definition of \(C^\infty\) map induced from \(C^\infty\) map from finite-product \(C^\infty\) manifold with boundary by fixing some domain components based on point
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
Target Context
- The reader will have a definition of \(C^\infty\) map induced from \(C^\infty\) map from finite-product \(C^\infty\) manifold with boundary by fixing some domain components based on point.
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:
\( \{M_1, ..., M_{n - 1}\}\): \(\subseteq \{\text{ the } C^\infty \text{ manifolds }\}\)
\( M_n\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\( M_1 \times ... \times M_n\): \(= \text{ the finite-product } C^\infty \text{ manifold with boundary }\)
\( M\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\( f\): \(: M_1 \times ... \times M_n \to M\), \(\in \{\text{ the } C^\infty \text{ maps }\}\)
\( m_0\): \(\in M_1 \times ... \times M_n\)
\( S\): \(= \{j_1, ..., j_s\} \subseteq \{1, ..., n\}\)
\(*f_{S, m_0}\): \(: M_1 \times ... \times M_n \to M, m = (m^1, ...m^n) \mapsto f (m_0^1, ..., m^{j_1}, ..., m^{j_s}, ..., m_0^n)\), which means that \(\{m_0^1, ..., m_0^n\}\) are used except \(\{m^{j_1}, ..., m^{j_s}\}\)
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Conditions:
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When \(S = \{j\}\), \(f_{S, m_0}\) is denoted also as \(f_{j, m_0}\), which is a typical case.
2: Note
Let us see that \(f_{S, m_0}\) is indeed \(C^\infty\).
Let \(m = (m^1, ..., m^n) \in M_1 \times ... \times M_n\) be any.
\(\widetilde{m} := (m_0^1, ..., m^{j_1}, ..., m^{j_s}, ..., m_0^n) = (\widetilde{m}^1, ..., \widetilde{m}^n)\).
As \(f\) is \(C^\infty\) at \(\widetilde{m}\), there are a chart around \(\widetilde{m}\), \((U_\widetilde{m} \subseteq M_1 \times ... \times M_n, \phi_\widetilde{m})\), and a chart around \(f (\widetilde{m})\), \((U_{f (\widetilde{m})} \subseteq M, \phi_{f (\widetilde{m})})\), such that \(f (U_\widetilde{m}) \subseteq U_{f (\widetilde{m})}\) and \(\phi_{f (\widetilde{m})} \circ f \circ {\phi_\widetilde{m}}^{-1}: \phi_\widetilde{m} (U_\widetilde{m}) \to \phi_{f (\widetilde{m})} (U_{f (\widetilde{m})})\) is \(C^\infty\) at \(\phi_\widetilde{m} (\widetilde{m})\).
By the definition of finite-product \(C^\infty\) manifold with boundary, \((U_\widetilde{m} \subseteq M_1 \times ... \times M_n, \phi_\widetilde{m})\) can be chosen as \(U_\widetilde{m} = U_{1, \widetilde{m}^1} \times ... \times U_{n, \widetilde{m}^n}\) and \(\phi_\widetilde{m} = \phi_{1, \widetilde{m}^1} \times ... \times \phi_{n, \widetilde{m}^n}\), where \((U_{j, \widetilde{m}^j} \subseteq M_j, \phi_{j, \widetilde{m}^j})\) is a chart for \(M_j\). \(U_\widetilde{m} = U_{1, {m_0}^1} \times ... \times U_{j_1, m^{j_1}} \times ... \times U_{j_s, m^{j_s}} \times ... \times U_{n, {m_0}^n}\) and \(\phi_\widetilde{m} = \phi_{1, {m_0}^1} \times ... \times \phi_{j_1, m^{j_1}} \times ... \times \phi_{j_s, m^{j_s}} \times ... \times \phi_{n, {m_0}^n}\).
Obviously, \({\phi_\widetilde{m}}^{-1} = {\phi_{1, {m_0}^1}}^{-1} \times ... \times {\phi_{j_1, m^{j_1}}}^{-1} \times ... \times {\phi_{j_s, m^{j_s}}}^{-1} \times ... \times {\phi_{n, {m_0}^n}}^{-1}\).
