# Realization of the $2$-form as the curvature of some bundle.

Suppose $$M$$ be a manifold, and $$\omega$$ be any smooth $$2$$-form. My question is about the existence of the Lie group $$G$$ which satisfies that there is always a principal $$G$$-bundles $$P$$(or its associated bundle with fixed fiber $$V$$) over $$M$$ such that its curvature $$2$$-form is given by $$a \otimes \omega$$ for some $$a$$ in $$\mathfrak{g}$$(or $$V$$) for any $$\omega \in \Omega^2(M)$$.

The similar result for integral cohomology class is well known. If we let $$G$$ be $$U(1)$$, then it's always possible($$a=2\pi i$$), by using good cover arguement and Cech's analysis on the covering. However, this couldn't be done for real cohomology class, since this is an equivalent condition for the existence.

We couldn't extend this theorem to the real coefficient case, even though there is the universal coefficient theorem and $$\mathbb{R}$$ has no torsion, because $$\omega$$ will not be recovered by just multiplyting the real number to some integral class.

So, I think it's natural to find such $$G$$, because only the integral cohomology class is possible to realize as the curvature is a little restrictive in some sense.