Alguns aspectos da geometria riemanniana das variedades de Hilbert

AUTOR(ES)

Leonardo Biliotti

DATA DE PUBLICAÇÃO

2002

RESUMO

The aim of this work is to formalize the local theory of infinite dimensional Riemannian manifold and to study the geometry/ topology when the sectional curvature is bounded by two positive constant. We compare this situation with the finite dimensional case and emphasize the difference. The local theory was already developed since 1960, so we describe, briefly, the basic facts of the theory as the existence and uniqueness of Levi Civita connection, Gauss lemma and existence of convex neighborhood. However, we proved that the fundamental theorem of tensor field is not verified and we introduced a class, that we called C?-weakly, for which the criterion holds. When we want to study the global properties, the fact that the manifold is complete is fundamental, as in finite dimensional case, but as the Hopf-Rinow theorem is not always verified, completeness is not always equivalent to geodesic completeness. These pathology is not verified by complete simply connected manifolds with constant sectional curvature and we have the same classification as the finite dimensional case. However, the class of infinite dimensional manifolds of constant positive curvatura is bigger than the respective class in the finite dimensional case. This fact is consequence of the study of the groups that could acts effective and properly discontinuosly, as isometry group, on the unitary sphere on infinite dimensional Hilbert spaces. The two basics facts that justify our are that any infinite dimensional Hilbert space and any group G without torsion, acts without fixed point and properly discontinuous, as isometry group, on the sphere in l2 (G). The extension of theorems of Rauch and Topogonov, is fundamental when we study the geometry of with sectional curvature bounded by two positive constants. The consequence are the extension of some classical result, that we have proved in chapter 6, like of Berger- Topogonov theorem about the maximal diameter and, when we assume that the radius of injectivity of the manifolds is at least ?, some results in the spirit of the pinching theorems

ASSUNTO(S)

espaço de curvatura geometria riemaniana hilbert

Documentos Relacionados

• Aspectos da geometria complexa das variedades bandeira
• Alguns aspectos da mecânica das alças de retração ortodôntica
• Teoria Cº vetorial em geometria riemanniana e decomposição em bolhas para aplicações de Palais-Smale
• Variedades riemannianas abertas : rigidez das fronteiras ideais e geometria das variedades com curvatura minimal radial assintoticamente não negativa
• Aritmetizando la geometría desde dentro: el cálculo de segmentos de David Hilbert