description/proof of that for \(C^\infty\) map from finite-product \(C^\infty\) manifold with boundary, induced map with set of components of domain fixed is \(C^\infty\)
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: Proof
Starting Context
- The reader knows a definition of finite-product \(C^\infty\) manifold with boundary.
- The reader knows a definition of \(C^k\) map between arbitrary subsets of \(C^\infty\) manifolds with boundary, where \(k\) includes \(\infty\).
- The reader knows a definition of product topology.
- The reader admits the proposition that for any Euclidean topological space, any lower-dimensional Euclidean topological space, the slicing map, the projection, and the inclusion, the inclusion after the projection after the slicing map equals the slicing map, and the projection after the slicing map of any open neighborhood of any point is an open neighborhood of the projection of the point.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) map from any finite-product \(C^\infty\) manifold with boundary, the induced map with any set of components of the domain fixed 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:
\(\{M_1, ..., M_{n - 1}\}\): \(\subseteq \{\text{ the } C^\infty \text{ manifolds }\}\)
\(M_n\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(M'_1\): \(= M_1 \times ... \times M_n = \text{ the product } C^\infty \text{ manifold with boundary }\)
\(d'\): \(= dim M'_1\)
\(M'_2\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(d'_2\): \(= dim M'_2\)
\(f'\): \(: M'_1 \to M'_2\), \(\in \{\text{ the } C^\infty \text{ maps }\}\)
\(J''\): \(\subset \{1, ..., n\}\)
\(M''_1\): \(= \times_{j \in J''} M_j\)
\(d''\): \(= dim M''_1\)
\(m_j\): \(\in M_j\) for each \(j \in \{1, ..., n\} \setminus J''\)
\(f''\): \(: M''_1 \to M'_2, m'' \mapsto f' (m'_1, ..., m'_n)\), where \(m'_j\) is the \(M_j\) component of \(m''\) for \(j \in J''\) and \(m'_j\) is \(m_j\) for \(j \in \{1, ..., n\} \setminus J''\)
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Statements:
\(f'' \in \{\text{ the } C^\infty maps\}\)
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2: Natural Language Description
For any \(C^\infty\) manifolds, \(M_1, ..., M_{n - 1}\), any \(C^\infty\) manifold with boundary, \(M_n\), the product \(C^\infty\) manifold with boundary, \(M'_1 = M_1 \times ... \times M_n\), \(d' = dim M'_1\), any \(C^\infty\) manifold with boundary, \(M'_2\), \(d'_2 = dim M'_2\), any \(C^\infty\) map, \(f': M'_1 \to M'_2\), any subset, \(J'' \subset \{1, ..., n\}\), the product \(C^\infty\) manifold with boundary, \(M''_1 := \times_{j \in J''} M_j\), \(d'' = dim M''_1\), any element, \(m_j \in M_j\), for each \(j \in \{1, ..., n\} \setminus J''\), and the map, \(f'': M''_1 \to M'_2, m'' \mapsto f' (m'_1, ..., m'_n)\), where \(m'_j\) is the \(M_j\) component of \(m''\) for \(j \in J''\) and \(m'_j\) is \(m_j\) for \(j \in \{1, ..., n\} \setminus J''\), \(f''\) is a \(C^\infty\) map.
