description/proof of that for \(C^\infty\) manifold, regular domain, inclusion, and 2 \(C^\infty\) vectors fields, Lie bracket of vectors fields is pulled-back Lie bracket of pushed-forward-and-extended vectors fields
Topics
About: \(C^\infty\) manifold
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Note 1
- 3: Proof
- 4: Note 2
Starting Context
- The reader knows a definition of regular domain of \(C^\infty\) manifold with boundary.
- The reader knows a definition of \(C^\infty\) vectors field on \(C^\infty\) manifold with boundary.
- The reader knows a definition of Lie bracket of \(C^\infty\) vectors fields on \(C^\infty\) manifold with boundary.
- 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, any embedded submanifold with boundary of the manifold, and any \(C^\infty\) vectors field over the submanifold with boundary, the differential by the inclusion after the vectors field is \(C^\infty\) over the submanifold with boundary.
- The reader admits the proposition that for any \(C^\infty\) manifold, its any regular domain, any \(C^\infty\) manifold with boundary, and any \(C^\infty\) map from the regular domain into the \(C^\infty\) manifold with boundary, the corresponding map with the domain regarded as the subset of the manifold is \(C^\infty\).
- The reader admits the proposition that for any \(C^\infty\) manifold with boundary, any embedded submanifold with boundary of the manifold with boundary is properly embedded if and only if it is closed.
- The reader admits the proposition that for any \(C^\infty\) vectors bundle, any \(C^\infty\) section along any closed subset of the base space can be extended to over the whole base space with the support contained in any open neighborhood of the subset.
- The reader admits the proposition that for any map from any subset of any \(C^\infty\) manifold with boundary into any subset of any \(C^\infty\) manifold \(C^k\) at any point, there is a \(C^k\) extension on an open-neighborhood-of-the-point domain.
- The reader admits the proposition that for any \(C^\infty\) map between any \(C^\infty\) manifolds with boundary, any the-map-related 2 \(C^\infty\) vectors fields on the domain and any 2 \(C^\infty\) vectors fields on the codomain, the Lie bracket of the vectors fields on the domain is the-map-related to the Lie bracket of the vectors fields on the codomain.
- The reader admits the proposition that for any \(C^\infty\) manifold with boundary and any regular domain, the differential of the inclusion at each point on the regular domain is a 'vectors spaces - linear morphisms' isomorphism.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) manifold, any regular domain of the \(C^\infty\) manifold, the inclusion of the regular domain into the \(C^\infty\) manifold, and any 2 \(C^\infty\) vectors fields over the regular domain, the Lie bracket of the vectors fields is the pulled-back Lie bracket of any pushed-forward-and-extended vectors fields.
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: Structured Description
Here is the rules of Structured Description.
Entities:
\(M'\): \(\in \{ \text{ the } d' \text{ -dimensional } C^\infty \text{ manifolds } \}\)
\(M\): \(\in \{\text{ the regular domains of } M'\}\)
\(\iota\): \(: M \to M'\), \(= \text{ the inclusion }\)
\(V\): \(: M \to TM\), \(\in \Gamma (TM)\)
\(W\): \(: M \to TM\), \(\in \Gamma (TM)\)
\(d \iota\): \(: TM \to TM'\), \(= \text{ the differential }\)
//
Statements:
\([V, W] = \lambda ([\tau V, \tau W])\), where \(\tau\) pushes-forward each vector on \(TM\) into \(TM'\) by \(d \iota\) and extends it smoothly over \(M'\) and \(\lambda\) pulls-back each vector pushed-forward by \(d \iota\)
//
2: Note 1
In order for \(\tau\) to make sense, \(d \iota \circ V: M \to TM'\) and \(d \iota \circ W: M \to TM'\) have to be proved to be \(C^\infty\) and \(d \iota \circ V\) and \(d \iota \circ W\) have to be proved to be able to be extended smoothly, which will be done in Proof.
In order for \(\lambda\) to make sense, \([\tau V, \tau W]\) at each point on \(\iota (M)\) has to be proved to be the pushed-forward of a vector on \(TM\), which will be done in Proof.
3: Proof
Whole Strategy: Step 1: see that \(d \iota \circ V: M \to TM'\) and \(d \iota \circ W: M \to TM'\) are some \(C^\infty\) maps; Step 2: see that \(d \iota \circ V: M \to TM'\) and \(d \iota \circ W: M \to TM'\) can be extended to some \(C^\infty\) \(\tau V \in \Gamma (TM')\) and \(\tau W \in \Gamma (TM')\); Step 3: see that \([V, W]\) is \(\iota\)-related to \([\tau V, \tau W]\), which means that \([V, W]\) can be pushed-forward-and-extended to \(\tau ([V, W]) = [\tau V, \tau W]\); Step 4: do \(\lambda \circ \tau ([V, W]) = \lambda ([\tau V, \tau W])\).
