Definition. Immersion, embedding, and submanifold [math-000Z]

Let $M, N$ be differentiable manifolds (dimensions are $m$ and $n$ respectively). A differentiable map $\varphi : M \to N$ is said to be an immersion if

is injective for all $p \in M$.

If in addition, $\varphi$ is a homeomorphism onto $\varphi(M) \subset N$, where $\varphi(M)$ has the subspace topology induced from $N$, then $\varphi$ is an embedding.

If $M \subset N$ and the inclusion $M \subset N$ is an embedding, then $M$ is a submanifold of $N$.