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 the category of diffeological spaces

Proposition

(manifolds with boundaries and corners form full subcategory of diffeological spaces)

The evident functor

SmthMfdWBdrCrnAAAADiffeologicalSpaces SmthMfdWBdrCrn \overset{\phantom{AAAA}}{\hookrightarrow} DiffeologicalSpaces

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

(Iglesias-Zemmour 13, 4.16, Gürer & Iglesias-Zemmour 19)

References

The full subcategory-embedding of manifolds with boundaries and corners into that of diffeological spaces is discussed in:

Last revised on September 11, 2019 at 09:57:29. See the history of this page for a list of all contributions to it.