Study of homogeneous DÀtri spaces, of the Jacobi operator on g.o. spaces and the locally homogeneous connections on 2-dimensional manifolds with the help of Mathematica.
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Study of homogeneous DÀtri spaces, of the Jacobi operator on g.o. spaces and the locally homogeneous connections on 2-dimensional manifolds with the help of Mathematica.

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Study of homogeneous DÀtri spaces, of the Jacobi operator on g.o. spaces and the locally homogeneous connections on 2-dimensional manifolds with the help of Mathematica.

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dc.contributor.advisor Kowalski, Oldrich es_ES
dc.contributor.advisor Martínez Naveira, Antonio es_ES
dc.contributor.author Arias Marco, Teresa es_ES
dc.contributor.other Universitat de València - GEOMETRIA I TOPOLOGIA es_ES
dc.date.accessioned 2010-07-07T08:00:59Z
dc.date.available 2010-07-07T08:00:59Z
dc.date.issued 2007 es_ES
dc.date.submitted 2007-06-04 es_ES
dc.identifier.uri http://hdl.handle.net/10550/14894
dc.description.abstract es_ES
dc.format.mimetype application/pdf es_ES
dc.language cat-en-es es_ES
dc.rights eng es_ES
dc.rights Copyright information available at source archive es_ES
dc.subject none es_ES
dc.title Study of homogeneous DÀtri spaces, of the Jacobi operator on g.o. spaces and the locally homogeneous connections on 2-dimensional manifolds with the help of Mathematica. es_ES
dc.type info:eu-repo/semantics/doctoralThesis es_ES
dc.description.abstractenglish Nowadays, the concept of homogeneity is one of the fundamental notions in geometry although its meaning must be always specified for the concrete situations. In this thesis, we consider the homogeneity of Riemannian manifolds and the homogeneity of manifolds equipped with affine connections. The first kind of homogeneity means that, for every smooth Riemannian manifold (M, g), its group of isometries I(M) is acting transitively on M. Part I of this thesis fits into this philosophy. Afterwards in Part II, we treat the homogeneity concept of affine connections. This homogeneity means that, for every two points of a manifold, there is an affine diffeomorphism which sends one point into another. In particular, we consider a local version of the homogeneity, that is, we accept that the affine diffeomorphisms are given only locally, i.e., from a neighborhood onto a neighborhood. More specifically, we devote the first Chapter of Part I to make a brief overview of some special kinds of homogeneous Riemannian manifolds which will be of special relevance in Part I and to show how the software MATHEMATICA© becomes useful. For that, we prove that the three-parameter families of flag manifolds constructed by N. R. Wallach in "Compact homogeneous Riemannian manifols with strictly positive curvature, Ann. of Math. 96 (1972), p. 276293" are DAtri spaces if and only if they are naturally reductive spaces. Thus, we improve the previous results given by DAtri, Nickerson and by Arias-Marco, Naveira. Moreover, in Chapter 2 we obtain the complete 4-dimensional classification of homogeneous spaces of type A. This allows us to prove correctly that every 4-dimensional homogeneous DAtri space is naturally reductive. Therefore, we correct, complete and improve the results presented by Podestà, Spiro, Bueken and Vanhecke. Chapter 3 is devoted to prove that the curvature operator has constant osculating rank over g.o. spaces. It is mean that a real number r exists such that under some assumptions, the higher order derivatives of the curvature operator from 1 to r are linear independent and from 1 to r + 1 are linear dependent. As a consequence, we also present a method valid on every g.o. space to solve the Jacobi equation. This method extends the method given by Naveira and Tarrío for naturally reductive spaces. In particular, we prove that the Jacobi operator on Kaplans example (the first known g.o. space that it is not naturally reductive) has constant osculating rank 4. Moreover, we solve the Jacobi equation along a geodesic on Kaplans example using the new method and the well-known method used by Chavel, Ziller and Berndt,Tricerri, Vanhecke. Therefore, we are able to present the main differences between both methods. In Part II, we classify (locally) all locally homogeneous affine connections with arbitrary torsion on two-dimensional manifolds. Therefore, we generalize the result given by Opozda for torsion-less case. Moreover, from our computations we obtain interesting consequences as the relation between the classifications given for the torsion less-case by Kowalski, Opozda and Vláek. In addition, we obtain interesting consequences about flat connections with torsion. In general, the study of these problems sometimes requires a big number of straightforward symbolic computations. In such cases, it is a quite difficult task and a lot of time consuming, try to make correctly this kind of computations by hand. Thus, we try to organize our computations in (possibly) most systematic way so that the whole procedure is not excessively long. Also, because these topics are an ideal subject for a computer-aided research, we are using the software MATHEMATICA©, throughout this work. But we put stress on the full transparency of this procedure. __________________________________________________________________________________________________ RESUMEN En esta tesis, se consideran dos tipos bien diferenciados de homogeneidad: la de las variedades riemannianas y la de las variedades afines. El primer tipo de homogeneidad se define como aquel que tiene la propiedad de que el grupo de isometrías actúa transitivamente sobre la variedad. La Parte I, recoge todos los resultados que hemos obtenido en esta dirección. Sin embargo, en la Parte II se presentan los resultados obtenidos sobre conexiones afines homogéneas. Una conexión afín se dice homogénea si para cada par de puntos de la variedad existe un difeomorfismo afín que envía un punto en otro. En este caso, se considera una versión local de homogeneidad. Más específicamente, la Parte I de esta tesis está dedicada a probar que "las familias 3-paramétricas de variedades bandera construidas por Wallach son espacios de D'Atri si y sólo si son espacios naturalmente reductivos". Más aún, en el segundo Capítulo, se obtiene la clasificación completa de los espacios homogéneos de tipo A cuatro dimensionales que permite probar correctamente que todo espacio de DAtri homogéneo de dimensión cuatro es naturalmente reductivo. Finalmente, en el tercer Capítulo se prueba que en cualquier g.o. espacio el operador curvatura tiene rango osculador constante y, como consecuencia, se presenta un método para resolver la ecuación de Jacobi sobre cualquier g.o. espacio. La Parte II se destina a clasificar (localmente) todas las conexiones afines localmente homogéneas con torsión arbitraria sobre variedades 2-dimensionales. Para finalizar el cuarto Capítulo, se prueban algunos resultados interesantes sobre conexiones llanas con torsión. En general, el estudio de estos problemas requiere a veces, un gran número de cálculos simbólicos aunque sencillos. En dichas ocasiones, realizarlos correctamente a mano es una tarea ardua que requiere mucho tiempo. Por ello, se intenta organizar todos estos cálculos de la manera más sistemática posible de forma que el procedimiento no resulte excesivamente largo. Este tipo de investigación es ideal para utilizar la ayuda del ordenador; así, cuando resulta conveniente, utilizamos la ayuda del software MATHEMATICA para desarrollar con total transparencia el método de resolución que más se adecua a cada uno de los problemas a resolver. es_ES

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