# Frobenius Theorem; Slices

Good Night. I am studying the Frobenius theorem. I'm reading the book Foundations of differentiable manifolds and Lie Groups; Frank Warner. In the first third part of the statement is written, "is a slice S $Y_ {1} = 0$", where $Y_ {1}$ is a function of a coordinated system of coordinates. I do not understand this! Also, I do not understand the phrase "The subspace S of M with the coordinate system {$x_ {i} | S: j = 1, ..., c\}$" which is said in the definition of Slice.

(Warner definition) 1.34 Slices Suppose that $(U,\varphi)$ is a coordinate system on $M$ with coordinate functions $x_{1} ..... x_{d}$, and that $c$ is an integer, $0\leq c \leq d$. Let $a\in \varphi(U)$, and let $S =\{q\in U:x_{i}(q)=r_{i}(a), i = c +i .... ,d\}$. The subspace $S$ of $M$ together with the coordinate system $\{x_{j}\mid S: j= 1,..., d\}$ forms a manifold which is a submanifod of $M$ called a slice of the coordinate system $(U, \varphi)$.

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You start with a chart $(U, \varphi)$ and a point in $a \in U$. The subspace $S$ is simply the subset of $U$ of the elements that have some particular coordinates in common with $a$.
Let's see an example in $\mathbb{R}^2$ (with the standard manifold structures). Let's be $U = \{(u,v)\in R^2\mid -1\le u,v\le 1\}$ and $a = (0,0)$. The slice with $x_1 = 0$ is simply the line segment between the points $(0,-1)$ and $(0,1)$.
EDIT 1: “Let $S$ be the slice $y_1 = 0$” means that $S$ is the subset of $V$ of points that have the first coordinate equal to $0$. In the Warner's definition, the points of a slice have the last $d-c$ coordinates equal. Nevertheless, you can choose them: you simply need to permute the basis and use it as a new chart. The second part of the definition simply said that if you fix $d-c$ coordinates you do not really need them and so you can ignore them. The slice depends only on the other $c$ coordinates. In my previous example, for example, the line segment depends merely on the value on the $y$ and it is in fact locally diffeomorphic to the real line.
P.S: $\{y_i|S \colon j = 1,\dots,n\}$ literally means the set of the restrictions on $S$ of the first $c$ coordinates functions. However, you can read it as the set of the restrictions of the coordinates that are not equal in all the elements of the slice.
Hello Vittorio. Thanks for your reply. When Warner Book "says": "Let $S$ be the slice $y_{1}=0$". How to see that according to Definition 1.34 (Warner's book), given in my question. Another thing, which means $\{x_{j}| S: j=1,...,c\}$ in the definition. thank you. –  Manoel Jun 2 '12 at 0:05