505: Pushforward Image of C∞ Vectors Field Along Curve on Regular Submanifold into Supermanifold Under Inclusion Is C∞

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description/proof of that pushforward image of $C^{\mathrm{\infty }}$ vectors field along curve on regular submanifold into supermanifold under inclusion is $C^{\mathrm{\infty }}$

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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: Note
• 3: Proof

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Starting Context

• The reader knows a definition of $C^{\mathrm{\infty }}$ vectors field along curve.
• The reader knows a definition of regular submanifold.
• The reader knows a definition of differential of $C^{\mathrm{\infty }}$ map between $C^{\mathrm{\infty }}$ manifolds with boundary at point.
• The reader admits the proposition that for any $C^{\mathrm{\infty }}$ manifold and any $C^{\mathrm{\infty }}$ curve over any open interval, any vectors field along the curve is $C^{\mathrm{\infty }}$ if and only if its operation result on any $C^{\mathrm{\infty }}$ function on the manifold 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, any its regular submanifold, and any $C^{\mathrm{\infty }}$ curve over any open interval on the regular submanifold, the pushforward image of any $C^{\mathrm{\infty }}$ vectors field along the curve on the regular submanifold into the supermanifold under the inclusion is $C^{\mathrm{\infty }}$.

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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, $M^{\prime }$, any regular submanifold, $M\subseteq M^{\prime }$, the inclusion, $\iota :M\to M^{\prime }$, any open interval, $I\subseteq \mathbb{R}$, any $C^{\mathrm{\infty }}$ curve, $c:I\to M$, and any $C^{\mathrm{\infty }}$ vectors field along $c$, $V:I\to c(I)\to TM$, the pushforward image of $V$ into $M^{\prime }$ under $\iota$, ${\iota }_{\ast }V:I\to \iota \circ c(I)\to T{M}^{\prime }$, $t\mapsto {\iota }_{\ast }V(t)$, is $C^{\mathrm{\infty }}$ as the vectors field along $\iota \circ c$.

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2: Note

Usual 'pushforward' (a.k.a differential) maps $T_{p}M$ into $T_{\iota (p)}M$; 'pushforward' here is constructing ${\iota }_{\ast }V:I\to T{M}^{\prime }$ from $V:I\to TM$.

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3: Proof

$\iota \circ c$ is a $C^{\mathrm{\infty }}$ curve on $M^{\prime }$, because for any point, $t_0\in I$, there are an adopted chart, $(U_{\iota \circ c(t_0)}^{\prime }\subseteq M^{\prime },{\varphi }_{\iota \circ c(t_0)}^{\prime }):p\mapsto (x^1,...,x^d,x^{d+1},...,x^{d^{\prime }})$, and the corresponding adopting chart, $(U_{c(t_0)}\subseteq M,{\varphi }_{c(t_0)}:p\mapsto (x^1,x^2,...,x^d)$, and the components function of $\iota \circ c$ is $(c^1(t),c^2(t),...,c^d(t),0,...,0)$, where $c^j(t)$ is $C^{\mathrm{\infty }}$.

For any $C^{\mathrm{\infty }}$ function, $f:M\to \mathbb{R}$, $(V(t)f)\circ c(t)$ is $C^{\mathrm{\infty }}$ by the proposition that for any $C^{\mathrm{\infty }}$ manifold and any $C^{\mathrm{\infty }}$ curve over any open interval, any vectors field along the curve is $C^{\mathrm{\infty }}$ if and only if its operation result on any $C^{\mathrm{\infty }}$ function on the manifold is $C^{\mathrm{\infty }}$.

For any $C^{\mathrm{\infty }}$ function, $f^{\prime }:M^{\prime }\to \mathbb{R}$, $({\iota }_{\ast }V)(t)f^{\prime }={\iota }_{\ast }V(t)f^{\prime }=V(t)(f^{\prime }\circ \iota )$, so, $(({\iota }_{\ast }V)(t)f^{\prime })\circ \iota \circ c(t)=(V(t)(f^{\prime }\circ \iota ))\circ c(t)$, which is $C^{\mathrm{\infty }}$ because $f^{\prime }\circ \iota$ is a $C^{\mathrm{\infty }}$ function over $M$. So, ${\iota }_{\ast }V$ is $C^{\mathrm{\infty }}$ as the vectors field along $\iota \circ c$, by the proposition that for any $C^{\mathrm{\infty }}$ manifold and any $C^{\mathrm{\infty }}$ curve over any open interval, any vectors field along the curve is $C^{\mathrm{\infty }}$ if and only if its operation result on any $C^{\mathrm{\infty }}$ function on the manifold is $C^{\mathrm{\infty }}$.

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References

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