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.

On the geometry of Einstein-type structures / Anselli, A.; Colombo, G.; Rigoli, M.. - In: NONLINEAR ANALYSIS. - ISSN 0362-546X. - 204:(2021), pp. 1-84. [10.1016/j.na.2020.112198]

On the geometry of Einstein-type structures

Colombo G.;
2021

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.
2021
On the geometry of Einstein-type structures / Anselli, A.; Colombo, G.; Rigoli, M.. - In: NONLINEAR ANALYSIS. - ISSN 0362-546X. - 204:(2021), pp. 1-84. [10.1016/j.na.2020.112198]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11588/938686
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