I will start by laying out the context.

Let $\pi : E\to M$ be a principal $G$-bundle. For subgroup $H$ of $G$ a "reduction of structure group" is defined to be a section of the $G/H$-fibre bundle $E_H := E \times_{G} G/H$. This is the same as giving a $G$ equivariant map from $E \to G/H$.

Consider now $M$ to be a compact orientable surface and $E$ to be a $G$-bundle with flat connection. A metric is defined to be reduction of structure group to the maximal compact subgroup $H$ of $G$.

Then one defines a energy operator as

$E(h) = \int (|Dh|^2.vol)$ where h is the metric seen as a section of $E_H \to M$

Now my question is :

What is the meaning of "norm $Dh$"?

Further one defines a harmonic metric as a critical point of the energy functional.

I don't understand what critical point means here?

I'll be really grateful if someone can explain with possibly a toy example also.

All involved spaces are smooth manifolds and groups are Lie groups



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