The aim of this thesis is to study the geometry of a Riemannian manifold M, with a special structure, called Einstein-type structure, depending on 3 real parameters, a smooth map phi into a target Riemannian manifold N, and a smooth function, called potential function, on M itself. We will occasionally let some of the parameters be smooth functions. The setting generalizes various previously studied situations:, Ricci solitons, almost Ricci-solitons, Ricci-harmonic solitons, quasi-Einstein manifolds and so on. By taking a constant potential function those structures reduces to harmonic-Einstein manifolds, that are a generalization of Einstein manifolds. The main ingredient of our analysis is the study of certain modified curvature tensors on M related to the map phi, called phi-curvatures, obtaining, for instance, their transformation laws under a conformal change of metric, and to develop a series of results for harmonic-Einstein manifolds that parallel those obtained for Einstein manifolds some times ago and also in the very recent literature. Einstein-type structures may be obtained, for some special values of the parameters involved, by a conformal deformation of a harmonic-Einstein manifold or even as the base of a warped product harmonic-Einstein manifold. The latter fact applies not only in the Riemannian but also in the Lorentzian setting and thus some Einstein-type structures are connected with solutions of the Einstein field equations, which are of particular interest in General Relativity. The main result of the thesis is the locally characterization, via a couple of integrability conditions and mild assumptions on the potential function, of Einstein-type structures with vanishing phi-Bach curvature (in the direction of the potential) as a warped product with harmonic-Einstein base and with an open real interval as fibre, extending in a very non trivial way a recent result for Bach flat Ricci solitons. Moreover the map phi depends only on the base of the warped product and not on the fibre . We also consider rigidity, triviality and non-existence results, both in the compact and non-compact cases. This is done via integral formulas and, in the non-compact case, via analytical tools, like the weak maximum principle and the classical results of Obata, Tashiro, Kanai.

PHI-CURVATURES, HARMONIC-EINSTEIN MANIFOLDS AND EINSTEIN-TYPE STRUCTURES / A. Anselli ; tutor: M. Rigoli; coordinatore: V. Mastropietro. - : . DIPARTIMENTO DI MATEMATICA "FEDERIGO ENRIQUES", 2020 Jan 28. ((32. ciclo, Anno Accademico 2019. [10.13130/anselli-andrea_phd2020-01-28].

PHI-CURVATURES, HARMONIC-EINSTEIN MANIFOLDS AND EINSTEIN-TYPE STRUCTURES

A. Anselli
2020

Abstract

The aim of this thesis is to study the geometry of a Riemannian manifold M, with a special structure, called Einstein-type structure, depending on 3 real parameters, a smooth map phi into a target Riemannian manifold N, and a smooth function, called potential function, on M itself. We will occasionally let some of the parameters be smooth functions. The setting generalizes various previously studied situations:, Ricci solitons, almost Ricci-solitons, Ricci-harmonic solitons, quasi-Einstein manifolds and so on. By taking a constant potential function those structures reduces to harmonic-Einstein manifolds, that are a generalization of Einstein manifolds. The main ingredient of our analysis is the study of certain modified curvature tensors on M related to the map phi, called phi-curvatures, obtaining, for instance, their transformation laws under a conformal change of metric, and to develop a series of results for harmonic-Einstein manifolds that parallel those obtained for Einstein manifolds some times ago and also in the very recent literature. Einstein-type structures may be obtained, for some special values of the parameters involved, by a conformal deformation of a harmonic-Einstein manifold or even as the base of a warped product harmonic-Einstein manifold. The latter fact applies not only in the Riemannian but also in the Lorentzian setting and thus some Einstein-type structures are connected with solutions of the Einstein field equations, which are of particular interest in General Relativity. The main result of the thesis is the locally characterization, via a couple of integrability conditions and mild assumptions on the potential function, of Einstein-type structures with vanishing phi-Bach curvature (in the direction of the potential) as a warped product with harmonic-Einstein base and with an open real interval as fibre, extending in a very non trivial way a recent result for Bach flat Ricci solitons. Moreover the map phi depends only on the base of the warped product and not on the fibre . We also consider rigidity, triviality and non-existence results, both in the compact and non-compact cases. This is done via integral formulas and, in the non-compact case, via analytical tools, like the weak maximum principle and the classical results of Obata, Tashiro, Kanai.
RIGOLI, MARCO
MASTROPIETRO, VIERI
Phi-curvatures; Harmonic-Einstein manifolds; Einstein-type structures; Conformal change of metric; Conformally harmonic-Einstein manifolds; Warped products; Rigidity results; Phi-Cotton; Phi-Weyl; Phi-Bach; Weak maximum principle; Vertical conformal vector fields; Codazzi tensors; Integrability conditions; Non-existence results; Volume estimates
Settore MAT/03 - Geometria
Settore MAT/05 - Analisi Matematica
PHI-CURVATURES, HARMONIC-EINSTEIN MANIFOLDS AND EINSTEIN-TYPE STRUCTURES / A. Anselli ; tutor: M. Rigoli; coordinatore: V. Mastropietro. - : . DIPARTIMENTO DI MATEMATICA "FEDERIGO ENRIQUES", 2020 Jan 28. ((32. ciclo, Anno Accademico 2019. [10.13130/anselli-andrea_phd2020-01-28].
Doctoral Thesis
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/703786
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