Made available in DSpace on 2015-03-26T13:45:33Z (GMT). No. of bitstreams: 1 texto completo.pdf: 445534 bytes, checksum: 6d9bec9dbcfd5842c82db998451fec95 (MD5) Previous issue date: 2011-02-23 / In this work we introduce the concept of curvature ellipse at one point of a surface immersed in Rn, n ≥ 4 to study the relationship between semiumbilics points (points that the curvature ellipse degenerated in a segment line), the existence of umbilic normal directions to the surface and hyperspherical surfaces. To achieve these goals we base our results especially in the articles “Umbilicity of surfaces with orthogonal asymptotic lines in R4” of M. C. Romero-Fuster and F. Sánchez-Bringas [24] and “Geometric Contacts of Surfaces Immersed in Rn, n ≥ 5” of S. I. R. Costa, S. M. Moraes and M. C. Romero-Fuster [5]. / Neste trabalho introduzimos o conceito de elipse de curvatura em um ponto de uma superfície imersa em Rn, n ≥ 4 com o objetivo de estudar as relações entre pontos semiumbílicos (pontos onde a elipse de curvatura se degenera em um segmento de reta), a existência de direções normais de umbilicidade à superfície e superfícies hiperesféricas. Para obter tais objetivos baseamos nossos resultados principalmente nos artigos “Umbilicity of surfaces with orthogonal asymptotic lines in IR4” de M. C. Romero-Fuster e F. Sánchez-Bringas [24] e “Geometric Contacts of Surfaces Immersed in IRn, n ≥ 5” de S. I. R. Costa, S. M. Moraes e M. C. Romero-Fuster [5].

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Neste trabalho, provamos novos resutados de rigidez para hipersuperfÃcies quase-Einsteins no espaÃo euclidiano, baseado-se nos resultados pinching do autovalor. EntÃo, nÃs deduzimos alguns resultados anÃlogos para hipersuperfÃcies quase-umbÃlicas e uma nova caracterizaÃÃo de esferas geodÃsicas / In this work, we prove new rigidity results for almost-Einstein hypersurfaces of the Euclidean space, based on previous eigenvalue pinching results. Then, we deduce some comparable result for almost-umbilic hypersurfaces and new characterizations of geodesic spheres.

Almost contact metric submersions constitute a class of Riemannian submersions whose total space is an almost contact metric manifold. Regarding the base space, two types are studied. Submersions of type I are those whose base space is an almost contact metric manifold while, when the base space is an almost Hermitian manifold, then the submersion is said to be of type II. After recalling the known notions and fundamental properties to be used in the sequel, relationships between the structure of the fibres with that of the total space are established. When the fibres are almost Hermitian manifolds, which occur in the case of a type I submersions, we determine the classes of submersions whose fibres are Kählerian, almost Kählerian, nearly Kählerian, quasi Kählerian, locally conformal (almost) Kählerian, Gi-manifolds and so on. This can be viewed as a classification of submersions of type I based upon the structure of the fibres. Concerning the fibres of a type II submersions, which are almost contact metric manifolds, we discuss how they inherit the structure of the total space. Considering the curvature property on the total space, we determine its corresponding on the fibres in the case of a type I submersions. For instance, the cosymplectic curvature property on the total space corresponds to the Kähler identity on the fibres. Similar results are obtained for Sasakian and Kenmotsu curvature properties. After producing the classes of submersions with minimal, superminimal or umbilical fibres, their impacts on the total or the base space are established. The minimality of the fibres facilitates the transference of the structure from the total to the base space. Similarly, the superminimality of the fibres facilitates the transference of the structure from the base to the total space. Also, it is shown to be a way to study the integrability of the horizontal distribution. Totally contact umbilicity of the fibres leads to the asymptotic directions on the total space. Submersions of contact CR-submanifolds of quasi-K-cosymplectic and quasi-Kenmotsu manifolds are studied. Certain distributions of the under consideration submersions induce the CR-product on the total space. / Mathematical Sciences / D. Phil. (Mathematics)

Almost contact metric submersions constitute a class of Riemannian submersions whose total space is an almost contact metric manifold. Regarding the base space, two types are studied. Submersions of type I are those whose base space is an almost contact metric manifold while, when the base space is an almost Hermitian manifold, then the submersion is said to be of type II. After recalling the known notions and fundamental properties to be used in the sequel, relationships between the structure of the fibres with that of the total space are established. When the fibres are almost Hermitian manifolds, which occur in the case of a type I submersions, we determine the classes of submersions whose fibres are Kählerian, almost Kählerian, nearly Kählerian, quasi Kählerian, locally conformal (almost) Kählerian, Gi-manifolds and so on. This can be viewed as a classification of submersions of type I based upon the structure of the fibres. Concerning the fibres of a type II submersions, which are almost contact metric manifolds, we discuss how they inherit the structure of the total space. Considering the curvature property on the total space, we determine its corresponding on the fibres in the case of a type I submersions. For instance, the cosymplectic curvature property on the total space corresponds to the Kähler identity on the fibres. Similar results are obtained for Sasakian and Kenmotsu curvature properties. After producing the classes of submersions with minimal, superminimal or umbilical fibres, their impacts on the total or the base space are established. The minimality of the fibres facilitates the transference of the structure from the total to the base space. Similarly, the superminimality of the fibres facilitates the transference of the structure from the base to the total space. Also, it is shown to be a way to study the integrability of the horizontal distribution. Totally contact umbilicity of the fibres leads to the asymptotic directions on the total space. Submersions of contact CR-submanifolds of quasi-K-cosymplectic and quasi-Kenmotsu manifolds are studied. Certain distributions of the under consideration submersions induce the CR-product on the total space. / Mathematical Sciences / D. Phil. (Mathematics)