine connections for which A is an affine Killing algebra. Olver’s tables are presented at the end of the paper. A remark: the authors used the software Maple V Release 4. Reviewer: Valentin Boju (Montreal) Cited in 3 ReviewsCited in 11 Documents MSC: 53B05 Linear and affine connections 53C30 Differential geometry of homogeneous manifolds Keywords:two-dimensional manifolds with affine connection; locally homogeneous connections; Lie algebras of vector fields; Killing vector fields; Olver’s classification; Opozda’s formula; software Maple V Release 4 Citations:Zbl 0837.58001 Software:Maple PDFBibTeX XMLCite \textit{O. Kowalski} et al., Cent. Eur. J. Math. 2, No. 1, 87--102 (2004; Zbl 1060.53013) Full Text: DOI References: [1] S. Kobayashi: Transformation Groups in Differential Geometry, Springer-Verlag, New York, 1972.; · Zbl 0246.53031 [2] S. Kobayashi, K. Nomizu: Foundations of Differential Geometry I, Interscience Publ., New York, 1963.; · Zbl 0119.37502 [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.; · Zbl 0834.53002 [7] P.J. Olver: Equivalence, Invariants and Symmetry Cambridge University Press, Cambridge, 1995.; · Zbl 0837.58001 [8] B. Opozda: “Classification of locally homogeneous connections on 2-dimensional manifolds”, to appear in Diff. Geom. Appl..; · Zbl 1063.53024 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.