Step 1:
Let us see that \(d \iota \circ V: M \to TM'\) is \(C^\infty\).
In fact, that has been proved by the proposition that for any \(C^\infty\) manifold, any embedded submanifold with boundary of the manifold, and any \(C^\infty\) vectors field over the submanifold with boundary, the differential by the inclusion after the vectors field is \(C^\infty\) over the submanifold with boundary.
\(d \iota \circ W: M \to TM'\) is \(C^\infty\), likewise.
Step 2:
While \(d \iota \circ V: M \to TM'\) is \(C^\infty\) with the domain regarded as the regular domain, it is \(C^\infty\) also with the domain regarded as the subset of \(M'\), by the proposition that for any \(C^\infty\) manifold, its any regular domain, any \(C^\infty\) manifold with boundary, and any \(C^\infty\) map from the regular domain into the \(C^\infty\) manifold with boundary, the corresponding map with the domain regarded as the subset of the manifold is \(C^\infty\).
\(\iota (M) \subseteq M'\) is closed on \(M'\), by the proposition that for any \(C^\infty\) manifold with boundary, any embedded submanifold with boundary of the manifold with boundary is properly embedded if and only if it is closed.
\(d \iota \circ V\) can be extended to a \(\tau V \in \Gamma (TM')\), by the proposition that for any \(C^\infty\) vectors bundle, any \(C^\infty\) section along any closed subset of the base space can be extended to over the whole base space with the support contained in any open neighborhood of the subset, while \(d \iota \circ V\) is indeed a \(C^\infty\) vectors field along \(\iota (M)\), by the proposition that for any map from any subset of any \(C^\infty\) manifold with boundary into any subset of any \(C^\infty\) manifold \(C^k\) at any point, there is a \(C^k\) extension on an open-neighborhood-of-the-point domain: the definition of \(C^\infty\) vectors field along subset is exactly that there is a \(C^\infty\) extension.
\(d \iota \circ W\) can be extended to a \(\tau V \in \Gamma (TM')\), likewise.
Step 3:
\(V\) is \(\iota\)-related to \(\tau V\), because \(d \iota V_p = (\tau V)_{\iota (p)}\).
\(W\) is \(\iota\)-related to \(\tau W\), likewise.
So, \([V, W]\) is \(\iota\)-related to \([\tau V, \tau W]\), by the proposition that for any \(C^\infty\) map between any \(C^\infty\) manifolds with boundary, any the-map-related 2 \(C^\infty\) vectors fields on the domain and any 2 \(C^\infty\) vectors fields on the codomain, the Lie bracket of the vectors fields on the domain is the-map-related to the Lie bracket of the vectors fields on the codomain.
That means that \([V, W]\) can be pushed-forward and extended to be \([\tau V, \tau W]\), which is denoted as \(\tau ([V, W]) = [\tau V, \tau W]\).
Step 4:
\(\lambda \circ \tau ([V, W])\) is valid because \(d \iota_p\) is bijective at each \(p \in M\), by the proposition that for any \(C^\infty\) manifold with boundary and any regular domain, the differential of the inclusion at each point on the regular domain is a 'vectors spaces - linear morphisms' isomorphism and \(\tau ([V, W])\) at each \(\iota (p)\) is the pushed-forward of the vector in \(T_pM\), \([V, W]_p\).
Let us do \(\lambda \circ \tau ([V, W]) = \lambda ([\tau V, \tau W])\), but \(\lambda \circ \tau = id\), the identity map, so, \([V, W] = \lambda \circ \tau ([V, W]) = \lambda ([\tau V, \tau W])\).
4: Note 2
An immediate corollary of this proposition is that \(\lambda ([\tau V, \tau W])\) does not depend on \(M'\) or the extension (because it equals \([V, W]\), which is ignorant of \(M'\) or the extension), which is in fact the immediate purpose of this proposition: the Lie derivative of \(W\) by \(V\) at any boundary point, \(p \in \partial M\), is defined to be the pulled-back Lie derivative of \(\tau W\) by \(\tau V\) at \(\iota (p)\), which equals \(\lambda ([\tau V, \tau W]_{\iota (p)})\), which needs to be proved to be independent of \(M'\) or the extension.
