Pullback of the Volume Form under a Conformal Transformation
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From: Xevious on 5 May 2010 17:58 Let (M,g) be a Riemannian manifold of dimension n. A diffeomorphism F is a conformal transformation if F*g = e^{2w} g for a smooth function w on M. The metric g gives rise to the volume form dV. It is not obvious (to me, anyway) that F*dV = e^{nw} dV. Using the definitions of pullback for g and dV yield functions that don't seem similar enough for this to be true, and I'm having trouble finding a proof of this. Can anyone help? |