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:


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



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> 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?
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**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 $S^3$?



A vector bundle associated to a codimension $1$ submanifold of a symplectic manifold
We have to choose a compatible Riemannian metric $g$ to define the normal bundle $N$. Then the almost complex structure $J$ defined by $\omega$ and $g$ induces an isomorphism between $E$ and $TM$.


As for your second question, assume that $M=f^{-1}(0)$ for some function $f$ that has $0$ as a regular value. Then $E\cap TM$ is perpendicular to the Hamiltonian vector field of $f$ acting on $M$.
Let $\alpha\in\Omega^1(M)$ be a (local) defining one-form, so $\ker\alpha=E\cap TM$. If $E\cap TM$ is integrable, we can find $\alpha$ such that $d\alpha=0$. Equivalently, we demand $\alpha\wedge d\alpha=0$, which is independent of the choice of $\alpha$.


On the other hand, if $\alpha\wedge(d\alpha)^{n-1}$ is a volume form (nonzero everywhere), then $E\cap TM$ is called a contact structure. Again, this condition does not depend on the choice of $\alpha$. This includes the case of the standard sphere $S^{2n-1}\subset\mathbb R^{2n}$.


