It is a type of submanifold.
It is an immersed manifold but now we require the map $f$ to be globally injective and a homeomorphism between $M$ and $f(M)$ (with the relative topology). It is also said that $f$ is an embedding.
The image of $f$ is called regular submanifold.
If $M$ is compact then we only have to require to be an injective immersion.
Very surprisingly, there are two important results relative to embeddings: the Whitney embedding theorem and the Nash embedding theorem. The latter refer specifically to isometric embeddings in $\mathbb R^N$, i.e., the original manifold $M$ is equipped with a Riemannian metric $g$ and we require that it concides with the pullback of the standard metric of $\mathbb R^N$.
Another less know result is the Cartan-Janet theorem.
A typical way to construct regular submanifolds is by mean of the preimage theorem.
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Author of the notes: Antonio J. Pan-Collantes
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