description/proof of that pushforward image of \(C^\infty\) vectors field along curve on regular submanifold into supermanifold under inclusion is \(C^\infty\)
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) vectors field along curve.
- The reader knows a definition of regular submanifold.
- The reader knows a definition of differential of \(C^\infty\) map between \(C^\infty\) manifolds with boundary at point.
- The reader admits the proposition that for any \(C^\infty\) manifold and any \(C^\infty\) curve over any open interval, any vectors field along the curve is \(C^\infty\) if and only if its operation result on any \(C^\infty\) function on the manifold is \(C^\infty\).
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) manifold, any its regular submanifold, and any \(C^\infty\) curve over any open interval on the regular submanifold, the pushforward image of any \(C^\infty\) vectors field along the curve on the regular submanifold into the supermanifold under the inclusion is \(C^\infty\).
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.
Main Body
1: Description
For any \(C^\infty\) manifold, \(M'\), any regular submanifold, \(M \subseteq M'\), the inclusion, \(\iota: M \to M'\), any open interval, \(I \subseteq \mathbb{R}\), any \(C^\infty\) curve, \(c: I \to M\), and any \(C^\infty\) vectors field along \(c\), \(V: I \to c (I) \to TM\), the pushforward image of \(V\) into \(M'\) under \(\iota\), \(\iota_* V: I \to \iota \circ c (I) \to TM'\), \(t \mapsto \iota_* V (t)\), is \(C^\infty\) as the vectors field along \(\iota \circ c\).
2: Note
Usual 'pushforward' (a.k.a differential) maps \(T_pM\) into \(T_{\iota (p)}M\); 'pushforward' here is constructing \(\iota_* V: I \to TM'\) from \(V: I \to TM\).
3: Proof
\(\iota \circ c\) is a \(C^\infty\) curve on \(M'\), because for any point, \(t_0 \in I\), there are an adopted chart, \((U'_{\iota \circ c (t_0)} \subseteq M', \phi'_{\iota \circ c (t_0)}): p \mapsto (x^1, ..., x^d, x^{d + 1}, ..., x^{d'})\), and the corresponding adopting chart, \((U_{c (t_0)} \subseteq M, \phi_{c (t_0)}: p \mapsto (x^1, x^2, ..., x^d)\), 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 \(C^\infty\).
For any \(C^\infty\) function, \(f: M \to \mathbb{R}\), \((V (t) f) \circ c (t)\) is \(C^\infty\) by the proposition that for any \(C^\infty\) manifold and any \(C^\infty\) curve over any open interval, any vectors field along the curve is \(C^\infty\) if and only if its operation result on any \(C^\infty\) function on the manifold is \(C^\infty\).
For any \(C^\infty\) function, \(f': M' \to \mathbb{R}\), \((\iota_* V) (t) f' = \iota_* V (t) f' = V (t) (f' \circ \iota)\), so, \(((\iota_* V) (t) f') \circ \iota \circ c (t) = (V (t) (f' \circ \iota)) \circ c (t)\), which is \(C^\infty\) because \(f' \circ \iota\) is a \(C^\infty\) function over \(M\). So, \(\iota_* V\) is \(C^\infty\) as the vectors field along \(\iota \circ c\), by the proposition that for any \(C^\infty\) manifold and any \(C^\infty\) curve over any open interval, any vectors field along the curve is \(C^\infty\) if and only if its operation result on any \(C^\infty\) function on the manifold is \(C^\infty\).
