description/proof of that for \(C^\infty\) manifold, embedded submanifold with boundary, and \(C^\infty\) vectors field over submanifold with boundary, differential by inclusion after vectors field is \(C^\infty\) over submanifold with boundary
Topics
About: \(C^\infty\) manifold
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
- 4: Proof
Starting Context
Target Context
- The reader will have a description and a proof of 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.
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:
\(M'\): \(\in \{ \text{ the } d' \text{ -dimensional } C^\infty \text{ manifolds } \}\)
\(M\): \(\in \{\text{ the embedded submanifolds with boundary of } M'\}\)
\(V\): \(: M \to TM\), \(\in \Gamma (TM)\)
\(\iota\): \(: M \to M'\), \(= \text{ the inclusion }\)
\(d \iota\): \(: TM \to TM'\), \(= \text{ the differential }\)
\(d \iota \circ V\): \(: M \to TM'\)
//
Statements:
\(d \iota \circ V \in \{\text{ the } C^\infty \text{ maps }\}\)
//
2: Natural Language Description
For any \(d'\)-dimensional \(C^\infty\) manifold, \(M'\), any embedded submanifold with boundary of \(M'\), \(M\), any \(C^\infty\) vectors field, \(V: M \to TM\), the inclusion, \(\iota: M \to M'\), the differential, \(d \iota: TM \to TM'\), and \(d \iota \circ V: M \to TM'\), \(d \iota \circ V\) is \(C^\infty\).
3: Note
Although "differential of \(V\)" usually makes people imagine a vectors field over \(M'\) (which is invalid as \(\iota\) is not diffeomorphic), we are not talking about that.
Anyway, \(d \iota\) is valid, because \(\iota\) is a \(C^\infty\) embedding, by the definition of embedded submanifold with boundary of \(C^\infty\) manifold with boundary.
4: Proof
Whole Strategy: Step 1: around any \(m \in M\), take an adopted chart, \((U'_m \subseteq M', \phi'_m)\), and the corresponding adopting chart, \((U_m := U'_m \cap M \subseteq M, \phi_m := \pi_J \circ \phi'_m \vert_{U_m})\); Step 2: see what \(V\)'s being \(C^\infty\) means with respect to \((U_m \subseteq M, \phi_m)\); Step 3: see what \(d \iota \circ V\)'s being \(C^\infty\) means with respect to \((U'_m \subseteq M', \phi'_m)\); Step 4: conclude the proposition.
Step 1:
Let \(m \in M\) be any.
As \(M\) is an embedded submanifold with boundary of \(M'\), there is an adopted chart, \((U'_m \subseteq M', \phi'_m)\), for \(M\).
There is the corresponding adopting chart, \((U_m := U'_m \cap M \subseteq M, \phi_m := \pi_J \circ \phi'_m \vert_{U_m})\), where \(\pi_J: \mathbb{R}^{d'} \to \mathbb{R}^d\) is the projection taking the \(J \subseteq \{1, ..., d'\}\) components.
Step 2:
Let us denote \(\pi: TM \to M\) and \(\pi': TM' \to M'\).
There are the induced charts using the standard bases for \(TM\) and \(TM'\), \((\pi^{-1} (U_m) \subseteq TM, \widetilde{\phi_m})\) and \((\pi'^{-1} (U'_m) \subseteq TM', \widetilde{\phi'_m})\).
As \(V\) is \(C^\infty\) at \(m\), \(\widetilde{\phi_m} \circ V \circ {\phi_m}^{-1}: \phi_m (U_m) \to \widetilde{\phi_m} (\pi^{-1} (U_m))\) is \(C^\infty\) at \(\phi_m (m)\).
Step 3:
Whether \(d \iota \circ V\) is \(C^\infty\) at \(m\) or not is about whether \(\widetilde{\phi'_m} \circ d \iota \circ V \circ {\phi_m}^{-1}: \phi_m (U_m) \to \widetilde{\phi'_m} (\pi'^{-1} (U'_m))\) is \(C^\infty\) at \(\phi_m (m)\) or not.
Step 4:
Let us denote the coordinates function of \(\iota\) as \(f := \phi'_m \circ \iota \circ {\phi_m}^{-1}: \phi_m (U_m) \to \phi'_m (U'_m)\).
In order for the simplicity of notations, we can choose \(\phi'_m\) such that \(f: (x^1, ..., x^d) \mapsto (x^1, ..., x^d, 0, ..., 0)\): we can just reorder the components of \(\phi'_m\) and translate \(\phi'_m\). Then, \(\partial f^k / \partial x^j = \delta^k_j\) for \(j, k \in \{1, ..., d\}\) and \(\partial f^k / \partial x^j = 0\) otherwise.
Let us denote the \(j\)-th component of \(V\) with respect to \((\pi^{-1} (U_m) \subseteq TM, \widetilde{\phi_m})\) as \(V^j\). Let us denote the \(j\)-th component of \(d \iota \circ V\) with respect to \((\pi'^{-1} (U'_m) \subseteq TM', \widetilde{\phi'_m})\) as \((d \iota \circ V)^j\).
\(\widetilde{\phi_m} \circ V \circ {\phi_m}^{-1}: (x^1, ..., x^d) \mapsto (x^1, ..., x^d, V^1, ..., V^d)\), which is \(C^\infty\) at \(\phi_m (m)\).
As is known well, \((d \iota \circ V)^k = \partial f^k / \partial x^j V^j\) with the Einstein convention, which is \(= V^k\) for each \(k \in \{1, ..., d\}\) and \(= 0\) otherwise. That means that \(\widetilde{\phi'_m} \circ d \iota \circ V \circ {\phi_m}^{-1}: (x^1, ..., x^d) \mapsto (x^1, ..., x^d, 0, ..., 0, V^1, ..., V^d, 0, ..., 0)\), which is \(C^\infty\) at \(\phi_m (m)\), obviously.
So, \(d \iota \circ V\) is \(C^\infty\) at \(m\).
As \(m \in M\) is arbitrary, \(d \iota \circ V\) is \(C^\infty\) all over \(M\).
