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A classification of locally homogeneous affine connections with skew-symmetric Ricci tensor on 2-dimensional manifolds. (English) Zbl 0993.53008
A classification à la Sophus Lie of the connections as in the title is obtained. The elaborate and laborious method used by the authors is quite different from the Y. C. Wong’s procedure [Monatsh. Math. 68, 175-184 (1964; Zbl 0141.19302)]; see also the authors [Colloq. Math. 81, No. 1, 123-139 (1999; Zbl 0942.53019)]. We note the surprisingly simple form of the integrability conditions for the six linear partial differential equations of a Killing vector field.
The motivation for such a classification arises from the special case of a 2-dimensional manifold with affine Osserman connection: in the case of a surface the connection \(D\) is affine Osserman iff the Ricci tensor of \(D\) is skew-symmetric. In [E. García-Río, D. N. Kupeli, M. E. Vásquez-Abal and R. Vásquez-Lorenzo, Differ. Geom. Appl. 11, No. 2, 145-153 (1999; Zbl 0940.53017)] the concept of affine Osserman connection was defined, which is useful to build new examples of pseudo-Riemannian Osserman spaces [N. Blazic, N. Bokan and P. Gilkey (1997); E. García-Río and D. N. Kupeli (1997); E. García-Río., M. E. Vásquez-Abal and R. Vásquez-Lorenzo (1998)]. A pseudo-Riemannian manifold is said to be Osserman if the eigenvalues of the Jacobi operators are constant on the unit tangent sphere bundle.

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