description/proof of that \(q\)-form over \(C^\infty\) manifold with boundary is \(C^\infty\) iff operation result on any \(C^\infty\) vectors fields is \(C^\infty\)
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(q\)-form over \(C^\infty\) manifold with boundary.
- The reader knows a definition of \(r'\)-\(r\)-open-balls charts pair around point on \(C^\infty\) manifold with boundary.
- The reader knows a definition of \(r'\)-\(r\)-open-half-balls charts pair around point on \(C^\infty\) manifold with boundary.
- The reader admits the proposition that any \((0, q)\)-tensors field over \(C^\infty\) manifold with boundary is \(C^\infty\) if and only if the operation result on any \(C^\infty\) vectors fields is \(C^\infty\).
- The reader admits the proposition that for any \(C^\infty\) manifold with boundary, each interior point has an \(r'\)-\(r\)-open-balls charts pair and each boundary point has an \(r'\)-\(r\)-open-half-balls charts pair for any positive \(r'\) and \(r\).
- 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.
Target Context
- The reader will have a description and a proof of the proposition that any \(q\)-form over any \(C^\infty\) manifold with boundary is \(C^\infty\) if and only if the operation result on any \(C^\infty\) vectors fields 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: Structured Description
Here is the rules of Structured Description.
Entities:
\(M\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(q\): \(\in \mathbb{N} \setminus \{0\}\)
\((T^0_q (TM), M, \pi)\): \(= (0, q) \text{ -tensors bundle over } M\)
\((\Lambda_q (TM), M, \pi)\): \(= q \text{ -covectors bundle over } M\)
\(f\): \(: M \to T^0_q (TM)\) such that \(Ran (f) \subseteq \Lambda_q (TM)\) or \(: M \to \Lambda_q (TM)\), \(\in \{\text{ the sections of } \pi\}\)
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Statements:
\(f \in \{\text{ the } C^\infty \text{ maps }\}\)
\(\iff\)
\(\forall V_1, ..., V_q \in \{\text{ the } C^\infty \text{ vectors fields over } M\} (f (V_1, ..., V_q): M \to \mathbb{R} \in \{\text{ the } C^\infty \text{ maps }\})\)
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2: Proof
Whole Strategy: Step 1: when \(: M \to T^0_q (TM)\), see that the proposition holds; Step 2: suppose that \(: M \to \Lambda_q (TM)\); Step 3: suppose that \(f (V_1, ..., V_q) \in \{\text{ the } C^\infty \text{ maps }\}\), and see that \(f\) is \(C^\infty\), by taking an \(r'\)-\(r\)-open-balls charts pair or an \(r'\)-\(r\)-open-half-balls charts pair around each \(m \in M\), \((U'_m \subseteq M, \phi'_m)\) and \((U_m \subseteq M, \phi_m)\), and the induced chart, \((\pi^{-1} (U_m) \subseteq \Lambda_q (TM), \widetilde{\phi_m})\), and seeing that the components function of \(f\) is \(C^\infty\); Step 4: suppose that \(f\) is \(C^\infty\), and see that \(f (V_1, ..., V_q)\) is \(C^\infty\) over a chart, \((U_m \subseteq M, \phi_m)\), around each \(m \in M\).
Step 1:
When \(f\) is \(: M \to T^0_q (TM)\), \(f\) is really just a special type of \((0, q)\)-tensors field, so, the proposition that any \((0, q)\)-tensors field over \(C^\infty\) manifold with boundary is \(C^\infty\) if and only if the operation result on any \(C^\infty\) vectors fields is \(C^\infty\) applies.
Step 2:
Let us suppose that \(f\) is \(: M \to \Lambda_q (TM)\).
Step 3:
Let us suppose that \(f (V_1, ..., V_q) \in \{\text{ the } C^\infty \text{ maps }\}\).
Let \(m \in M\) be any.
