• Source: Neat submanifold

In differential topology, an area of mathematics, a neat submanifold of a manifold with boundary is a kind of "well-behaved" submanifold.
To define this more precisely, first let




M


{\displaystyle M}

be a manifold with boundary, and




A


{\displaystyle A}

be a submanifold of



M


{\displaystyle M}

.
Then



A


{\displaystyle A}

is said to be a neat submanifold of



M


{\displaystyle M}

if it meets the following two conditions:

The boundary of



A


{\displaystyle A}

is a subset of the boundary of



M


{\displaystyle M}

. That is,



∂
A
⊂
∂
M


{\displaystyle \partial A\subset \partial M}

.
Each point of



A


{\displaystyle A}

has a neighborhood within which



A


{\displaystyle A}

's embedding in



M


{\displaystyle M}

is equivalent to the embedding of a hyperplane in a higher-dimensional Euclidean space.
More formally,



A


{\displaystyle A}

must be covered by charts



(
U
,
ϕ
)


{\displaystyle (U,\phi )}

of



M


{\displaystyle M}

such that



A
∩
U
=

ϕ

−
1


(


R


m


)


{\displaystyle A\cap U=\phi ^{-1}(\mathbb {R} ^{m})}

where



m


{\displaystyle m}

is the dimension of



A


{\displaystyle A}

. For instance, in the category of smooth manifolds, this means that the embedding of



A


{\displaystyle A}

must also be smooth.




See also


Local flatness




References

Kata Kunci Pencarian:

• Neat submanifold
• Submanifold
• Glossary of differential geometry and topology
• Local flatness
• Non-Euclidean geometry
• Noether's theorem
• Quasitoric manifold