Suppose $M$ is a smooth manifold. I am intersted in the different types of data I can use to specify a smooth submanifold $N\subseteq M$.
For example, if $f:M\to \mathbb{R}^n$ is a smooth map, then under some conditions on $f$ we can define $N=f^{-1}(0)$ with $\dim N=\dim M-n$.
Alternatively, if $N_0$ is a smooth manifold and $\varphi: N_0\to M$ is a smooth map, then under some conditions on $\varphi$ we can define $N=\varphi(N_0)$, with $N$ diffeomorphic to $N_0$.
Finally, we can use coordinate charts on $M$ in which $N$ lies along a canonical $\mathbb{R}^{\dim N}\subseteq \mathbb{R}^{\dim M}$.
Are there any other general ways of specifying a submanifold? I'm mainly interested in local definitions. For example, defining a curve in $M$ as a flow line of a vector field from a given point isn't local because a local deformation of the vector field will lead to a global change in the curve.