Let us take any \(r'\)-\(r\)-open-balls charts pair or any \(r'\)-\(r\)-open-half-balls charts pair around \(m\), \((U'_m \subseteq M, \phi'_m)\) and \((U_m \subseteq M, \phi_m)\), which is possible by the proposition that for any \(C^\infty\) manifold with boundary, each interior point has an \(r'\)-\(r\)-open-balls charts pair and each boundary point has an \(r'\)-\(r\)-open-half-balls charts pair for any positive \(r'\) and \(r\), and take the induced chart, \((\pi^{-1} (U_m) \subseteq \Lambda_q (TM), \widetilde{\phi_m})\).
Let us take \(V_j = V_j^{l_j} \partial / \partial x^{l_j}\) over \(U'_m\) as \(V_j^{l'_j} \equiv 1\) and \(V_j^{l_j} \equiv 0\) for each \({l_j} \neq l'_j\) where \(l'_1 \lt ... \lt l'_q\). \(V_j\) is \(C^\infty\) over \(U'_m\). \(V_j\) is \(C^\infty\) over \(\overline{U_m} \subseteq U'_m\). 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, \(V_j\) is extended to over \(M\). The extended \(V_j\) equals the original \(V_j\) over \(\overline{U_m}\) especially over \(U_m\).
Let \(m' \in U_m\) be any.
\(f (m') = \sum_{j_1 \lt ... \lt j_q} f_{j_1, ..., j_q} (m') d x^{j_1} \wedge ... \wedge d x^{j_q}\).
\(f (m') (V_1, ..., V_q) = \sum_{j_1 \lt ... \lt j_q} f_{j_1, ..., j_q} (m') d x^{j_1} \wedge ... \wedge d x^{j_q} (V_1^{l_1} \partial / \partial x^{l_1}, ..., V_q^{l_q} \partial / \partial x^{l_q}) = \sum_{j_1 \lt ... \lt j_q} f_{j_1, ..., j_q} (m') q! Asym d x^{j_1} \otimes ... \otimes d x^{j_q} (V_1^{l_1} \partial / \partial x^{l_1}, ..., V_q^{l_q} \partial / \partial x^{l_q}) = \sum_{j_1 \lt ... \lt j_q} f_{j_1, ..., j_q} (m') q! 1 / q! \sum_{\sigma \in P_{\{1, ..., q\}}} sgn \sigma d x^{j_1} \otimes ... \otimes d x^{j_q} (V_{\sigma_1}^{l_{\sigma_1}} \partial / \partial x^{l_{\sigma_1}}, ..., V_{\sigma_q}^{l_{\sigma_q}} \partial / \partial x^{l_{\sigma_q}}) = \sum_{j_1 \lt ... \lt j_q} f_{j_1, ..., j_q} (m') \sum_{\sigma \in P_{\{1, ..., q\}}} sgn \sigma d x^{j_1} (V_{\sigma_1}^{l_{\sigma_1}} \partial / \partial x^{l_{\sigma_1}}) ... d x^{j_q} (V_{\sigma_q}^{l_{\sigma_q}} \partial / \partial x^{l_{\sigma_q}}) = \sum_{j_1 \lt ... \lt j_q} f_{j_1, ..., j_q} (m') \sum_{\sigma \in P_{\{1, ..., q\}}} sgn \sigma V_{\sigma_1}^{l_{\sigma_1}} \delta^{j_1}_{l_{\sigma_1}} ... V_{\sigma_q}^{l_{\sigma_q}} \delta^{j_q}_{l_{\sigma_q}} = \sum_{j_1 \lt ... \lt j_q} f_{j_1, ..., j_q} (m') \sum_{\sigma \in P_{\{1, ..., q\}}} sgn \sigma V_{\sigma_1}^{j_1} ... V_{\sigma_q}^{j_q} = \sum_{j_1 \lt ... \lt j_q} f_{j_1, ..., j_q} (m') \sum_{\sigma \in P_{\{1, ..., q\}}} sgn \sigma V_1^{j_{\sigma^{-1} (1)}} ... V_q^{j_{\sigma^{-1} (q)}}\): \(V_{\sigma_1}^{j_1} ... V_{\sigma_q}^{j_q}\) is reordered to \(V_1^{m_1} ... V_q^{m_q}\), then, \(V_{\sigma_n}^{j_n} = V_s^{m_s}\), which means that \(\sigma_n = s\), so, \(n = \sigma^{-1} (s)\).
