Let $M$ and $N$ be two smooth finite dimensional manifolds and $C^\infty(M)$ as well as $C^\infty(N)$ their smooth function algebras.

Is the following true:

Let $\imath: M \to N$ be an embedding. Then the algebra morphism $\imath^* : C^\infty (N) \to C^\infty (M)$ defined by $\imath^*(f)(m) = f(\imath(m))$ for all $m \in M$ and $f \in C^\infty(N)$ is surjective.

Let $\pi: N \to M$ be a surjective submersion. Then the algebra morphism $\pi^* : C^\infty (M) \to C^\infty (N)$ defined by $\pi^*(f)(n) = f(\pi(n))$ for all $n \in N$ and $f \in C^\infty(M)$ is injective.