Mundet i Riera, IgnasiLlorens Giralt, Quim2019-09-182019-09-182019-01-18https://hdl.handle.net/2445/140379Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2019, Director: Ignasi Mundet i Riera[en] This work presents two important subjects of modern mathematics, Lie Groups and semi-Riemannian Geometry, and shows a beautiful theorem that arises as a combination of both matters: the isometry group of a semi-Riemannian manifold is a Lie group. The structure of the proof presented is as follows. First, we introduce a theorem by Palais [1], which gives a sufficient condition for a group G of diffeomorphisms acting on a smooth manifold M to be a Lie group: that the set of all vector fields on M which generate global 1-parameters subgroups of G generates a finite-dimensional Lie algebra. Then we show that this result can be applied to the isometry group of semi-Riemannian manifolds, by proving that the set of all complete Killing vector fields generates a finite-dimensional Lie algebra.65 p.application/pdfengcc-by-nc-nd (c) Quim Llorens Giralt, 2019http://creativecommons.org/licenses/by-nc-nd/3.0/es/Grups de LieTreballs de fi de grauGeometria de RiemannGeometria diferencial globalVarietats diferenciablesLie groupsBachelor's thesesRiemannian geometryGlobal differential geometryDifferentiable manifoldsinfo:eu-repo/semantics/bachelorThesisinfo:eu-repo/semantics/openAccess