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Definition Checking II. Manifolds.

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
Hello MHB,
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.