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.


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