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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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up vote 4 down vote accepted

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.

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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
I edited the answer because the explanation was too long for a simple comment. – Vittorio Patriarca Jun 2 '12 at 1:39

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