Bach-flat isotropic gradient Ricci solitons

Abstract

We construct examples of Bach-flat gradient Ricci solitons which are neither half conformally flat nor conformally Einstein.2010 Mathematics Subject Classification. 53C25, 53C20, 53C44.

Half conformally flat generalized quasi-Einstein manifolds of metric signature (2,2)

“…Affine surfaces supporting a parallel nilpotent (1, 1)-tensor field In this section, we give examples of anti-self-dual quasi-Einstein manifolds which do not fit into the classification of Theorem 12, thus emphasizing the role of the Walker orientation. We will examine a special family of affine surfaces (Σ, D) which admit a parallel nilpotent (1, 1)-tensor field T (DT = 0, T 2 = 0); we refer to [13] for details. We assume the system of local coordinates (x 1 , x 2 ) is chosen so that: T ∂ x 1 = ∂ x 2 and T ∂ x 2 = 0.…”

“…We adopt the notation of Equation ( 6). The modified Riemannian extension g D,0,T,T = ιT • ιT + g D is never self-dual, but it is anti-self-dual if ω satisfies ω(ker T ) = 0 (see [13] for details). These affine surfaces are strongly projectively flat.…”

Conformal geometry of non‐reductive four‐dimensional homogeneous spaces

“…It is important to emphasize that although any four‐dimensional locally conformally Einstein metric is Bach flat, there are examples of strictly Bach flat manifolds, i.e., they are neither half conformally flat nor locally conformally Einstein (see, for example , , and references therein). Kozameh, Newman and Tod (see also ) obtained the following necessary conditions for any four‐dimensional conformally Einstein metric.…”

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]).…”

“…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. We use the Cauchy-Kovalevski Theorem to show that any such modified Riemannian extension can be locally deformed to a Bach flat one in the real analytic setting.…”