Let $N \to X$ be normal bundle of a submanifold $X$ of $Y$.
How can I prove that $TN|_{TX}$ is isomorphic to the normal bundle of the inclusion $TX\to TY$? And why this vector bundle is isomorphic to the pullback of $N\oplus N$ by the projection $TX \to X$?
To give a little background, this question came up when I was reading this paper:
The first statement is just a consequence of the tubular neighborhood theorem, I believe. If $ h : N_e \to Y $ is a diffeomorphism of the open disk-bundle of $N$ (radius $e$ disks) to a neighborhood of $X$ in $Y$, with $h|X = id_X$, i.e., the thing whose existence is guaranteed by the tubular neighborhood theorem, then $dH : TN_e \to TY$ is a nice bundle map whose restriction to $TX$ is what you need.
I suspect the second part is similar, but I confess I can't work out the details just now.