Existence of a smooth map to its embedded submanifold

Question

Assume $M$ is a manifold, and $N$ is an embedded submanifold of $M$. Is there always a smooth map $f$ from $M$ to $N$, s.t. $f|_N=\text{id}_N$?

Answer 1

There does not exist even a continuous map $f$ from $M = \mathbb S^2$ to it's equator $N$ such that $f|_N = id$: If $f$ is such a map, then

$$f_* : \{0\} = \pi _1(M) \to \pi_1(N)$$

would sends $[N] = 0\in \pi_1(M)$ to the generator of $\pi_1(N)$.