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

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?

sorry pressed "post" when I wanted to press "preview"