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 use the theory of isoparametric functions to investigate gradient Ricci solitons with constant scalar curvature. We show rigidity of gradient Ricci solitons with constant scalar curvature under some conditions on the Ricci tensor, which are all sa
tisfied if the manifold is curvature homogeneous. This leads to a complete description of four- and six-dimensional Kaehler gradient Ricci solitons with constant scalar curvature.
We show that the scalar curvature of a steady gradient Ricci soliton satisfying that the ratio between the square norm of the Ricci tensor and the square of the scalar curvature is bounded by one half, is boundend from below by the hyperbolic secant
of one half the distance function from a fixed point.
