dg.differential geometry - Is an injective smooth map an immersion?

Suppose $M$ and $N$ are smooth manifolds. An immersion is a smooth map $f: M rightarrow N$ whose pushforward is injective at each point.

Is a smooth injective map an immersion?

We can actually simplify the question further.

Suppose $f : M rightarrow N$ is a smooth injective map. Suppose $(U, phi)$ and $(V, psi)$ are smooth charts for $M$ and $N$ respectively. Fix $p in U$. Then

$$f_ast = ( psi^{-1}circ psi circ f circ phi^{-1} circ phi)_{ast} = (psi^{-1})_ast circ (psi circ f circ phi^{-1})_ast circ phi_ast$$

As $phi$ and $psi$ are diffeomorphisms, $phi_ast$ and $(psi^{-1})_ast$ are linear isomorphisms.

Therefore, if $(psi circ f circ phi^{-1})_ast$ is injective then $f_ast$ is injective.

This shows that if every smooth injective map between open subsets of euclidean space is an immersion, then every smooth injective map between smooth manifolds is an immersion.

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