322: Regular Submanifold of Regular Submanifold Is Regular Submanifold of Base C^\infty Manifold of Specific Codimension

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A description/proof of that regular submanifold of regular submanifold is regular submanifold of base $C^{\mathrm{\infty }}$ manifold of specific codimension

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Topics

About: $C^{\mathrm{\infty }}$ manifold

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof

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Starting Context

• The reader knows a definition of $C^{\mathrm{\infty }}$ manifold.
• The reader knows a definition of regular submanifold.
• The reader admits the proposition that any chart on any regular submanifold of any $C^{\mathrm{\infty }}$ manifold is an extension of an adapting chart.

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Target Context

• The reader will have a description and a proof of the proposition that any regular submanifold of any regular submanifold of any $C^{\mathrm{\infty }}$ manifold is a regular submanifold of the base manifold, of codimension of the codimension of the child submanifold plus the codimension of the grandchild submanifold with respect to the child submanifold.

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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.

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Main Body

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1: Description

For any $C^{\mathrm{\infty }}$ manifold, $M$, of dimension $n$, any regular submanifold of $M$, $M_1\subseteq M$, of dimension $n_1$, and any regular submanifold of $M_1$, $M_2\subseteq M_1$, of dimension $n_2$, $M_2$ is a regular submanifold of $M$ of codimension of $(n-n_1)+(n_1-n_2)=n-n_2$.

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2: Proof

As $M_2$ is a regular submanifold of $M_1$, around any point, $p\in M_2$, there is an adopted chart on $M_1$, $(U_p^{\prime },{\varphi }^{\prime })$, where $U_p^{\prime }\subseteq M_1$ and $(U_p^{\prime }\cap M_2,{\varphi }^{\prime }{|}_{U_p^{\prime }\cap M_2})$ is the adopting chart on $M_2$.

By the proposition that any chart on any regular submanifold of any $C^{\mathrm{\infty }}$ manifold is an extension of an adapting chart, $(U_p^{\prime },{\varphi }^{\prime })$ is an extension of an adapting char, $(U_p^{″}\cap M_1,{\varphi }^{″}{|}_{U_p^{″}\cap M_1})$ where $U_p^{″}\subseteq M$.

${\varphi }^{″}(U_p^{″})$ with the $n_1+1$ ~ $n$ components vanished equals ${\varphi }^{″}(U_p^{″}\cap M_1)$, but as $(U_p^{″}\cap M_1,{\varphi }^{″}{|}_{U_p^{″}\cap M_1})$ is a restriction of $(U_p^{\prime },{\varphi }^{\prime })$, ${\varphi }^{″}(U_p^{″}\cap M_1\cap M_2)$ equals ${\varphi }^{″}(U_p^{″}\cap M_1)$ with the $n_2+1$ ~ $n_1$ coordinates vanished, but as ${\varphi }^{″}(U_p^{″}\cap M_1)$ equals ${\varphi }^{″}(U_p^{″})$ with the $n_1+1$ ~ $n$ components vanished, ${\varphi }^{″}(U_p^{″}\cap M_1\cap M_2)$ equals ${\varphi }^{″}(U_p^{″})$ with the $n_2+1$ ~ $n_1,n_1+1$ ~ $n$ components vanished, because any $p^{\prime }\in {\varphi }^{″}(U_p^{″})$ whose one of the $n_2+1$ ~ $n$ components is not 0 is not on ${\varphi }^{″}(U_p^{″}\cap M_1\cap M_2)$ and any $p^{\prime }\in {\varphi }^{″}(U_p^{″})$ whose all the $n_2+1$ ~ $n$ components are 0 is on ${\varphi }^{″}(U_p^{″}\cap M_1\cap M_2)$.

As around any $p\in M_2$, there is an adopting chart, $(U_p^{″}\cap M_2,{\varphi }^{″}{|}_{U_p^{″}\cap M_2})$, where $(U_p^{″},{\varphi }^{″})$ is the adopted chart on $M$, $M_2$ is a regular submanifold of $M$ whose codimension is $n-n_2=(n-n_1)+(n_1-n_2)$.

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References

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