In Folland's Introduction to Partial Differential Equations:
A subset $S$ of ${\mathbb R}^n$ is called a hypersurface of class $C^k$($1\leq k\leq\infty$) if for every $x_0\in S$ there is an open set $V\subset{\mathbb R}^n$ containing $x_0$ and a real-valued function $\phi\in C^k(V)$ such that $\nabla\phi$ is nonvanishing on $S\cap V$ and $ S\cap V=\{x\in V:\phi(x)=0\}.$
In this case, by the implicit function theorem we can solve the equation $\phi(x)=0$ near $x_0$ for some coordinate $x_i$---for convenience, say $i=n$---to obtain $x_n=\psi(x_1,\dots,x_{n-1})$ for some $C^k$ function $\psi$. A neighborhood of $x_0$ in $S$ can then be mapped to a piece of the hyperplane $x_n=0$ by the $C^k$ transformation x\to(x',x_n-\psi(x'))\qquad (x'=(x_1,\dots,x_{n-1})) This same neighborhood can also be represented in parametric form as the image of an open set in ${\mathbb R}^{n-1}$(with coordinate $x'$) under the map x'\to(x',\psi(x')).
Here is my question:
Is the above definition equivalent to say that $S$ is a $C^k$-differentiable manifold?
I learned from S.S. Chern 's Lectures on Differential Geometry the definition as following:
Suppose $M$ is an m-dimensional topological manifold. If a given set of coordinate charts ${\mathcal A} = \{(U,\phi_U),(V,\phi_V),(W,\phi_W),\cdots\}$ on $M$ satisfies the following conditions, then we call ${\mathcal A}$ a $C^r$-differentiable structure on $M$:
1). $\{U,V,W,\cdots\}$ is an open covering of $M$;
2). any two coordinate charts in ${\mathcal A}$ are $C^r$-compatible;
3). ${\mathcal A}$ is maximal, i.e., if a coordinate chart $(\tilde{U},\phi_{\tilde{U}})$ is $C^r$-compatible with all coordinate charts in ${\mathcal A}$, then $(\tilde{U},\phi_{\tilde{U}})\in{\mathcal A}$.
If a $C^r$-differentiable structure is given on $M$, then $M$ is called a $C^r$-differentiable manifold.
1) is not hard to find, 3) can be obtained once one has a covering by compatible charts. I am not able to get 2).
For some particular case, e.g., $S=S^1$ in ${\mathbb R}^{2}$, the answer is yes. However, for arbitrary $S$, I don't know how to find a covering of $S$ by compatible charts.