ajb Posted September 6, 2006 Share Posted September 6, 2006 We know that the set of all Killing vectors on a Riemannian manifold [math] (M,g)[/math] form a lie subalgebra of the Lie algebra of vector fields on [math] (M,g)[/math]. This is the Lie algebra of isometries of [math](M,g)[/math]. For example, the Killing fields on [math]S^{2}[/math] with the standard metric [math]g = d\theta \otimes d\theta + \sin \theta ^{2} d \phi \otimes d \phi[/math] form the Lie algebra [math]\mathfrak{so}(3)[/math]. My question is, does anyone know what the conformal Killing vector fields on [math] S^{2}[/math] (again with the standard metric) are? What is the Lie algebra of conformal isometries of [math](S^{2}, g)[/math]? I ask because I have been "playing" with the conformal Killing equation on [math]S^{2}[/math] and have got no where. I am positive that someone has calculated these things. Do you know the references I should be reading? Cheers Link to comment Share on other sites More sharing options...
the tree Posted September 6, 2006 Share Posted September 6, 2006 We know that the set of all Killing vectors on a Riemannian manifold [math] (M,g)[/math] form a lie subalgebra of the Lie algebra of vector fields on [math] (M,g)[/math]. This is the Lie algebra of isometries of [math](M,g)[/math]. For example, the Killing fields on [math]S^{2}[/math] with the standard metric [math']g = d\theta \otimes d\theta[/math]If "we know" that, then you haven't given new infomation. And you don't appear to have asked a question either.So how are we meant to respond? Link to comment Share on other sites More sharing options...
ajb Posted September 6, 2006 Author Share Posted September 6, 2006 sorry pressed "post" when I wanted to press "preview" Link to comment Share on other sites More sharing options...
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