# Integrable subbundle

Let $$D\subset TM$$ is a integrable smooth regular subbundle and $$f_{1},...f_{k}$$ is smooth local frame for $$Ann(D)$$. Why $$\omega=f_{1}\wedge ....\wedge f_{k}$$ is closed form?

It is not necessarily closed. For example, take $$M=\mathbb R^3$$ with coordinates $$(x,y,z)$$, and let $$D$$ be the subbundle spanned everywhere by $$\partial/\partial z$$. We could take $$f_1 = dx$$ and $$f_2 = (z^2 +1) dy$$, and then $$f_1\wedge f_2$$ is not closed.
What is true is that in a neighborhood of each point, it is always possible to find $$1$$-forms $$f_1,\dots,f_k$$ whose wedge product is closed. In fact, it's always possible to have each $$f_i$$ individually closed. This follows from the Frobenius theorem -- in a neighborhood of each point, there are coordinates $$(x^1,\dots,x^n)$$ such that $$D$$ is annihilated by $$dx^1,\dots,dx^k$$.