= Solution
An embedded smooth $n$-dimensional <manifold> $X\subseteq\mathbb R^N$ has, around every point, a parametrization by an open subset of $\mathbb R^n$ which is a homeomorphism onto a relative neighbourhood, has rank $n$, and has smooth coordinate changes. Equivalently, after smooth ambient coordinates it is locally a coordinate $n$-plane. For a <manifold with boundary> use relatively open subsets of $\mathbb H^n=\{u_n\geq0\}$ instead, with parametrizations and transitions smoothly extendible across the boundary. Points corresponding to $u_n=0$ form the boundary. Interior charts are ordinary manifold charts; a boundary point is not a corner.
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