Master's Dissertation
DOI
https://doi.org/10.11606/D.45.2011.tde-20230727-113335
Document
Author
Full name
Leandro Augusto Lichtenfelz
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2011
Supervisor
Title in Portuguese
Automorfismos de G-estruturas
Keywords in Portuguese
Fibrados Vetoriais
Grupos De Lie
Abstract in Portuguese
Dada uma variedade diferenciavel M, para cada subgrupo de Lie G 2286 GL(n), pode-se contemplar a reducao do grupo estrutural do GL(n)-fibrado principal de referenciais sobre M a G. Quando existe, tal reducao se chama uma G-estrutura. Dentre todas as G-estruturas, ha uma classe favoravel delas, chamadas G-estruturas de tipo finito, para as quais o grupo G satisfaz uma certa condicao algebrica, a saber, que o k-esimo prolongamento da sua algebra de Lie, g, e o espaco vetorial nulo. Para estas G-estruturas, mostramos que seu grupo de automorfismos, que consiste dos difeomorfismos de M que mandam referenciais da G-estrutura sobre referenciais da G-estrutura, e um grupo de Lie. Casos particulares incluem grupos de isometrias Riemannianas, grupos de isometrias Lorentzianas e grupos conformes.
Title in English
not available
Abstract in English
Given a differentiable manifold M, for each group G 2286 GL(n), one might consider the reduction of the structure group of the GL(n)-principal bundle of frames over M to G. When such a reduction exists, it is called a G-structure over M. Among all G-structures, there exists a more tractable class, called G-structures of finite type, for which the group G satisfies a certain algebraic condition, namely, that the kth prolongation of its Lie algebra, g, is the null vector space. We prove, for such G-structures, that their automorphism group, which consists of all diffeo- morpshisms of M onto itself sending frames from the G-structure into frames again belonging to the G-structure, is a Lie group. Some special cases include isometry groups of Riemannian manifolds, isometry groups of Lorentzian manifolds and conformal groups.
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Publishing Date
2023-07-27