- Thread starter
- #1

- Mar 10, 2012

- 834

I need to get the following definition checked. The is a result of me riding on two different boats at almost the same time. One boat is Spivak's

*Calculus on Manifolds*and the other is Munkres'

*Analysis on Manifolds*.

Let $\pi_k$ denote the projection on the $k$-th component.

A subset $M$ of $\mathbb R^n$ is a $k$-

**manifold**of class $C^r$ if for every point $x\in M$ there is a neighborhood $U\subseteq \mathbb R^n$ of $x$, an open set $V\subseteq \mathbb R^n$, and a $C^r$-diffeomorphism $h:U\to V$ such that $\pi_k(h(x))=0$ and:

either

\begin{equation*}

h(U\cap M)=V\cap (\mathbb R^k \times \{0\})\tag{1}

\end{equation*}

or

\begin{equation*}

h(U\cap M)=V\cap \{x\in \mathbb R^n:x_k\geq 0, x_{k+1}=x_{k+2}=\cdots=0\}\tag{2}

\end{equation*}

The set of all the points $x$ in $M$ which satisfy $(1)$ is called the

**interior**of $M$ and the set of all the points $x$ in $M$ for which $(2)$ is satisfied is called the

**boundary**of $M$ and is denoted by $\partial M$.

Thanks.