Let $Y$ be a submanifold of a smooth manifold $X$. Let $NY \rightarrow Y$ denotes the normal bundle of $X$ m, I've read that at a point $y \in Y$, we have $T_y(NY)= T_yY \oplus N_yY$, how can we prove this ?
Is there a generalization of this result to an arbitrary vector bundle, i.e if $\pi: E \rightarrow M$ is a vector bundle with fiber $F$, and $x \in M$, do we have $T_xE = T_{\pi(x)} M \oplus T_x F $?