\(V_1^{j_{\sigma^{-1} (1)}} ... V_q^{j_{\sigma^{-1} (q)}}\) can be nonzero only for \(\sigma = id\), because \(l'_1 \lt ... \lt l'_q\).
So, \(= \sum_{j_1 \lt ... \lt j_q} f_{j_1, ..., j_q} (m') V_1^{j_1} ... V_q^{j_q} = f_{l'_1, ..., l'_q} (m')\).
\(f (V_1, ..., V_q): U_m \to \mathbb{R}\) is \(C^\infty\) by the supposition, so, \(f_{l'_1, ..., l'_q}: U_m \to \mathbb{R}\) is \(C^\infty\).
So, each \(f_{j_1, ..., j_q}\) is \(C^\infty\) over \(U_m\), which means that the components function of \(f\) with respect to \((U_m \subseteq M, \phi_m)\) and \((\pi^{-1} (U_m) \subseteq \Lambda_q (TM), \widetilde{\phi_m})\), \(\widetilde{\phi_m} \circ f \circ {\phi_m}^{-1}\) whose components are \(f_{j_1, ..., j_q} \circ {\phi_m}^{-1}\) s, is \(C^\infty\).
So, \(f\) is \(C^\infty\).
Step 4:
Let us suppose that \(f\) is \(C^\infty\).
Let \(m \in M\) be any.
Let us take any chart around \(m\), \((U_m \subseteq M, \phi_m)\), and the induced chart, \((\pi^{-1} (U_m) \subseteq \Lambda_q (TM), \widetilde{\phi_m})\).
Over \(U_m\), \(f = f_{j_1, ..., j_q} d x^{j_1} \wedge... \wedge d x^{j_q}\), where \(f_{j_1, ..., j_q}: U_m \to \mathbb{R}\) is \(C^\infty\), because it is a component of \(\widetilde{\phi_m} \circ f \circ {\phi_m}^{-1} \circ \phi_m\), while the components function of \(f\) with respect to \((U_m \subseteq M, \phi_m)\) and \((\pi^{-1} (U_m) \subseteq \Lambda_q (TM), \widetilde{\phi_m})\), \(\widetilde{\phi_m} \circ f \circ {\phi_m}^{-1}\), is \(C^\infty\) and \(\phi_m\) is \(C^\infty\).
\(V_j = V_j^{l_j} \partial / \partial x^{l_j}\), where \(V_j^{l_j}: U_m \to \mathbb{R}\) is \(C^\infty\).
\(f (V_1, ..., V_q) = \sum_{j_1 \lt ... \lt j_q} f_{j_1, ..., j_q} d x^{j_1} \wedge ... \wedge d x^{j_q} (V_1^{l_1} \partial / \partial x^{l_1}, ..., V_q^{l_q} \partial / \partial x^{l_q}) = \sum_{j_1 \lt ... \lt j_q} f_{j_1, ..., j_q} (m') \sum_{\sigma \in P_{\{1, ..., q\}}} sgn \sigma V_{\sigma_1}^{j_1} ... V_{\sigma_q}^{j_q}\) as before, which is \(C^\infty\).
As \(f (V_1, ..., V_q)\) is \(C^\infty\) over a neighborhood of each \(m \in M\), \(f (V_1, ..., V_q)\) is \(C^\infty\) over \(M\).
