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A classification of locally homogeneous connections on 2-dimensional manifolds via group-theoretical approach. (English) Zbl 1060.53013

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.

MSC:

53B05 Linear and affine connections
53C30 Differential geometry of homogeneous manifolds

Citations:

Zbl 0837.58001

Software:

Maple
PDFBibTeX XMLCite
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
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[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
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