An n-manifold with corners is topologically an n-manifold with boundary, but with a smooth structure that makes it locally diffeomorphic to [0,∞)n instead of [0,∞)×Rn−1. See also:

- J. Lee, Introduction to Smooth Manifolds (Chapter 16, Integration on Manifolds)
- D. Joyce, On manifolds with corners (http://arxiv.org/abs/0910.3518)
- http://ncatlab.org/nlab/show/manifold+with+boundary
The filled cube is naturally a 3-manifold with corners, with every vertex having a neighborhood that is diffeomorphic to a subset of [0,∞)3. It does not seem possible to me that we can map a neighborhood of the top vertex of the filled square pyramid to [0,∞)3 since it has four adjacent faces instead of three.

Can we conclude that the filled square pyramid is not a manifold with corners? If yes, what is the point in excluding this seemingly useful structure by such a strict definition?

**Answer**

If there were a diffeomorphism taking (a neighborhood of) the top vertex of the pyramid to a (neighborhood of) the vertex in [0,∞)3, taking its derivative it would in particular preserve the tangent cone (the set of tangent vectors v∈TpM such that v=γ′(0) for some curve γ:[0,1]→M with γ(0) the vertex). If, for convenience, we set the top point of the pyramid to be 0∈R3, the tangent cone of the top vertex of the pyramid is the set of scalar multiples of points in the pyramid; the tangent cone of the vertex of the cube is [0,∞)3. So we’ve reduced this to a linear algebra problem: showing a linear isomorphism cannot take the first cone to the second. But, as you predicted, a linear isomorphism preserves the number of faces of the tangent cone. (I do not want to write down the details, so I leave this to you.)

As for why we don’t want to include the pyramid: I haven’t ever used manifolds with corners much myself, so I can only speculate, but I suspect it’s because much of the interest is in a) developing a theory that works with simplices, so that we can integrate along chains; and b) developing a theory that will allow us to define cobordisms between manifolds with boundaries. Such cobordisms would, near the ‘boundary’, look like M×I, where M is a manifold with boundary; this gives us the local model [0,∞)2×Rn−1. Now we’re going to want cobordisms between these, etc, inductively saying we should want to include all local models [0,∞)k×Rn−k. In doing so we never really needed to have corners like the top of a square pyramid.

**Attribution***Source : Link , Question Author : Friedrich , Answer Author : Community*