Bach-flat manifolds and conformally Einstein structures
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Title: | Bach-flat manifolds and conformally Einstein structures |
Author: | Gutiérrez Rodríguez, Ixchel Dzohara |
Advisor: | Calviño Louzao, Esteban García Río, Eduardo Vázquez Lorenzo, Ramón |
Affiliation: | Universidade de Santiago de Compostela. Centro Internacional de Estudos de Doutoramento e Avanzados (CIEDUS) Universidade de Santiago de Compostela. Escola de Doutoramento Internacional en Ciencias e Tecnoloxía Universidade de Santiago de Compostela. Programa de Doutoramento en Matemáticas |
Subject: | Bach tensor | Conformally Einstein manifolds | Ricci solitons | |
Date of Issue: | 2019 |
Abstract: | Einstein manifolds, being critical for the Hilbert-Einstein functional, are central in Riemannian Geometry and Mathematical Physics. A strategy to construct Einstein metrics consists on deforming a given metric by a conformal factor so that the resulting metric is Einstein. In the present Thesis we follow this approach with special emphasis in dimension four. This is the first non-trivial case where the conformally Einstein condition is not tensorial and there are topological obstructions to the existence of Einstein metrics. The conformally Einstein condition is given by a overdetermined PDE-system. Hence the consideration of necessary conditions to be conformally Einstein are of special relevance: the Bach-flat condition is central. In this Thesis we classify four-dimensional homogeneous conformally Einstein manifolds and provide a large family of strictly Bach-flat gradient Ricci solitons. We show the existence of Bach-flat structures given as deformations of Riemannian extensions by means of the Cauchy-Kovalevskaya theorem. |
URI: | http://hdl.handle.net/10347/19468 |
Rights: | Attribution-NonCommercial-NoDerivatives 4.0 Internacional |
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