Bach-flat manifolds and conformally Einstein structures
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...
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| Format: | doctoral thesis |
| Publication Date: | 2019 |
| Country: | España |
| Institution: | Universidad de Santiago de Compostela (USC) |
| Repository: | Minerva. Repositorio Institucional de la Universidad de Santiago de Compostela |
| Language: | English |
| OAI Identifier: | oai:minerva.usc.gal:10347/19468 |
| Online Access: | http://hdl.handle.net/10347/19468 |
| Access Level: | Open access |
| Keyword: | Materias::Investigación::12 Matemáticas::1204 Geometría::120404 Geometría diferencial Materias::Investigación::12 Matemáticas::1204 Geometría::120411 Geometría de Riemann |
| Summary: | 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. |
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