Let us take any chart around \(m\), \((U_m \subseteq M_1 \times ... \times M_n, \phi_m)\), such that \(U_m := U_{1, m^1} \times ... \times U_{j_1, m^{j_1}} \times ... \times U_{j_s, m^{j_s}} \times ... U_{n, m^n}\) and \(\phi_m = \phi_{1, m^1} \times ... \times \phi_{j_1, m^{j_1}} \times ... \times \phi_{j_s, m^{j_s}} \times ... \times \phi_{n, m^n}\), where \((U_{j, m^j} \subseteq M_j, \phi_{j, m^j})\) is a chart for \(M_j\), while \(U_{j_1, m^{j_1}}, ..., U_{j_s, m^{j_s}}\) and \(\phi_{j_1, m^{j_1}}, ..., \phi_{j_s, m^{j_s}}\) are the ones introduced above.
\(f_{S, m_0} (U_m) \subseteq U_{f (\widetilde{m})}\), because for each \(m' = (m'^1, ..., m'^n) \in U_m\), \(f_{S, m_0} (m') = f ((m_0^1, ..., m'^{j_1}, ..., m'^{j_s}, ..., m_0^n))\), but \((m_0^1, ..., m'^{j_1}, ..., m'^{j_s}, ..., m_0^n) \in U_{1, {m_0}^1} \times ... \times U_{j_1, m^{j_1}} \times ... \times U_{j_s, m^{j_s}} \times ... \times U_{n, {m_0}^n} = U_\widetilde{m}\) while \(f (U_\widetilde{m}) \subseteq U_{f (\widetilde{m})}\).
Let us think of \(\phi_{f (\widetilde{m})} \circ f_{S, m_0} \circ {\phi_m}^{-1}: \phi_m (U_m) \to \phi_{f (\widetilde{m})} (U_{f (\widetilde{m})})\).
It is \(\phi_{f (\widetilde{m})} \circ f_{S, m_0} \circ {\phi_{1, m^1}}^{-1} \times ... \times {\phi_{j_1, m^{j_1}}}^{-1} \times ... \times {\phi_{j_s, m^{j_s}}}^{-1} \times ... \times {\phi_{n, m^n}}^{-1}: (x'^1, ..., x'^n) \mapsto \phi_{f (\widetilde{m})} \circ f_{S, m_0} ((m'^1, ..., m'^n)) = \phi_{f (\widetilde{m})} \circ f ((m_0^1, ..., m'^{j_1}, ..., m'^{j_s}, ..., m_0^n))\).
That is in fact same with \(\phi_{f (\widetilde{m})} \circ f \circ {\phi_\widetilde{m}}^{-1}\) where \((\phi_{\widetilde{m}^1} (m_0^1), ..., \widehat{x'^{j_1}}, ..., \widehat{x'^{j_s}}, ..., \phi_{\widetilde{m}^n} (m_0^n))\) is fixed.
As \(\phi_{f (\widetilde{m})} \circ f \circ {\phi_\widetilde{m}}^{-1}\) is \(C^\infty\) at \(\phi_\widetilde{m} (\widetilde{m})\), it is \(C^\infty\) with respect to \((x'^{j_1}, ..., x'^{j_s})\), so, \(\phi_{f (\widetilde{m})} \circ f_{S, m_0} \circ {\phi_m}^{-1}\) is \(C^\infty\) with respect to \((x'^{j_1}, ..., x'^{j_s})\).
\(\phi_{f (\widetilde{m})} \circ f_{S, m_0} \circ {\phi_m}^{-1}\) is constant with respect to \((x'^1, ..., \widehat{x'^{j_1}}, ..., \widehat{x'^{j_s}}, ..., x'^n)\), so, is \(C^\infty\) with respect to \((x'^1, ..., \widehat{x'^{j_1}}, ..., \widehat{x'^{j_s}}, ..., x'^n)\).
So, \(\phi_{f (\widetilde{m})} \circ f_{S, m_0} \circ {\phi_m}^{-1}\) is \(C^\infty\) with respect \(x'^1, ..., x'^n\).
So, \(f_{S, m_0}\) is \(C^\infty\) at \(m\).
