Smooth manifold with corners - Imported Knowlpedia Content

Definition

An $n$ -dimensional smooth manifold with corners is a Hausdorff, second-countable space $M$ with a maximal compatible atlas of charts onto open subsets of

\mathbb R_k^n=[0,\infty)^k\times\mathbb R^{n-k},\qquad 0\leq k\leq n.

Two corner charts are compatible when each transition map and its inverse locally extend to smooth maps between open subsets of $\mathbb R^n$ . The integer $k$ may vary between charts; at a particular point, only the number of vanishing boundary coordinates is intrinsic. Ordinary smooth manifolds use $k=0$ , while manifolds with boundary require at most one vanishing boundary coordinate locally.

Depth strata

The number of vanishing boundary coordinates at a point is independent of the chart and is its depth. Points of depth $r$ form a smooth $(n-r)$ -dimensional stratum without boundary. This canonical stratification distinguishes the interior, open boundary faces, and higher-codimension corners.

Products and examples

Products of manifolds with corners inherit product corner charts, and depths add:

\operatorname{depth}_{M\times N}(x,y) =\operatorname{depth}_M(x)+\operatorname{depth}_N(y).

The cube $[0,1]^n$ , a product of compact intervals, is the basic example. A closed disk is a manifold with boundary and hence also a manifold with corners, but its boundary points all have depth one and it has no higher-depth points.

Conventions and scope

References

Dominic Joyce, “On Manifolds with Corners,” final preprint version, 2010. arXiv record. Relevant: §2, Definition 2.1 and the comparison of conventions; §3, smooth maps.