I'm reading a lecture note (in French). They define the following:
Let $S$ be a subset of $\mathbb R^n$ and $d \in \mathbb N$. $S$ is a sub-variety of class $\mathcal C^1$ and of dimension $d$ if for every point $a \in S$, there exist an open set $U \subset \mathbb R^n$ containing $a$ and a function $\mathcal C^1$-diffeomorphism $\phi: U \to \phi(U)$ such that: $\phi (S \cap U) = E \cap \phi(U)$, where $E$ is a sub vector space of $R^n$, of dimension $d$.
Could anyone explain the motivation (or intuition) of this definition? Why a sub-variety of $\mathbb R^n$ is a local diffeomorphism to a sub vector space?