Homogeneous affine surfaces: Moduli spaces

Abstract

Abstract. We analyze the moduli space of non-flat homogeneous affine connections on surfaces. For Type A surfaces, we write down complete sets of invariants that determine the local isomorphism type depending on the rank of the Ricci tensor and examine the structure of the associated moduli space. For Type B surfaces which are not Type A we show the corresponding moduli space is a simply connected real analytic 4-dimensional manifold with second Betti number equal to 1.

The affine quasi-Einstein Equation for homogeneous surfaces

“…We say that two Type A surface models are linearly isomorphic if there exists T ∈ GL(2, R) intertwining the two structures. One has (see Lemma 2.1 in [5]) the following result. Remark 2.3.…”

“…The remaining cases correspond to the generic situation where dim{K(M)} = 2. As we wish to describe the eigenspaces E(µ, ∇) very explicitly, we shall proceed rather combinatorially and rely on previous results [4,5,8] concerning different models for homogeneous affine surfaces.…”

The moduli space of Type A surfaces with torsion and non-singular symmetric Ricci tensor

“…We now turn to the question of invariants and, for the sake of completeness, present some results from M. Brozos-Vázquez et al [6]. We work in the torsion free setting.…”

Affine Killing complete and geodesically complete homogeneous affine surfaces

“…for different values of the parameters involved; for example, we may interchange the coordinates x 1 ↔ x 2 to see that M 2 1 (a 1 , a 2 ) is linearly equivalent to M 2 1 (a 2 , a 1 ). Giving a precise description of the identifications describing the relevant moduli spaces is somewhat difficult and we refer for [4,9] for further details as it will play no role here. The notation is chosen so that dim{K(M j i (·))} = j.…”