The isometry group of semi-Riemannian manifolds

Abstract

[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.