What is a regular submanifold?

Question

Definition: Let $M$ be a non-empty subset of $\mathbb{R}^n$ and consider $m\in\mathbb{N}$ such that $1\leqslant m\leqslant n$. M is named a regular sub manifold of $\mathbb{R}^n$ of dimension $m$ if it verifies the following condition: For each point $p$ of $M$ there is an open set $V$ of $\mathbb{R}^n$ and a function $F:V\to\mathbb{R}^{n-m},x\to(F_1(x),...,F_{n-m}(x))$ of $\mathscr{C}^{\infty}$ such that:

(i) $F(p)=0$

(ii) $F^{-1}(0)\cap M$ is an open set of $M$.

(iii) $F$ is a submersion in each point $x\in F^{-1}(0)\cap M$, such as $rank(J(F,x))=n-m\:\:\:\:\forall x\in F^{-1}(0)\cap M$

J stands for the jacobian matrix

I was presented this definition of a regular sub manifold in an real analysis course. It has been difficult to get an intuition of what is sub manifold, since I cannot picture it. I have been looking in real analysis books for this definition in a hope that it could be explained in a more intuitive way.

Question:

Can someone give me an intuitive explanation of what a sub manifold is? I would appreciate if someone could give a source of where I can find geometric pictures of this definition. The teacher did not define manifold in the material it has provided.

Thanks in advance!

Answer 1

A regular submanifold of a manifold $N$ is commonly defined as the image of an immersion $f: M \to N$ (i.e. the induced map $T_pM \to T_{f(p)}N$ on tangent spaces is injective for all $p \in M$) whose topology is compatible with the subspace topology in $N$; i.e. $f$ is a diffeomorphism of $M$ onto its image. So although $M$ may be defined intrinsically, we may think of it as lying inside $N$ without losing any information. For example, the Whitney embedding theorem says that every manifold is a regular submanifold of some $\mathbb{R}^n$.