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Estudo do tensor de curvatura ó longo de xodésicas e circulos en variedades de Walker
The notion of curvature is one of the most important concepts in differential geometry. As a consequence of the difficulty of working with all the information that curvature tensor encodes, usually the study focusses in some operators defined in a natural way from the curvature tensor. In this memory we are going to focuss in two of them: Jacobi operator (R¡) and skew-symmetric curvature operator (Rc). These two operators give a big quantity of information about geodesics and circles, that are objects of main interest in mathematics and also in physics. The Riemannian locally symmetric spaces can be characterized by the fact that the Jacobi operator along every geodesic ¡ has constant eigenvalues (C) and parallel eigenspaces (P). Two natural generalizations of locally symmetric spaces appear when we consider conditions C and P separately. Also Riemannian locally symmetric spaces can be characterized by the fact that for every unit circle c, the skew-symmetric curvature operator has constant eigenvalues (O) and there exists a Jordan parallel basis for Rc along c (T). Our objective in this work is to investigate all these properties in Lorentzian geometry focussing in Walker metrics as a first step towards a description of the corresponding manifolds. The geometry of Walker manifolds, as the study of some of their specific characteristics and its influence in the study of Lorentzian geometry, will be fundamental aspects that we will use along this work. Since the point of view of Walker geometry, Walker manifolds of dimension 3 are the first non trivial case to consider. Also, the fact that all geometric information is codified in an unique function makes this geometry more tractable than the higher dimension cases.
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The notion of curvature is one of the most important concepts in differential geometry. As a consequence of the difficulty of working with all the information that curvature tensor encodes, usually the study focusses in some operators defined in a natural way from the curvature tensor. In this memory we are going to focuss in two of them: Jacobi operator (R¡) and skew-symmetric curvature operator (Rc). These two operators give a big quantity of information about geodesics and circles, that are objects of main interest in mathematics and also in physics. The Riemannian locally symmetric spaces can be characterized by the fact that the Jacobi operator along every geodesic ¡ has constant eigenvalues (C) and parallel eigenspaces (P). Two natural generalizations of locally symmetric spaces appear when we consider conditions C and P separately. Also Riemannian locally symmetric spaces can be characterized by the fact that for every unit circle c, the skew-symmetric curvature operator has constant eigenvalues (O) and there exists a Jordan parallel basis for Rc along c (T). Our objective in this work is to investigate all these properties in Lorentzian geometry focussing in Walker metrics as a first step towards a description of the corresponding manifolds. The geometry of Walker manifolds, as the study of some of their specific characteristics and its influence in the study of Lorentzian geometry, will be fundamental aspects that we will use along this work. Since the point of view of Walker geometry, Walker manifolds of dimension 3 are the first non trivial case to consider. Also, the fact that all geometric information is codified in an unique function makes this geometry more tractable than the higher dimension cases.
