# Equivalences of definitions of submanifolds in $\mathbb{R^n}$

A subset $$M \in \mathbb{R}^n$$ is called a smooth k-dimensional submanifold of $$\mathbb{R}^n$$, $$k \leq n$$, if any point $$x \in M$$ has a neighborhood $$O_x$$ in $$\mathbb{R}^n$$ in which $$M$$ is described in one of the following ways:

• (1) there exists a smooth vector-function $$F:O_x \to \mathbb{R}^{n-k}, \text{ } rank \frac{dF}{dx}|_q=n-k$$ such that $$O_x \cap M = F^{-1}(0)$$

• (2) there exists a smooth vector-function $$f:V_0 \to \mathbb{R}^{n}$$ from a neighborhood of the origin $$0 \in V_0 \subset \mathbb{R}^k$$ with $$f(0)= x \text{ and } rank \frac{df}{dx}|_0=k$$ such that $$f: V_0 \to O_x \cap M$$ is homeomorphism.

• (3) there exists a smooth vector-function $$\Phi:O_x \to O_0 \subset \mathbb{R}^{n},$$ onto a neighborhood of the origin $$0 \in O_0 \subset \mathbb{R}^n$$ with $$\text{ } rank \frac{d \Phi}{dx}|_x=n$$ such that $$\Phi(O_x \cap M)= \mathbb{R}^k \cap O_0$$

I am trying to show the equivalence of these definitions and my first question is why is each definition imposing itself on the rank?