Embedded manifold

It is a type of submanifold.
It is an immersed manifold but now we require the map $f$ to be globally injective and it is 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 refers, 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 coincides 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.

Alternative definition

@baez1994gauge. I guess it could be shown to be equivalent to the previous one.
Given a subset $S$ of an $n$-manifold $M$, we say that $S$ is a $k$-dimensional submanifold of $M$ if, for each point $p\in S$, there exists an open set $U\subset M$ and a chart $\phi :U\to {\mathbb{R}}^n$ such that $S\cap U={\phi }^{-1}({\mathbb{R}}^k)$.