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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

$$ f_\ast = ( \psi^{-1}\circ \psi \circ f \circ \phi^{-1} \circ \phi){\ast} phi)_{\ast} = (\psi^{-1})\ast \psi^{-1})_\ast \circ (\psi \circ f \circ \phi^{-1})\ast phi^{-1})_\ast \circ \phi\ast phi_\ast $$

As $\phi$ and $\psi$ are diffeomorphisms, $\phi_\ast$ and $(\psi^{-1})\ast$ (\psi^{-1})_\ast$ are linear isomorphisms.

Therefore, if $(\psi \circ f \circ \phi^{-1})\ast$ 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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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 an 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

$$f_\ast $ f_\ast = ( \psi^{-1}\circ \psi \circ f \circ \phi^{-1} \circ \phi){\ast} = (\psi^{-1})\ast \circ (\psi \circ f \circ \phi^{-1})\ast \circ \phi\ast $$

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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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 an 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.