Manifold with corners

Summary

A smooth manifold with corners is a space modelled near each point on an open subset of the closed orthant $[0, \infty)^n$. Its changes of chart are smooth. It allows boundary points where several boundary faces meet at an angle, like the vertices of a square or a cube. Products of manifolds with boundary are manifolds with corners. This is one reason the notion is needed, for example in cobordism theory and in Stokes' theorem on cubes.