321: Chart on Regular Submanifold Is Extension of Adapting Chart|T.B.P.

<The previous article in this series | The table of contents of this series | The next article in this series>

A description/proof of that chart on regular submanifold is extension of adapting chart

Topics

About: $C^{\infty}$ manifold

Starting Context

The reader knows a definition of $C^{\infty}$ manifold.
The reader knows a definition of regular submanifold.
The reader admits the proposition that for any map between $C^{\infty}$ manifolds, its continuousness in the topological sense equals its continuousness in the norm sense for the coordinates functions.
The reader admits the invariance of domain theorem: any injective continuous map from any open set on any Euclidean topological space to any Euclidean topological space is an open map.

Target Context

The reader will have a description and a proof of the proposition that any chart on any regular submanifold of any $C^{\infty}$ manifold is an extension of an adapting chart.

Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

Main Body

1: Description

For any

C^{\infty}

manifold, M, of dimension

n

, and any regular submanifold,

M_{1} \subseteq M

, of dimension

n_{1}

, any chart around any point,

p \in M_{1}

, on

M_{1}

(U_{p}, ϕ)

where

U_{p} \subseteq M_{1}

, is an extension of an adopting chart,

(U_{p}^{'} \cap M_{1}, π \circ ϕ^{'} |_{U_{p}^{'} \cap M_{1}})

where

(U_{p}^{'}, ϕ^{'})

is the adopted chart such that

U_{p}^{'} \subseteq M

and

π

is the vanishing of the always '0',

n_{1} + 1

n

coordinates.

2: Proof

Around

p

, there is an adopting chart,

(U_{p}^{″} \cap M_{1}, π \circ ϕ^{″} |_{U_{p}^{″} \cap M_{1}})

where

(U_{p}^{″}, ϕ^{″})

is the adopted chart such that

U_{p}^{″} \subseteq M

. As

U_{p}

is open in the subspace topology,

U_{p} = U_{p}^{‴} \cap M_{1}

where

U_{p}^{‴} \subseteq M

is open on

M

. So,

U_{p} \cap U_{p}^{″} \cap M_{1} = U_{p}^{‴} \cap M_{1} \cap U_{p}^{″} \cap M_{1} = U_{p}^{″} \cap U_{p}^{‴} \cap M_{1}

, and

(U_{p}^{″} \cap U_{p}^{‴} \cap M_{1}, π \circ ϕ^{″} |_{U_{p}^{″} \cap U_{p}^{‴} \cap M_{1}})

is an adopting chart with

(U_{p}^{″} \cap U_{p}^{‴}, ϕ^{″} |_{U_{p}^{″} \cap U_{p}^{‴}})

as the adopted chart. On the other hand,

(U_{p} \cap U_{p}^{″} \cap M_{1} = U_{p}^{″} \cap U_{p}^{‴} \cap M_{1}, ϕ |_{U_{p}^{″} \cap U_{p}^{‴} \cap M_{1}})

is a chart on

M_{1}

.

There is a diffeomorphism

f : π \circ ϕ^{″} (U_{p}^{″} \cap U_{p}^{‴} \cap M_{1}) \to ϕ (U_{p}^{″} \cap U_{p}^{‴} \cap M_{1})

, because the 2 charts belong to the same atlas, and we define an injective map,

f^{'} : ϕ^{″} (U_{p}^{″} \cap U_{p}^{‴}) \to R^{n}

, such that the 1 ~

n_{1}

components are mapped by

f

and the remaining components are mapped identically. By the proposition that for any map between

C^{\infty}

manifolds, its continuousness in the topological sense equals its continuousness in the norm sense for the coordinates functions,

f^{'}

is continuous, and by the invariance of domain theorem: any injective continuous map from any open set on any Euclidean topological space to any Euclidean topological space is an open map,

f^{'}

is an open map, so,

f^{'} (ϕ^{″} (U_{p}^{″} \cap U_{p}^{‴}))

is open, and

f^{'} : ϕ^{″} (U_{p}^{″} \cap U_{p}^{‴}) \to f^{'} (ϕ^{″} (U_{p}^{″} \cap U_{p}^{‴}))

is a homeomorphism, in fact, obviously, a diffeomorphism. So,

(U_{p}^{″} \cap U_{p}^{‴}, f^{'} \circ ϕ^{″})

is a chart, and

f^{'} \circ ϕ^{″} (U_{p}^{″} \cap U_{p}^{‴} \cap M_{1})

equals

f^{'} \circ ϕ^{″} (U_{p}^{″} \cap U_{p}^{‴})

whose

n_{1} + 1

n

components vanished, because any point on

f^{'} \circ ϕ^{″} (U_{p}^{″} \cap U_{p}^{‴})

whose one of the

n_{1} + 1

n

coordinates is not 0 is not on

f^{'} \circ ϕ^{″} (U_{p}^{″} \cap U_{p}^{‴} \cap M_{1})

and any point on

f^{'} \circ ϕ^{″} (U_{p}^{″} \cap U_{p}^{‴})

whose all the

n_{1} + 1

n

coordinates are 0 is on

f^{'} \circ ϕ^{″} (U_{p}^{″} \cap U_{p}^{‴} \cap M_{1})

. So,

(U_{p}^{″} \cap U_{p}^{‴} \cap M_{1}, π \circ f^{'} \circ ϕ^{″} |_{U_{p}^{″} \cap U_{p}^{‴} \cap M_{1}})

is the adopting char of the adopted chart,

(U_{p}^{″} \cap U_{p}^{‴}, f^{'} \circ ϕ^{″} |_{U_{p}^{″} \cap U_{p}^{‴}}

, but

π \circ f^{'} \circ ϕ^{″} |_{U_{p}^{″} \cap U_{p}^{‴} \cap M_{1}} = ϕ |_{U_{p}^{″} \cap U_{p}^{‴} \cap M_{1}} = ϕ |_{U_{p} \cap U_{p}^{″} \cap M_{1}}

, of which the chart,

(U_{p}, ϕ)

, is an extension.

References

The Bias Planet
Definitions and Propositions
School Mathematics from Higher Viewpoints
To Disentangle Confusing Terms or Discourses
Information Tables
Exploiting LibreOffice or OpenOffice
Exposing the Java Programming Language
Exposing C++
Exposing Python
Exposing C#
Exposing Gradle
Exposing Git
UNO Dispatch Commands

2022-07-24