3: Proof
Whole Strategy: Step 1: define \(\pi_{J''}: M'_1 \to M''_1\), which takes the \(J''\) components, and \(\tau_{J''}: M''_1 \to M'_1\), which adds the \(\{1, ..., n\} \setminus J''\) components as \(m_j\) s; for each \(K'' \subset \{1, ..., d'\}\) such that \(\vert K'' \vert = d''\) and \(r' \in \mathbb{R}^{d'}\), define \(\lambda_{K'', r'}: Pow (\mathbb{R}^{d'}) \to Pow (\mathbb{R}^{d'}), S \mapsto \{s \in S \vert \forall j \in \{1, ..., d'\} \setminus K'' (s^j = r'^j)\}\), \(\pi_{K''}: \mathbb{R}^{d'} \to \mathbb{R}^{d''}\), which takes the \(K''\) components, and \(\tau_{K'', r'}: \mathbb{R}^{d''} \to \mathbb{R}^{d'}\), which adds the \(\{1, ..., d'\} \setminus K''\) components as those of \(r'\); Step 2: for each \(m' = (m'_1, ..., m'_n) \in M'_1\), choose a chart, \((U'_{m'} = U_{m'_1} \times ... \times U_{m'_n} \subseteq M'_1, \phi'_{m'} = \phi_{m'_1} \times ... \times \phi_{m'_n})\), and a chart, \((U'_{f' (m')} \subseteq M'_2, \phi'_{f' (m')})\), such that \(f' (U'_{m'}) \subseteq U'_{f' (m')}\); Step 3: see that for any \(m'' \in M''_1\), \(m' := \tau_{J''} (m'')\), and the chart, \((U''_{m''} = \times_{j \in J''} U_{m'_j} \subseteq M''_1, \phi''_{m''} = \times_{j \in J''} \phi_{m'_j})\), \(\phi'_{f' (m')} \circ f'' \circ {\phi''_{m''}}^{-1}: \phi''_{m''} (U''_{m''}) \to \phi'_{m'} (U'_{f' (m')})\) is \(C^\infty\).
Step 1:
As a preparation, let us define some maps, which are used frequently later.
Let us define \(\pi_{J''}: M'_1 \to M''_1\), which takes the \(J''\) components.
Let us define \(\tau_{J''}: M''_1 \to M'_1\), which adds the \(\{1, ..., n\} \setminus J''\) components as \(m_j\) s.
As \(M'_1\) is \(d'\)-dimensional, the codomain of each chart map is in \(\mathbb{R}^{d'}\), while as \(M''_1\) is \(d''\)-dimensional, the codomain of each chart map is in \(\mathbb{R}^{d''}\). Let \(K'' \subset \{1, ..., d'\}\) be the indexes set that corresponds to \(\mathbb{R}^{d''}\): for example, when \(dim M_1 = 1, dim M_2 = 2, dim M_3 = 3\) and \(J'' = \{1, 3\}\), \(d' = 6\) and \(K'' = \{1, 4, 5, 6\}\) and \(d'' = 4\): \(2, 3\) are missing because \(M_2\) is left out.
Let \(r' \in \mathbb{R}^{d'}\) be any.
Let us define \(\lambda_{K'', r'}: Pow (\mathbb{R}^{d'}) \to Pow (\mathbb{R}^{d'}), S \mapsto \{s \in S \vert \forall j \in \{1, ..., d'\} \setminus K'' (s^j = r'^j)\}\), which chooses points whose \(\{1, ..., d'\} \setminus K''\) components equal those of \(r'\).
Let us define \(\pi_{K''}: \mathbb{R}^{d'} \to \mathbb{R}^{d''}\), which takes the \(K''\) components.
Let us define \(\tau_{K'', r'}: \mathbb{R}^{d''} \to \mathbb{R}^{d'}\), which adds the \(\{1, ..., d'\} \setminus K''\) components as those of \(r'\).
Step 2:
Let \(m' = (m'_1, ..., m'_n) \in M'_1\) be any such that \(m'_j = m_j\) for each \(j \in \{1, ..., n\} \setminus J''\).
