Andrea Anselli | Searcharxiv

arXiv2020

On the rigidity of harmonic-Ricci solitons

In this paper we introduce the notion of rigidity for harmonic-Ricci solitons and we provide some characterizations of rigidity, generalizing some known results for Ricci solitons. In the compact case we are able to deal with not necessarily gradient solitons while, in the complete non-compact case, we restrict our attention to steady and shrinking gradient solitons. We show that the rigidity can be traced back to the vanishing of certain modified curvature tensors that take into account the geometry a Riemannian manifold equipped with a smooth map $φ$, called $φ$-curvatures, which are a natural generalization of the standard curvature tensors in the setting of harmonic-Ricci solitons.

math.DG↗

arXiv2020

On the Bach and Einstein 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 $M$ in presence of field $φ$, that in this context is given by a smooth map with source $M$ 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 $φ$-Bach tensor, that in absence of the field $φ$ reduces to the classic Bach tensor. We provide a variational characterization for $φ$-Bach flat manifolds and we do the same also for harmonic-Einstein manifolds, i.e., solutions of the Einstein field equations with source the conservative field $φ$. We take the opportunity to discuss a generalization of some related topics: the Yamabe problem, the image of the scalar curvature map, warped product solutions and static manifolds.

math.DG↗