Normal bundle in tangent bundle Let's consider the normal bundle $NM$ of zero section in $TM$. Is it true that $NM \cong TM$?
There is exact sequence $$0 \rightarrow TM \rightarrow TE|_M \rightarrow NM \rightarrow 0$$ for the normal bundle, but if $E=TM$... it's not exact.
 A: Actually a  much more general result is true:  given a manifold $M$ and an arbitrary vector bundle $E\stackrel {\pi}{\to} M$ on $M$, the normal vector bundle $N_E(M)$   of $M$ (identified with the zero section of $E$) in $E$ is isomorphic to $E$.
 In other words we have a direct sum decomposition of vector bundles  on $M$: 
$$T(E)|M =T(M)\oplus   E                     $$ 
The key to understanding this is to consider the case where $M$ is reduced to a point: it is then the result which says that the tangent  space at the origin $T_0E$ of a vector space $E$      can be identified with the vector space $E$ itself.
Indeed, if  tangent vectors are defined  (say) as derivations, then  the vector $v\in E$ is identified with  the derivation $\partial _v|_0$ which sends the function $f$ to its directional derivative $$\partial _vf(0)=\lim_{t\to 0} \frac {f(tv)-f(0)}{t}$$ 
Edit
It might be of interest to notice that associated to any vector bundle $E\stackrel {\pi}{\to} M$  we have a canonical  exact sequence of bundles on $\textbf E$ : $$0\to\pi^*(E)\to T(E)\to \pi^*(T(M))\to 0  $$ The vector bundle $\pi^*(E)=:T_{vert}(E)$ is the subvector bundle of $T(E)$ consisting of vectors tangent to the fibers of $\pi$.
Restricting this exact sequence to the zero section of the bundle identified with $M$ we get the canonical exact sequence of bundles on $\textbf M$: $$0\to E\to T(E)\vert M\to T(M)\to 0$$ 
This exact sequence can non canonically  be split and we obtain the already mentioned decomposition $T(E)|M =T(M)\oplus E$ .
A: The dimension count argument I gave is wrong. If $Z:M\hookrightarrow TM$, then $NM = Z^*(TTM)/TM$ does indeed have the right dimension. The intuition is wrong because the directions in $TM$ "perpendicular" to $M$ are exactly the tangent directions.
