Quasihomogeneous Analytic Affine Connections on Surfaces

Abstract

ABSTRACT. We classify torsion-free real-analytic affine connections on compact oriented real-analytic surfaces which are locally homogeneous on a nontrivial open set, without being locally homogeneous on all of the surface. In particular, we prove that such connections exist. This classification relies in a local result that classifies germs of torsion-free real-analytic affine connections on a neighborhood of the origin in the plane which are quasihomogeneous, in the sense that they are locally homogeneous on an open set containing the origin in its closure, but not locally homogeneous in the neighborhood of the origin.

On distinguished local coordinates for locally homogeneous affine surfaces

“…3.4. The proof of Lemma 2.2 (4). Throughout this section, we will not use the normalizations of Lemma 3.1.…”

“…The special case of locally symmetric affine surfaces was addressed in [14], where it is shown that any locally symmetric affine surface is either modeled on a surface of constant curvature with the Levi-Civita connection or, up to linear equivalence, on one of two affine surfaces which have the form given in Theorem 1.1-(1). Theorem 1.1 has been useful in many works on affine surfaces, including but not limited to [4,5,10,12]. 1 We also refer to Kowalski et al [11] for another proof of Theorem 1.1 in the torsion free setting.…”

Homogeneous affine surfaces: Moduli spaces

“…A slightly weaker definition of homegeneity was considered in [7]: a manifold is called quasihomogeneous if it is locally homogeneous on a nontrivial open set, but not on the whole surface. There, a classification of torsion-free real-analytic quasihomogeneous affine connections on compact orientable surfaces was given.…”

Quasihomogeneous three-dimensional real-analytic Lorentz metrics do not exist

“…In a recent joint work with A. Guillot, the first author obtained the classification of germs of quasihomogeneous, real-analytic, torsion free, affine connections on surfaces [9]. The article [9] also classifies the quasihomogeneous germs of real-analytic, torsion free, affine connections which extend to compact surfaces.…”

“…The article [9] also classifies the quasihomogeneous germs of real-analytic, torsion free, affine connections which extend to compact surfaces. In particular, such germs of quasihomogeneous connections do exist.…”