nLab
manifold with boundary

Contents

Contents

Idea

A manifold is a topological space that is locally isomorphic to a Cartesian space n\mathbb{R}^n.

A manifold with boundary is a topological space that is locally isomorphic either to an n\mathbb{R}^n or to a half-space H n={x n|x n0}H^n = \{ \vec x \in \mathbb{R}^n | x^n \geq 0\}.

A manifold with corners is a topological space that is locally isomorphic to an H i n={x n|x i,x i+1,,x n0}H^n_i = \{ \vec x \in \mathbb{R}^n | x^i , x^{i+1}, \cdots, x^n \geq 0\} for 0in0 \leq i \leq n.

For details see at manifold.

Properties

Collar neighbourhood theorem

Embedding into diffeological spaces

Proposition

(manifolds with boundary form full subcategory of diffeological spaces)

The evident functor

SmthMfdWBdrAAAADiffeologicalSpaces SmthMfdWBdr \overset{\phantom{AAAA}}{\hookrightarrow} DiffeologicalSpaces

from the category of smooth manifolds with boundary to that of diffeological spaces is fully faithful, hence is a full subcategory-embedding.

(Iglesias-Zemmour 13, 4.16)

References

The full subcategory-embedding of manifolds with boundary into that of diffeological spaces is discussed in

Related discussion to the case of manifolds with corners is in

Last revised on June 17, 2019 at 07:24:28. See the history of this page for a list of all contributions to it.