Ricci almost solitons with associated projective vector field

A Ricci almost soliton whose associated vector field is projective is shown to have vanishing Cotton tensor, divergence-free Bach tensor and Ricci tensor as conformal Killing. For the compact case, a sharp inequality is obtained in terms of scalar curvature.We show that every complete gradient Ricci soliton is isometric to the Riemannian product of a Euclidean space and an Einstein space. A complete K-contact Ricci almost soliton whose associated vector field is projective is compact Einstein and Sasakian.

Bach and Einstein’s equations in presence of a field

The aim of this paper is to introduce and justify a possible generalization of the classic Bach field equations on a four-dimensional smooth manifold [Formula: see text] in the presence of field [Formula: see text], given by a smooth map with source [Formula: see text] and target another Riemannian manifold. Those equations are characterized by the vanishing of a two times covariant, symmetric, traceless and conformally invariant tensor field, called [Formula: see text]-Bach tensor, that in absence of the field [Formula: see text] reduces to the classic Bach tensor, and by the vanishing another tensor related to the bi-energy of [Formula: see text]. Since solutions of the Einstein-massless scalar field equations, or more generally, of the Einstein field equations with source the wave map [Formula: see text] solves those generalized Bach’s equations, we include the latter in our analysis providing a systematic study for them, relying on the recent concept of [Formula: see text]-curvatures. We take the opportunity to discuss the related topic of warped product solutions.

Evolution of some geometric quantities along the cotton flow

In this paper we have addressed the behaviour of Yamabe constant along the Cotton flow. We have also studied the evolution of ADM mass along the Cotton flow and it is shown that the ADM mass is conserved along the Cotton flow. Among others evolution of Bach tensor under Cotton flow is derived. It is shown that if the metric of a local conformally flat 3-manifold evolves under the Cotton flow, then the Bach tensor satisfies the heat equation.

The Cotton Tensor and the Ricci Flow

We compute the evolution equation of the Cotton and the Bach tensor under the Ricci flow of a Riemannian manifold, with particular attention to the three dimensional case, and we discuss some applications.