As \(f'\) is \(C^\infty\), there are a chart, \((U'_{m'} = U_{m'_1} \times ... \times U_{m'_n} \subseteq M'_1, \phi'_{m'} = \phi_{m'_1} \times ... \times \phi_{m'_n})\), and a chart, \((U'_{f' (m')} \subseteq M'_2, \phi'_{f' (m')})\), such that \(f' (U'_{m'}) \subseteq U'_{f' (m')}\) and \(\phi'_{f' (m')} \circ f' \circ {\phi'_{m'}}^{-1}: \phi'_{m'} (U'_{m'}) \to \phi'_{f' (m')} (U'_{f' (m')})\) is \(C^\infty\) at \(\phi'_{m'} (m')\), by the definition of \(C^k\) map between arbitrary subsets of \(C^\infty\) manifolds with boundary, where \(k\) includes \(\infty\). Note that the chart for \(M'_1\) can be taken like that as a basic open set, because the basic open sets generates the product topology by the definition of product topology.
That means that there are an open neighborhood of \(\phi'_{m'} (m')\), \(U'_{\phi'_{m'} (m')} \subseteq \mathbb{R}^{d'}\), and a \(C^\infty\) map, \(g': U'_{\phi'_{m'} (m')} \to \mathbb{R}^{d'_2}\), such that \(g' \vert_{U'_{\phi'_{m'} (m')} \cap \phi'_{m'} (U'_{m'})} = \phi'_{f' (m')} \circ f' \circ {\phi'_{m'}}^{-1} \vert_{U'_{\phi'_{m'} (m')} \cap \phi'_{m'} (U'_{m'})}\).
Step 3:
For each \(m'' \in M''_1\), there is the \(m' := \tau_{J''} (m'') \in M'_1\). \(m'' = \pi_{J''} (m')\).
\((U''_{m''} = \times_{j \in J''} U_{m'_j} \subseteq M''_1, \phi''_{m''} = \times_{j \in J''} \phi_{m'_j})\) is a chart around \(m''\) for \(M''_1\).
\(\tau_{J''} (U''_{m''}) \subseteq U'_{m'}\), because for each \(j \in \{1, ..., n\} \setminus J''\), \(m'_j \in U_{m'_j}\).
\(\tau_{K'', \phi'_{m'} (m')} (\phi''_{m''} (U''_{m''})) \subseteq \phi'_{m'} (U'_{m'})\), because \(\phi'_{m'} (\tau_{J''} (U''_{m''})) \subseteq \phi'_{m'} (U'_{m'})\), but \(\phi'_{m'} (\tau_{J''} (U''_{m''})) = \tau_{K'', \phi'_{m'} (m')} (\phi''_{m''} (U''_{m''}))\), because for each \(p \in U''_{m''}\), \(\tau_{J''}\) adds the \(\{1, ..., n\} \setminus J''\) components as \(m_j\) s and \(\phi'_{m'}\) maps the \(\{1, ..., n\} \setminus J''\) components to the \(\{1, ..., d'\} \setminus K''\) components as \(\phi_{m'_j} (m_j)\) s, while \(\tau_{K'', \phi'_{m'} (m')}\) adds the \(\{1, ..., d'\} \setminus K''\) components as \(\phi_{m'_j} (m_j)\) s, and \(\phi'_{m'} \circ \tau_{J''}\) and \(\tau_{K'', \phi'_{m'} (m')} \circ \phi''_{m''}\) map the \(J''\) components the same way.
\(f'' (U''_{m''}) = f' (\tau_{J''} (U''_{m''})) \subseteq f' (U'_{m'}) \subseteq U'_{f' (m')}\).
So, we can take the chart, \((U'_{f' (m')} \subseteq M'_2, \phi'_{f' (m')})\), in order to check that \(f''\) is \(C^\infty\).
What we need to show is that \(\phi'_{f' (m')} \circ f'' \circ {\phi''_{m''}}^{-1}: \phi''_{m''} (U''_{m''}) \to \phi'_{m'} (U'_{f' (m')})\) is \(C^\infty\) at \(\phi''_{m''} (m'')\), which is about choosing an open neighborhood of \(\phi''_{m''} (m'')\), \(U''_{\phi''_{m''} (m'')} \subseteq \mathbb{R}^{d''}\) and a \(C^\infty\) map, \(g'': U''_{\phi''_{m''} (m'')} \to \mathbb{R}^{d'_2}\), such that \(g'' \vert_{U''_{\phi''_{m''} (m'')} \cap \phi''_{m''} (U''_{m''})} = \phi'_{f' (m')} \circ f'' \circ {\phi''_{m''}}^{-1} \vert_{U''_{\phi''_{m''} (m'')} \cap \phi''_{m''} (U''_{m''})}\).
