Affine Surfaces Which are Kähler, Para-Kähler, or Nilpotent Kähler

The affine quasi-Einstein Equation for homogeneous surfaces

“…Assume therefore that ρ s = 0. It follows from Lemma 5.6 in [9] that if a Type B surface admits a Kähler structure, then J = t 1 1 1 t 2 1 −t 1 1 ∈ M 2 (R), and the Christoffel symbols satisfy C 11 1 = C 22 1 t 2 1 + 2(C 22 2 + 2C 22 1 t 1 1 )t 1 1 , C 12 1 = C 22 2 + 2C 22 1 t 1 1 , C 11 2 = (C 22 2 + 2C 22 1 t 1 1 )t 2 1 , C 12 2 = C 22 1 t 2 1 , C 22 1 = 0.…”

“…and it follows from Lemma 3.6 in [9] that these are necessary and sufficient conditions for (M, ∇, J) to be a Kähler affine surface.…”

“…Assume therefore that ρ s = 0. It follows from Lemma 5.6 in [9] that if a Type B surface admits a Kähler structure, then…”

Constructing Bach flat manifolds of signature (2, 2) using the modified Riemannian extension

“…A modification of the classical Patterson-Walker Riemannian extension [20] was used in [7] to provide a new source of strictly Bach flat metrics which support gradient Ricci solitons. This construction requires the existence of a background affine surface admitting a parallel nilpotent tensor field, which is a rather restrictive condition (see [5]). In this paper, we shall generalize the construction of [7] to characterize Bach flat Riemannian extensions of affine surfaces admitting a nilpotent structure.…”