Locally homogeneous rigid geometric structures on surfaces

Abstract

Abstract. We study locally homogeneous rigid geometric structures on surfaces. We show that a locally homogeneous projective connection on a compact surface is flat. We also show that a locally homogeneous unimodular affine connection ∇ on a two dimensional torus is complete and, up to a finite cover, homogeneous.Let ∇ be a unimodular real analytic affine connection on a real analytic compact connected surface M . If ∇ is locally homogeneous on a nontrivial open set in M , we prove that ∇ is locally homogeneous on all of M .

Homogeneous affine surfaces: Moduli spaces

“…We have Γ 11 2 Γ 22 1 = 0. If Γ 22 1 = 0, we obtain the point (6,5) which was missing from the boundary curve discussed in Case 3.1 above. On the other hand, if Γ 22 1 is non-zero, we can normalize Γ 22 1 = 1 and obtain Ψ 3 = 5 − 4xy −2 = 5 − 4x 3 where x = 0.…”

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

“…The following result was first proved in the torsion free setting by Opozda [26] and subsequently extended to surfaces with torsion by Arias-Marco and Kowalski [4], see also [4,11,16,21,22,27] for related work. Theorem 1.1.…”

Quasihomogeneous Analytic Affine Connections on Surfaces

“…This was also proved for three dimensional real-analytic Lorentz metrics [Dum08] and, in higher dimension, for complete real-analytic Lorentz metrics having semisimple Killing Lie algebra [Mel09]. In [Dum12], the first author proved that a real-analytic unimodular affine connection on a real-analytic compact surface which is locally homogeneous on a nontrivial open set is locally homogeneous on all of the surface, and asks about the extent to which the unimodularity hypothesis is necessary.…”

“…Surprisingly, for some specific geometric structures, if the subset of local homogeneity is not empty, a maximal open set of local homogeneity is all of the (connected) manifold, even if we drop the assumption that the automorphism group acts with a dense orbit. This is known to be true in the Riemannian setting [PTV96], as a consequence of the fact that all scalar invariants are constant (see also section 3 in [Dum12]). This was also proved for three dimensional real-analytic Lorentz metrics [Dum08] and, in higher dimension, for complete real-analytic Lorentz metrics having semisimple Killing Lie algebra [Mel09].…”