Let $X$ be a real $n$ dimensional manifold. One knows that it can be embedded into $\mathbb{R}^{2n}$ by the Whitney embedding theorem. The normal bundle for such an embedding will be a rank $n$ real vector bundle. I would be interested in understanding the connection between this embedding, its normal bundle and the rank $n$ bundles on $X$. 1. How does the normal bundle change under the isotopy of this embedding? 2. Is it possible to obtain all the isomorphism classes of rank $n$ vector bundles by such isotopies? 3. If the answer for 2. is negative, is it still possible to find an embedding of $X$ into $\mathbb{R}^{2n}$ for each vector bundle of rank $n$, such that the corresponding normal bundle is isomorphic to it? I was thinking that the answer should have something to do with the self-intersection of $X$ in the vector bundles and its self-intersection in $\mathbb{R}^{2n}$, but don't really see how to use that.