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Dg Errata

Post Icon Posted: Submitted by Newtonswig on 9 November 2007 - 1:56pm.

Joined: 2006-10-01
Posts: 427
Sorry all, the path metric function for our punctured disc should have been:
$$\frac{1}{|z|log(|z|)}|dz| $$

You will recall that we sincerely hoped our form invariant under indeces, we note that any non trivial linear combination of these would give us non zero roots, so we need only worry about poincare isometries of the form $ u=e^{ib}z^a $. We can ignore the exponential at the start as it will easily get lost among the moduli.

Thus we may take $ u=z^a $, so
$$\frac{1}{|u|log(|u|)}.|du|= \frac{1}{|z^a|log(|z^a|)}.|dz^a|= \frac{1}{|z|^a.a.log(|z|)}.a|z|^{a-1}|dz|=\frac{1}{|z|log(|z|)}.|dz|$$

Hence our differential form is invariant under indeces as required.

Er... tada!...

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