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.