Characterizing a manifold up to isometry is a challenging task. A manifold is a topological space. One may equip a manifold with a metric, and generally speaking, this metric determines how the manifold “looks". An example of this would be the unit sphere in R3. While we typically envision the standard metric on this sphere to give it its familiar shape, one could define a different metric on this set of points, distorting distances within this set to make it seem perhaps more ellipsoidal, something not isometric to the standard round sphere. In an effort to distinguish manifolds up to isometry, we wish to compute meaningful invariants. For example, the Riemann curvature tensor and its surrogates are examples of invariants one could construct. Since these objects are generally too complicated to compare and are not real valued, we construct scalar invariants from these objects instead. This thesis will explore these invariants and exhibit a special family of manifolds that are not flat on which all of these invariants vanish. We will go on to properly define, and gives examples of, manifolds, metrics, tangent vector fields, and connections. We will show how to compute the Christoffel symbols that define the Levi-Civita connection, how to compute curvature, and how to raise and lower indices so that we can produce scalar invariants. In order to construct the curvature operator and curvature tensor, we use the miracle of pseudo-Riemannian geometry, i.e., the Levi-Civita connection, the unique torsion free and metric compatible connection on a manifold. Finally, we examine Generalized Plane Wave Manifolds, and show that all scalar invariants of Weyl type on these manifolds vanish, despite the fact that many of these manifolds are not flat.

Thesis (Ph. D.)--University of Oregon, 2000. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 123-128). Also available for download via the World Wide Web; free to University of Oregon users.

Submitted by Erika Demachki (erikademachki@gmail.com) on 2018-06-29T18:43:00Z No. of bitstreams: 2 Tese - Marcos Tulio Alves de Carvalho - 2018.pdf: 2579945 bytes, checksum: 29a08a3db199f6061cf6020d90ce9213 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-07-03T15:18:43Z (GMT) No. of bitstreams: 2 Tese - Marcos Tulio Alves de Carvalho - 2018.pdf: 2579945 bytes, checksum: 29a08a3db199f6061cf6020d90ce9213 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-07-03T15:18:43Z (GMT). No. of bitstreams: 2 Tese - Marcos Tulio Alves de Carvalho - 2018.pdf: 2579945 bytes, checksum: 29a08a3db199f6061cf6020d90ce9213 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-06-12 / Fundação de Amparo à Pesquisa do Estado de Goiás - FAPEG / In this work we study two problems: the first one involving the prescribed Schouten tensor and the second one the prescribed curvature operator. The first problem was inspired by the works Deturck and Yang, [6], which consist of: Given a tensor T of order 2 in the pseudo-Euclidean space ( ,g), n ≥ 3, with coordinates x = ( ), and metric g, where , = ±1, find a metric as = g, such that the tensor of Schouten be T. The second problem is the problem of the prescribed curvature tensor consist of: Let Euclidean space ( ;g), n ≥ 3, with coordinates x = ( ), is , the R a tensor of order 4 of the form , where T = , with differentiable functions.We want to find a metric = g, such that = , where is the tensor curvature of the metric . Considering that the solutions are invariant by translation and rotation, we find necessary and sufficient conditions for both problems to have solution. / Neste trabalho estudamos dois problemas: o primeiro envolvendo o tensor de Schouten prescrito e o segundo o tensor curvatura prescrito. O primeiro problema foi inspirado no trabalho de Deturck e Yang, [6], que consiste em: Dado um tensor T de ordem 2 no espaço pseudo-Euclidiano ( ;g), n ≥ 3, com coordenadas x = ( ), e métrica g, onde , = ±1, encontrar uma métrica conforme = g, tal que o tensor de Schouten da métrica seja T. O segundo problema é o problema do tensor curvatura prescrito que consiste em: Seja o espaço Euclidiano ( ;g), n ≥ 3, com coordenadas x = ( ), e , e R um tensor de ordem 4 da forma onde T = , com funções diferenciáveis. Queremos encontrar uma métrica = g, tal que = , onde é o tensor curvatura da métrica . Considerando que as soluções sejam invariantes por translação e rotação, encontramos condições necessárias e suficientes para que ambos os problemas tenham solução.

We discuss two topics in this talk. First we study compact Ricci-flat four dimensional manifolds without boundary and obtain point wise restrictions on curvature( not involving global quantities such as volume and diameter) which force the metric to be flat. We obtain the same conclusion for compact Ricci-flat K¨ahler surfaces with similar but weaker restrictions on holomorphic sectional curvature. Next we study the reaction ODE associated to the evolution of the Riemann curvature operator along the Ricci flow. We analyze the behavior of this ODE near algebraic curvature operators of certain special type that includes the Riemann curvature operators of various(locally) symmetric spaces. We explicitly show the existence of some solution curves to the ODE connecting the curvature operators of certain symmetric spaces. Although the results of these two themes are different, the underlying common feature is the reaction ODE which plays an important role in both.