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Let $E \rightarrow F$ be a principal $G$-bundle. Let $\alpha$ be a connection 1-form with values in the Lie algebra of $G$. Let $\omega$ denote the curvature 2-form of connection $\alpha$.

We know that $\omega =0$ everywhere i.e., $\alpha$ is a flat connection if and only if the distribution $\ker (\alpha)=\{v_x \in T_x E | \alpha(v_x)=0 \ ; x\in E \} $ on $E$ is completely integrable.

Now, suppose we have a norm on the space of 2-forms. We start with an "almost flat connection $\alpha$ " i.e., the curvature form $|\omega| < \epsilon$ everywhere, for sufficiently small $\epsilon$. Is it true that the distribution $\ker(\alpha)$ on $E$ is "close" to a completely integrable distribution? and "close" in what sense?

I have a feeling that with appropriate notion of "closeness" of distributions the above question has an affirmative answer. While I am trying to show that it is so, I have a difficulty deducing any useful information about connection 1-form from bounds on curvature. Thanks in advance for bringing in any new insight.

I have posted this question on MO also. http://mathoverflow.net/questions/68794/almost-flat-connections-on-principal-g-bundles

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Note, the OP has also cross-posted the question to MO. –  Willie Wong Jun 25 '11 at 18:23

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