Let $M$ be a manifold and $V$ be an oriented vector bundle. It's well known that if the Euler class of $V$ is non zero, then $V$ can't have a non-vanishing section. The converse is not true, see [Vanishing of Euler class](https://mathoverflow.net/questions/31376/vanishing-of-euler-class) , or the answer by John Klein below, given to the first version of this question. So:


**Question.** Recall that the normal bundle of a smooth orieantable submanifold in $\mathbb R^n$ always has zero Euler class. Does such a normal bundle always has a non-vanishing section? If yes, how to prove this? If no, what is a counterexample? 


**PS.** It turns out that this question is a subquestion of the following much more informed one: 
[Embeddings without nonvanishing normal vector fields](https://mathoverflow.net/questions/80758/embeddings-without-nonvanishing-normal-vector-fields)



Can one disjoin any submanifold in $\mathbb R^n$ from itself by a $C^{\infty}$-small isotopy?
Here's a counter example. Take any embedding of $\mathbb{C} P^2$ in $\mathbb{R}^7$ (such embeddings exist by 


*Steer, B.*, [**On the embedding of projective spaces in Euclidean space**](http://dx.doi.org/10.1112/plms/s3-21.3.489), Proc. Lond. Math. Soc., III. Ser. 21, 489-501 (1970). [ZBL0206.25501](https://zbmath.org/?q=an:0206.25501);


the construction is summarised [in this answer on MSE](https://math.stackexchange.com/a/1927022/31046)).


If such an embedding admits a normal vector field, then by the [Compression Theorem of Rourke and Sanderson](https://arxiv.org/abs/math/0301356) it is isotopic to an embedding with normal field parallel to the last coordinate of $\mathbb{R}^7=\mathbb{R}^6\times\mathbb{R}$, and then the projection is an immersion $\mathbb{C} P^2\looparrowright \mathbb{R}^6$. Such immersions cannot exist, by András Szűcs' answer [here](https://mathoverflow.net/a/136924/8103).


