Soluções explícitas para variedades tipo Einstein e tensor de Einstein prescrito conforme ao espaço pseudo Euclidiano
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2021-12-16
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Universidade Federal de Goiás
Resumo
In this thesis we study two problems involving Ricc i curvature. Initially, we start by considering
gradient Einstein type solitons conformal to an $n --$dimensional pseudo Euclidean space. We present the
associeated system of differential equations partial and, for such a system, solutions invariant under th e
action of the pseudo orthogonal group, are considered. This approach allows us to provide explicit
examples of $n --$dimensional gradient Einstein type manifolds, as well as, a rigidity result. We prove that a
large class of gradient Riemannian Einstein ty pe manifolds conformal to an Euclidean space, rotationally
symmetric, is isometric to $
symmetric, is isometric to $\\mathbb{S}^{nmathbb{S}^{n--1} 1} \\times times \\mathds{R}$.mathds{R}$.
Posteriorly, we extend our analysis to prescribed tensors. When considering the space pseudoPosteriorly, we extend our analysis to prescribed tensors. When considering the space pseudo--Euclidean $(Euclidean $(\\mathds{R}^n,g)$, $n mathds{R}^n,g)$, $n \\geq geq 3$, with coordinates $x=3$, with coordinates $x=\\left(x_1,...,x_nleft(x_1,...,x_n\\right)$ and metric right)$ and metric components $g_{ij} = components $g_{ij} = \\delta_{ij}delta_{ij}\\epsilon_i$, $1epsilon_i$, $1\\leq i, jleq i, j\\leq n$, where $leq n$, where $\\varepsilon_i=varepsilon_i=\\pm1$ and one pm1$ and one diagonal $(0,2)$diagonal $(0,2)$--tensors of the form $tensors of the form $\\mathcal{T}=mathcal{T}=\\sum_isum_i\\epsilon_i{h_i(x)dx_i^2}$, we obtain nepsilon_i{h_i(x)dx_i^2}$, we obtain necessary ecessary and sufficient conditions for the existence of a metric $and sufficient conditions for the existence of a metric $\\overline{g}$, conformal to $g$, such that overline{g}$, conformal to $g$, such that $Ric_{$Ric_{\\overline{g}}overline{g}}--\\displaystyledisplaystyle\\frac{frac{\\overline{overline{\\mathcal{K}}}{2} mathcal{K}}}{2} \\overline{g} =overline{g} =\\mathcal{T}$, where mathcal{T}$, where $Ric_{$Ric_{\\overline{g}}$ and $overline{g}}$ and $\\overline{overline{\\mathcal{K}mathcal{K}}$ are the Ricci tensor and scalar curvature of the metric }$ are the Ricci tensor and scalar curvature of the metric $$\\overline{g}$, respectively. Using the results obtained, we construct an example of a static perfect fluid overline{g}$, respectively. Using the results obtained, we construct an example of a static perfect fluid spacetime. Similar problems are considered for locally conformally flat manifolds.spacetime. Similar problems are considered for locally conformally flat manifolds.
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Citação
MENEZES, I. F. Soluções explícitas para variedades tipo Einstein e tensor de Einstein prescrito conforme ao espaço pseudo Euclidiano. 2021. 81 f. Tese (Doutorado em Matemática) - Universidade Federal de Goiás, Goiânia, 2021.