360: C^\infty Vectors Field on Regular Submanifold Is C^\infty as Vectors Field Along Regular Submanifold on Supermanifold

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A description/proof of that $C^{\mathrm{\infty }}$ vectors field on regular submanifold is $C^{\mathrm{\infty }}$ as vectors field along regular submanifold on supermanifold

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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.
• The reader knows a definition of $C^{\mathrm{\infty }}$ vectors field along regular submanifold on super $C^{\mathrm{\infty }}$ manifold.
• The reader admits the proposition that any vectors field is $C^{\mathrm{\infty }}$ if and only if its components function on any chart is $C^{\mathrm{\infty }}$.

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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 $C^{\mathrm{\infty }}$ vectors field on the regular submanifold is $C^{\mathrm{\infty }}$ as a vectors field along the regular submanifold on the supermanifold.

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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, $M_1$, and any regular submanifold, $M_2\subseteq M_1$, any $C^{\mathrm{\infty }}$ vectors field, $V\in \mathfrak{X}(M_2)$, is a $C^{\mathrm{\infty }}$ vectors field along $M_2$ on $M_1$, which is ${\pi }_{\ast }V\in \mathrm{\Gamma }(T{M}_1|M_2)$ where $\pi :M_2\to M_1$ is the inclusion.

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2: Proof

There is a adopted chart on $M_1$ and the corresponding adopting chart on $M_2$. On the adopting chart, $V=V^i{\partial }_i$ where $0\le i\le d_2$ where $d_2$ is the dimension of $M_2$ and $V^i$ is a $C^{\mathrm{\infty }}$ function, by the proposition that any vectors field is $C^{\mathrm{\infty }}$ if and only if its components function on any chart is $C^{\mathrm{\infty }}$. The vectors field along $M_2$ on $M_1$ is the push-forward, ${\pi }_{\ast }V$, and ${\pi }_{\ast }V=V^i{\partial }_i+\sum _{j}0{\partial }_j$ by the adopted chart where $d_2+1\le j\le d_1$ where $d_1$ is the dimension of $M_1$. For any $C^{\mathrm{\infty }}$ function, $f\in C^{\mathrm{\infty }}(M_1)$, $({\pi }_{\ast }V)(f)=V^i{\partial }_{i}f$, which is a $C^{\mathrm{\infty }}$ function on $M_2$, which means that ${\pi }_{\ast }V$ is a $C^{\mathrm{\infty }}$ vectors field along $M_2$ on $M_1$ by the definition.

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References

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