On the geometry of Einstein-type structures. (March 2021)

• Record Type: Journal Article
• Authors: Anselli, Andrea; Colombo, Giulio; Rigoli, Marco
• Is Part Of: Nonlinear analysis. Volume 204(2021)
• Publication Date: 2021-03
• Subjects: 53C25 -- 53C21 -- 53C24 -- 53B20 -- 58J60; φ-curvatures -- Harmonic-Einstein manifolds -- Conformally harmonic-Einstein manifolds -- Rigidity results -- Codazzi tensors -- Warped products -- Integrability conditions -- Non-existence results -- Uniqueness results -- Curvature restrictions -- Volume estimates -- Weak maximum principle; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2020.112198
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: The aim of the paper is to study the geometry of a Riemannian manifold M, with a special structure depending on 3 real parameters, a smooth map φ into a target Riemannian manifold N, and a smooth function f on M itself. We will occasionally let some of the parameters be smooth functions. For a special value of one of them, the structure is obtained by a conformal deformation of a harmonic-Einstein manifold. The setting generalizes various previously studied situations; for instance, Ricci solitons, Ricci harmonic solitons, generalized quasi-Einstein manifolds and so on. One main ingredient of our analysis is the study of certain modified curvature tensors on M, related to the map φ, and to develop a series of results for harmonic-Einstein manifolds that parallel those obtained for Einstein manifolds both some time ago and in the very recent literature. We then turn to locally characterize, via a couple of integrability conditions and mild assumptions on f, the manifold M as a warped product with harmonic-Einstein fibers extending in a very non trivial way a recent result for Ricci solitons. We then consider rigidity and non existence, both in the compact and non-compact cases. This is done via integral formulas and, in the non-compact case, via analytical tools previously introduced by the authors.