We consider the standard symplectic structure $\omega=\sum dx_i\wedge dy_i$ on $\mathbb{R}^{2n}$. To every codimension $1$ submanifold $M\subset \mathbb{R}^{2n}$ we associate a vector bundle $E$ over $M$ as follows:

$$E=\{(x,v)\in M\times \mathbb{R}^{2n}\mid \omega(v,N_x)=0,\;\;N_x\perp T_x M\}$$

Does the structure of this bundle $E$, or at least its characteristic classes, depend on the way $M$ is embedded in $\mathbb{R}^{2n}$? What can be said about orientability of this bundle? What is the structure of this bundle for the standard odd dimensional spheres?Is it trivial?is it isomorphic to the tangent bundle of sphere?

**Note:** One can extend this question by replacing $\mathbb{R}^{2n}$ with an arbitrary symplectic Riemannian manifold. Moreover the intersection of this bundle $E$ with $TM$ gives us a codimension $1$ distribution of $M$. So it would be interesting to study the dependency (and integrability and the structure of the foliation generated by) of this distribution on the way of embedding of $M$ in the ambient symplectic manifold. As an example: What is this distribution for $S^3$?

Euclideannormal bundle, or the symplectic one? In the first case, you should change the title of your question. In the second, $E$ is just the tangent bundle. $\endgroup$ – Sebastian Goette May 5 at 11:17