We give a new short self-contained proof of the result of Opozda [B. Opozda, A classification of locally homogeneous connections on 2-dimensional manifolds, Differential Geom. Appl. 21 (2004), 173-198.] classifying the locally homogeneous torsion fre
e affine surfaces and the extension to the case of surfaces with torsion due to Arias-Marco and Kowalski [T. Arias-Marco and O. Kowalski, Classification of locally homogeneous linear connections with arbitrary torsion on 2-dimensional manifolds, Monatsh. Math. 153 (2008), 1-18.]. Our approach rests on a direct analysis of the affine Killing equations and is quite different than the approaches taken previously in the literature.
We use the modified Riemannian extension of an affine surface to construct Bach flat manifolds. As all these examples are VSI (vanishing scalar invariants), we shall construct scalar invariants which are not of Weyl type to distinguish them. We illus
trate this phenomena in the context of homogeneous affine surfaces.
If $mathcal{M}=(M, abla)$ is an affine surface, let $mathcal{Q}(mathcal{M}):=ker(mathcal{H}+frac1{m-1}rho_s)$ be the space of solutions to the quasi-Einstein equation for the crucial eigenvalue. Let $tilde{mathcal{M}}=(M,tilde abla)$ be another affin
e structure on $M$ which is strongly projectively flat. We show that $mathcal{Q}(mathcal{M})=mathcal{Q}(tilde{mathcal{M}})$ if and only if $ abla=tilde abla$ and that $mathcal{Q}(mathcal{M})$ is linearly equivalent to $mathcal{Q}(tilde{mathcal{M}})$ if and only if $mathcal{M}$ is linearly equivalent to $tilde{mathcal{M}}$. We use these observations to classify the flat Type~$mathcal{A}$ connections up to linear equivalence, to classify the Type~$mathcal{A}$ connections where the Ricci tensor has rank 1 up to linear equivalence, and to study the moduli spaces of Type~$mathcal{A}$ connections where the Ricci tensor is non-degenerate up to affine equivalence.
We examine the space of solutions to the affine quasi--Einstein equation in the context of homogeneous surfaces. As these spaces can be used to create gradient Yamabe solitions, conformally Einstein metrics, and warped product Einstein manifolds usin
g the modified Riemannian extension, we provide very explicit descriptions of these solution spaces. We use the dimension of the space of affine Killing vector fields to structure our discussion as this provides a convenient organizational framework.
Motivated by the construction of Bach flat neutral signature Riemannian extensions, we study the space of parallel trace free tensors of type $(1,1)$ on an affine surface. It is shown that the existence of such a parallel tensor field is characterize
d by the recurrence of the symmetric part of the Ricci tensor.
We examine the local geometry of affine surfaces which are locally symmetric. There are 6 non-isomorphic local geometries. We realize these examples as Type A, Type B, and Type C geometries using a result of Opozda and classify the relevant geometrie
s up to linear isomorphism. We examine the geodesic structures in this context. Particular attention is paid to the Lorentzian analogue of the hyperbolic plane and to the pseudosphere.
We analyze the moduli space of non-flat homogeneous affine connections on surfaces. For Type $mathcal{A}$ surfaces, we write down complete sets of invariants that determine the local isomorphism type depending on the rank of the Ricci tensor and exam
ine the structure of the associated moduli space. For Type $mathcal{B}$ surfaces which are not Type $mathcal{A}$ we show the corresponding moduli space is a simply connected real analytic 4-dimensional manifold with second Betti number equal to $1$.
We give a general Lie-theoretic construction for anti-invariant almost Hermitian Riemannian submersions, anti-invariant quaternion Riemannian submersions, anti-invariant para-Hermitian Riemannian submersions, anti-invariant para-quaternion Riemannian
submersions, and anti-invariant octonian Riemannian submersions. This yields many compact Einstein examples.
We use the solution space of a pair of ODEs of at least second order to construct a smooth surface in Euclidean space. We describe when this surface is a proper embedding which is geodesically complete with finite total Gauss curvature. If the associ
ated roots of the ODEs are real and distinct, we give a universal upper bound for the total Gauss curvature of the surface which depends only on the orders of the ODEs and we show that the total Gauss curvature of the surface vanishes if the ODEs are second order. We examine when the surfaces are asymptotically minimal.
We derive the Chern-Gauss-Bonnet Theorem for manifolds with smooth non-degenerate boundary in the pseudo-Riemannian context from the corresponding result in the Riemannian setting by examining the Euler-Lagrange equations associated to the Pfaffian o
f a complex metric on the tangent space and then applying analytic continuation.
