# Confusion regarding definition of energy operator/harmonic metric

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