YAMABE PROBLEM ON RICCI SOLITONS, AND POISSON STRUCTURES ON SINGULAR FIBRATIONS
We study the Yamabe Problem in a special class of Riemannian manifolds, the Ricci solitons. We also explore contact and symplectic manifolds admitting a compatible Ricci soliton, where we obtain some observations. With respect to Poissons geometry, we provide local expressions for Poisson bivectors...
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| Format: | doctoral thesis |
| Status: | Versión aceptada para publicación |
| Publication Date: | 2017 |
| Country: | México |
| Institution: | Centro de Investigación en Matemáticas |
| Repository: | Repositorio Institucional CIMAT |
| OAI Identifier: | oai:cimat.repositorioinstitucional.mx:1008/567 |
| Online Access: | http://cimat.repositorioinstitucional.mx/jspui/handle/1008/567 |
| Access Level: | Open access |
| Keyword: | info:eu-repo/classification/MSC/Yamabe info:eu-repo/classification/MSC/Estructura de pisón info:eu-repo/classification/cti/1 info:eu-repo/classification/cti/12 info:eu-repo/classification/cti/1299 info:eu-repo/classification/cti/129999 |
| Summary: | We study the Yamabe Problem in a special class of Riemannian manifolds, the Ricci solitons. We also explore contact and symplectic manifolds admitting a compatible Ricci soliton, where we obtain some observations. With respect to Poissons geometry, we provide local expressions for Poisson bivectors and the corresponding symplectic forms with broken Lefschetz and wrinkled singularities in dimensions 4, and 6, and discuss the higher dimension case. With respect to the Yamabe Problem on compact Ricci solitons the main result that we obtained is: Theorem There exists a unique U(2)-invariant solution to the Yamabe equation on CP2#-CP2 with the Koiso-Cao metric. We also explore the Ricci soliton equation. We use Hamiltonian and Liouville vector fields to derive some results concerning a Ricci soliton with a compatible symplectic structure. The main result obtained: Theorem Let (M, w) be a symplectic manifold of dimension greater than 2. If w has compatible Ricci soliton g determined by a Hamiltonian or a Liouville holomorphic vector field, then g is Einstein. We also constructed singular Poisson structures on manifolds of dimension 4 and 6, where the singularties are given by a broken Lefschetz fibration or a wrinkled fibration. The main results are: Theorem A closed, orientable, smooth 4-manifold equipped with a wrinkled fibration admits a complete Poisson structure. The fibres of the fibration are leaves of the symplectic foliation and both structures share the same singularities. Theorem A generalized broken Lefschetz fibration admits a Poisson structure compatible with the fibration structure. Also, generalized wrinkled fibrations in dimension 6 admit compatible Poisson structures. |
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