In the book **Geometric theory of foliations** by Camacho and Neto, the following question is posed:

- Let $G$ be a smooth transversaly orientable foliation. Let $F$ be a compact leaf of $G$. Prove that $F$ has tubular neighborhoord diffeomorphic to a product. Hint: Prove that the normal fibration of $F$ is diffeomorphic to a product.

I tried to prove that $F$ has trivial normal bundle. But it is possible to ensure only that the submanifold $F$ possesses an atlas with transition functions $g_{ij}$ locally constant. There are nontrivial vector bundles with flat connections.

Is the exercise false?