# Locally trivial line bundle

I am reading the book "An Introduction to Contact Topology" by Geiges.

In the proof of Lemma 1.1 where he proves that any co-dimension 1 hyperplane field distribution is locally kernel of a 1-form, he obtains the line bundle $$TM/\xi=\xi^\perp$$ and states that $$\xi^\perp$$ is locally trivial. I can not see why this is true, why is $$\xi^\perp$$ locally trivial? Thank you for any help.

Here $$M$$ is the manifold, $$TM$$ its tangent bundle, $$\xi$$ is the hyperplane field decomposition and $$\xi^\perp$$ is the orthogonal complement of $$\xi$$ with respect to a Riemannian metric on M.

Let $$x\in M$$ you can construct a vector field $$X$$ on a neighborhood $$U$$ of $$x$$ such that $$X(x)$$ is not $$\xi_x$$ (suppose that $$U$$ is contractible, $$TM_{\mid U}$$ is trivial, take any $$u\in T_xM$$ which is not in $$\xi_x$$, $$y\rightarrow(y,v)$$ is $$X$$). This implies that there exists a neighborhood $$V\subset U$$ such that $$X(y)$$ is not an element of $$\xi_y$$. There exists an isomorphism $$V\times\mathbb{R}\rightarrow (TM/\xi)_V$$ defined by $$(y,u)\rightarrow p(uX(y))$$ where $$p:TM\rightarrow TM/\xi$$ is the quotient map.
The results follows from the fact that $$TM/\xi$$ is isomorphic to $$\xi^{\perp}$$.