369: C^\infty Function on C^\infty Manifold Is C^\infty on Regular Submanifold

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A description/proof of that $C^{\mathrm{\infty }}$ function on $C^{\mathrm{\infty }}$ manifold is $C^{\mathrm{\infty }}$ on regular submanifold

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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 $C^{\mathrm{\infty }}$ function on $C^{\mathrm{\infty }}$ manifold.
• The reader knows a definition of regular submanifold of $C^{\mathrm{\infty }}$ manifold.

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Target Context

• The reader will have a description and a proof of the proposition that for any $C^{\mathrm{\infty }}$ manifold and its any regular submanifold, any $C^{\mathrm{\infty }}$ function on the super manifold is $C^{\mathrm{\infty }}$ on the regular 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, $M^{\prime }$, and any regular submanifold, $M\subseteq M^{\prime }$, any $C^{\mathrm{\infty }}$ function, $f:M^{\prime }\to \mathbb{R}$, is $C^{\mathrm{\infty }}$ on $M$, which means that the restriction, $f{|}_M:M\to \mathbb{R}$ is $C^{\mathrm{\infty }}$.

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2: Proof

For any $p\in M$, there is an adopted chart, $(U_p^{\prime }\subseteq M^{\prime },{\varphi }_p^{\prime })$, and $f\circ {{\varphi }_p^{\prime }}^{-1}$ is $C^{\mathrm{\infty }}$. There is the corresponding adopting chart, $(U_p,{\varphi }_p)$, and $f{|}_M\circ {{\varphi }_p}^{-1}$ is $C^{\mathrm{\infty }}$, because $f{|}_M\circ {{\varphi }_p}^{-1}(r^1,r^2.,...,r^d)=f\circ {{\varphi }_p^{\prime }}^{-1}(r^1,r^2.,...,r^d,0,...,0)$.

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References

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