regular submanifold I have some problem to understand this .

if this means of regular submanifold is a subset $S$ of a manifold $N$ of dimension $n$is regular submanifold of dimension $k$ if for every $p\in S$ there is a coordinate nieghborhood $(U,\phi)=(U,x^1,x^2,...,x^n)$of $p$ in the atlas of $N$ such that $U \cap S$ is defined by the vanishing of $n-k$ of the coordinate function.

is the $U\cap S=S$?why?



in this example if $V=(-1,0)\times (-1,1)$ then $(V,x,y)$ would be adapted chart relative to $S$?
thank you
 A: The linguistically incorrect formulation of the definition you quoted seems to be the cause of misunderstanding. Here's a definition as stated in Jeffrey M. Lee's book:

A subset $S$ of a smooth $n$-manifold $M$ is called a regular submanifold of dimension $k$ if every point $p \in S$ is in the domain of a chart $(U,\ \phi)$ that has the following regular submanifold property with respect to $S$: $$\phi(U \cap S) = \left( \mathbb{R}^{k} \times\{c\} \right) \;\; \mbox{for some } c\in\mathbb{R}^{n-k}.$$

Usually $c$ is chosen to be $0$, which can always be accomplished by composition with a translation of $\mathbb{R}^{n}$, and so you should be able to see immediately that the definition is equivalent to the one you'd given.
Since a chart (and so its domain) is given locally, it doesn't have to be (an usually it's not) that $U \cap S = S$. Consider the following example of a unit sphere $\mathbb{S}^{n} \subset \mathbb{R}^{n+1}$:
$$U^{\pm}_{i} := \{(a^{1},\ldots,a^{n+1}) \in \mathbb{R}^{n+1} \; \mid \; \pm a^{i} > 0 \}$$
and $$\psi^{\pm}_{i} : U^{\pm}_{i} \longrightarrow \psi^{\pm}_{i}(\mathbb{R}^{n+1}) \; \mbox{ where } \; \psi^{\pm}_{i} :=\left(a^{1},\ldots,a^{i-1},a^{i+1},\ldots,a^{n+1},||a||-1 \right)$$
so that $\psi^{\pm}_{i}(U^{\pm}_{i} \cap \mathbb{S}^{n}) =  \psi^{\pm}_{i}(U) \cap \left(\mathbb{R}^{n} \times \{0\} \right)$, which answers both your first and second question.
