On distinguished local coordinates for locally homogeneous affine surfaces

Abstract

We give a new short self-contained proof of the result of Opozda [15] classifying the locally homogeneous torsion free affine surfaces and the extension to the case of surfaces with torsion due to Arias-Marco and Kowalski [1]. Our approach rests on a direct analysis of the affine Killing equations and is quite different than the approaches taken previously in the literature.2010 Mathematics Subject Classification. 53C21.

Symmetric affine surfaces with torsion

“…Opozda [18] classified the locally homogeneous affine surfaces without torsion. Subsequently, Arias-Marco and Kowalski [1] extended this classification to the more general setting; a different proof of this result has been given recently by Brozos-Vázquez et al [2]. Previous studies of locally homogeneous surfaces in the torsion free setting include [13,14].…”

“…The proof of Theorem 2 (2). Let M be a symmetric affine surface with parallel non-zero torsion which is not flat.…”

Spaces of locally homogeneous affine surfaces

“…Kowalski and Sekizawa [10] used it to examine Riemannian extensions of affine surfaces, Vanzurova [13] used it to study the metrizability of locally homogeneous affine surfaces, and Dǔsek [5] used it to study homogeneous geodesics. It plays a central role in the study of locally homogeneous connections with torsion of Arias-Marco and Kowalski [1] (see also [2] for a unified treatment independently of the torsion tensor). Although we will work with the local theory, the compact setting has been examined in [8,12].…”