Formula to calculate the scalar curvature

Hello there,

Suppose $\mathcal{M}_1$ is the Yang-Mills moduli space of the self-dule connections over 4-manifold $M=S^4$ and $\overline{g}$ is a Riemannian metric on $\mathcal{M}_1$ with new coordinates $\{u^i\} $. Dose anyone know good formula to calculate the scalar curvature?

Thanks.

edited Jul 11 2011 at 2:50

asked Jul 11 2011 at 1:54

Maths lover
19●1●2

Please see mathoverflow.net/howtoask, in particular, what on earth is 'a moduli space over 4-manifold $M$'? What are 'new coordinates'? – David Roberts Jul 11 2011 at 2:30

Do you mean the natural hyperkaehler metric on the ADHM moduli space of instantons over the round 4-sphere? HK metrics are Ricci-flat, hence scalar-flat. – Tim Perutz Jul 11 2011 at 3:44

@Tim , I meant the metric that defined by Groisser and Parker in their paper springerlink.com/content/g4661lr7317685lp – Maths lover Jul 11 2011 at 19:04

Ah, I see: the point is that the $L^2$-metric on the space of connections on a Riemannian 4-manifold induces a metric on the non-singular part of the instanton moduli space. Do edit your question: it's a perfectly reasonable one, but wasn't clear before. (ADHM is really about instantons on $\mathbb{R}^4$, not $S^4$; but I was struggling to interpret your question.) – Tim Perutz Jul 12 2011 at 0:59

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