We choose \(U''_{\phi''_{m''} (m'')} = \pi_{K''} \circ \lambda_{K'', \phi'_{m'} (m')} (U'_{\phi'_{m'} (m')})\) and \(g'' = g' \circ \tau_{K'', \phi'_{m'} (m')}\).
\(\pi_{K''} (\phi'_{m'} (m')) = \phi''_{m''} (m'')\), because \(m''\) is the \(J''\) components of \(m'\) and \(\phi''_{m''} (m'')\) is the \(K''\) components of \(\phi'_{m'} (m')\), \(U''_{\phi''_{m''} (m'')}\) is indeed an open neighborhood of \(\phi''_{m''} (m'')\) on \(\mathbb{R}^{d''}\), and \(g''\) is valid, by the proposition that for any Euclidean topological space, any lower-dimensional Euclidean topological space, the slicing map, the projection, and the inclusion, the inclusion after the projection after the slicing map equals the slicing map, and the projection after the slicing map of any open neighborhood of any point is an open neighborhood of the projection of the point: \(\tau_{K'', \phi'_{m'} (m')} (U''_{\phi''_{m''} (m'')}) \subseteq U'_{\phi'_{m'} (m')}\), because \(\tau_{K'', \phi'_{m'} (m')} \circ \pi_{K''} \circ \lambda_{K'', \phi'_{m'} (m')} (U'_{\phi'_{m'} (m')}) = \lambda_{K'', \phi'_{m'} (m')} (U'_{\phi'_{m'} (m')}) \subseteq U'_{\phi'_{m'} (m')}\).
We will check that they satisfy the conditions.
\(g' \circ \tau_{K'', \phi'_{m'} (m')}\) is \(C^\infty\), because \(\tau_{K'', \phi'_{m'} (m')}\) just adds some constant components and \(g'\) is \(C^\infty\).
\(g'' \vert_{U''_{\phi''_{m''} (m'')} \cap \phi''_{m''} (U''_{m''})} = g' \circ \tau_{K'', \phi'_{m'} (m')} \vert_{U''_{\phi''_{m''} (m'')} \cap \phi''_{m''} (U''_{m''})}\), but for each \(p \in U''_{\phi''_{m''} (m'')} \cap \phi''_{m''} (U''_{m''})\), \(g' \circ \tau_{K'', \phi'_{m'} (m')} (p) = \phi'_{f' (m')} \circ f' \circ {\phi'_{m'}}^{-1} (\tau_{K'', \phi'_{m'} (m')} (p))\): note that \(\tau_{K'', \phi'_{m'} (m')} (p) \in U'_{\phi'_{m'} (m')} \cap \phi'_{m'} (U'_{m'})\) (has been shown in some earlier paragraphs as \(\tau_{K'', \phi'_{m'} (m')} (\phi''_{m''} (U''_{m''})) \subseteq \phi'_{m'} (U'_{m'})\) and \(\tau_{K'', \phi'_{m'} (m')} (U''_{\phi''_{m''} (m'')}) \subseteq U'_{\phi'_{m'} (m')}\)), \(= \phi'_{f' (m')} \circ f'' \circ {\phi''_{m''}}^{-1} (p)\), because \(\tau_{K'', \phi'_{m'} (m')}\) adds the \(\{1, ..., d'\} \setminus K''\) components as those of \(\phi'_{m'} (m')\), which become the \(\{1, ..., n\} \setminus J''\) components as \(m_j\) s under \({\phi'_{m'}}^{-1}\), and \(f'\) on it equals \(f''\).
So, they satisfy the conditions.
