Nonlinear Analysis 204 (2021) 112198 Contents lists available at ScienceDirect Nonlinear Analysis www.elsevier.com/locate/na On the geometry of Einstein-type structures Andrea Anselli, Giulio Colombo ∗ , Marco Rigoli Dipartimento di Matematica, Università degli Studi di Milano, Via Cesare Saldini 50, 20133 Milano, Italy article info Article history: Received 25 April 2020 Accepted 9 November 2020 Communicated by Andrea Mondino MSC: 53C25 53C21 53C24 53B20 58J60 Keywords: 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 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. © 2020 Elsevier Ltd. All rights reserved. 1. Introduction The aim of this paper is to study the geometry of connected, complete, possibly compact, Riemannian manifolds (M, ⟨ , ⟩) with a (gradient) Einstein-type structure, if any, of the form { Ric φ + Hess(f ) − µdf ⊗ df = λ⟨ , ⟩ τ (φ)= dφ(∇f ), (1.1) ∗ Corresponding author. E-mail addresses: andrea.anselli@unimi.it (A. Anselli), giulio.colombo@unimi.it (G. Colombo), marco.rigoli55@gmail.com (M. Rigoli). https://doi.org/10.1016/j.na.2020.112198 0362-546X/© 2020 Elsevier Ltd. All rights reserved.