370: For Euclidean C^\infty Manifold and Its Regular Submanifold, Vectors Field Along Regular Submanifold Is C^\infty iff Its Components w.r.t. Standard Chart Are C^\infty on Regular Submanifold

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A description/proof of that for Euclidean $C^{\mathrm{\infty }}$ manifold and its regular submanifold, vectors field along regular submanifold is $C^{\mathrm{\infty }}$ iff its components w.r.t. standard chart are $C^{\mathrm{\infty }}$ 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 $C^{\mathrm{\infty }}$ vectors field along regular submanifold.
• The reader admits the proposition that any $C^{\mathrm{\infty }}$ function on any $C^{\mathrm{\infty }}$ manifold is $C^{\mathrm{\infty }}$ on any regular submanifold of the $C^{\mathrm{\infty }}$ manifold.

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Target Context

• The reader will have a description and a proof of the proposition that for any Euclidean $C^{\mathrm{\infty }}$ manifold and its any regular submanifold, any vectors filed along the regular submanifold is $C^{\mathrm{\infty }}$ if and only if the components of the vectors field with respect to the standard chart on the Euclidean $C^{\mathrm{\infty }}$ manifold are $C^{\mathrm{\infty }}$ 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 Euclidean $C^{\mathrm{\infty }}$ manifold, ${\mathbb{R}}^n$, and its any regular submanifold, $M\subseteq {\mathbb{R}}^n$, any vectors field along $M$, $V=V^i(p\in M){\partial }_i$ where $\{{\partial }_i\}$ is the canonical basis of $T_p{\mathbb{R}}^n$ by the standard chart on ${\mathbb{R}}^n$ is $C^{\mathrm{\infty }}$ if and only if $V^i(p\in M)$ is $C^{\mathrm{\infty }}$ on $M$.

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2: Proof

Let us suppose that $V$ is $C^{\mathrm{\infty }}$. For any $C^{\mathrm{\infty }}$ function, $f$, on ${\mathbb{R}}^n$, $Vf$ is $C^{\mathrm{\infty }}$ on $M$, by the definition of $C^{\mathrm{\infty }}$ vectors field along regular submanifold. $Vf=V^i(p\in M)\frac{\partial f}{{\partial }_i}$, but $f$ can be taken to be the coordinate function, $r^i$, then, $V{r}^i=V^i(p\in M)$, which is $C^{\mathrm{\infty }}$ on $M$.

Let us suppose that $V^i(p\in M)$ is $C^{\mathrm{\infty }}$ on $M$. Then, $Vf=V^i(p\in M)\frac{\partial f}{{\partial }_i}$ is $C^{\mathrm{\infty }}$ on $M$, because $\frac{\partial f}{{\partial }_i}$ is $C^{\mathrm{\infty }}$ on ${\mathbb{R}}^n$, and is $C^{\mathrm{\infty }}$ on $M$, by the proposition that any $C^{\mathrm{\infty }}$ function on any $C^{\mathrm{\infty }}$ manifold is $C^{\mathrm{\infty }}$ on any regular submanifold of the $C^{\mathrm{\infty }}$ manifold.

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References

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