description/proof of that for \(C^\infty\) vectors bundle, union of \(k\)-dimensional vectors subspaces of fibers that allows local \(C^\infty\) frames is \(C^\infty\) vectors subbundle with subspace topology and adopting atlas
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) vectors subbundle of rank \(k\) of \(C^\infty\) vectors bundle of rank \(k'\).
- The reader admits the proposition that for any \(C^\infty\) vectors bundle and any set of local \(C^\infty\) sections over any open domain that is linearly independent, of each point of the domain, there is a possibly smaller open neighborhood over which there is a local \(C^\infty\) frame that contains the restricted sections set.
- The reader admits the proposition that for any \(C^\infty\) vectors bundle, any \(C^\infty\) frame exists over and only over any trivializing open subset.
- The reader admits the proposition that for any \(C^\infty\) vectors bundle, a trivializing open subset is not necessarily a chart open subset, but there is a possibly smaller chart trivializing open subset at each point on any trivializing open subset.
- The reader admits the proposition that for any \(C^\infty\) vectors bundle, the trivialization of any chart trivializing open subset induces the canonical chart map.
- The reader admits the proposition that any subset of any \(C^\infty\) manifold with boundary that satisfies the local-slice condition is an embedded submanifold with boundary of the manifold with boundary with the subspace topology and the adopting atlas.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) vectors bundle, the union of any \(k\)-dimensional vectors subspaces of the fibers that allows any local \(C^\infty\) frames is a \(C^\infty\) vectors subbundle with the subspace topology and the adopting atlas.
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:
\((E', M, \pi')\): \(\in \{\text{ the } C^\infty \text{ vectors bundles of rank } k'\}\)
\(k\): \(\in \mathbb{N} \setminus \{0\}\) such that \(k \le k'\)
\(\{E_m \in \{\text{ the } k \text{ -dimensional vectors subspaces of } \pi'^{-1} (m)\} \vert m \in M\}\)
\(E\): \(= \cup_{m \in M} E_m \subseteq E'\) with the subspace topology
\(\pi\): \(= \pi' \vert_{E}: E \to M\)
//
Statements:
\(\forall m \in M (\exists U_m \subseteq M \in \{\text{ the open neighborhoods of } m\}, \exists s_1, ..., s_k \in \{\text{ the } C^\infty \text{ sections of } \pi' \vert_{\pi'^{-1} (U_m)}\} (\forall m' \in U_m (\{s_1 (m'), ..., s_k (m')\} \in \{\text{ the bases for } E_{m'}\})))\)
\(\implies\)
(
\(E\) with the adopting atlas \(\in \{\text{ the embedded submanifolds with boundary of } E'\}\)
\(\land\)
\((E, M, \pi) \in \{\text{ the } C^\infty \text{ vectors subbundles of } (E', M, \pi')\}\)
)
//
2: Proof
Whole Strategy: Step 1: take a trivializing chart around \(m\), \((V_m \subseteq M, \phi_m)\), such that \(V_m \subseteq U_m\) and a local \(C^\infty\) frame over \(V_m\) on \(E'\), \(\{s_1, ..., s_k, s_{k + 1}, ..., s_{k'}\}\); Step 2: take the corresponding chart for \(E'\), \((\pi'^{-1} (V_m) \subseteq E', \widetilde{\phi_m})\), and see that the chart is an adopted chart for \(E\), so, give \(E\) the adopting atlas making \(E\) an embedded submanifold with boundary of \(E'\); Step 3: see that \((E, M, \pi)\) is a \(C^\infty\) vectors bundle; Step 4: see that \((E, M, \pi)\) is a \(C^\infty\) vectors subbundle of \((E', M, \pi')\).
Step 1:
There are an open neighborhood of \(m\), \(V'_m \subseteq M\), such that \(V'_m \subseteq U_m\) and a local \(C^\infty\) frame over \(V'_m\) on \(E'\), \(\{s_1, ..., s_k, s_{k + 1}, ..., s_{k'}\}\), by the proposition that for any \(C^\infty\) vectors bundle and any set of local \(C^\infty\) sections over any open domain that is linearly independent, of each point of the domain, there is a possibly smaller open neighborhood over which there is a local \(C^\infty\) frame that contains the restricted sections set.
\(V'_m\) is a trivializing open subset for \(E'\), by the proposition that for any \(C^\infty\) vectors bundle, any \(C^\infty\) frame exists over and only over any trivializing open subset, with the trivialization, \(\Phi'_m: \pi'^{-1} (V'_m) \to V'_m \times \mathbb{R}^{k'}, v = s^j s_j \mapsto (\pi' (v), s^1, ..., s^k, s^{k + 1} ..., s^{k'})\).
There is a trivializing chart around \(m\), \((V_m \subseteq M, \phi_m)\), such that \(V_m \subseteq V'_m\), with the trivialization, \(\Phi_m \vert_{\pi'^{-1} (V_m)}: \pi'^{-1} (V_m) \to V_m \times \mathbb{R}^{k'}, v = s^j s_j \mapsto (\pi' (v), s^1, ..., s^k, s^{k + 1} ..., s^{k'})\), by the proposition that for any \(C^\infty\) vectors bundle, a trivializing open subset is not necessarily a chart open subset, but there is a possibly smaller chart trivializing open subset at each point on any trivializing open subset.
Step 2:
Let us take the chart induced by the trivialization, \((\pi'^{-1} (V_m) \subseteq E', \widetilde{\phi_m})\), where \(\widetilde{\phi_m}: \pi'^{-1} (V_m) \to \mathbb{R}^{d + k'} \text{ or } \mathbb{H}^{d + k'}, v \mapsto (\pi_2 (\Phi_m (v)), \phi_m (\pi' (v)))\), which is possible, by the proposition that for any \(C^\infty\) vectors bundle, the trivialization of any chart trivializing open subset induces the canonical chart map.
Let us see that the chart is an adopted chart for \(E\) around any \(e \in E_m\).
It is about that there are a \(J \subseteq \{1, ..., k' + d\}\) and a \(u \in \pi'^{-1} (V_m)\) such that \(\pi'^{-1} (V_m) \cap E = S_{J, u} (\pi'^{-1} (V_m))\).
Let \(J = \{1, ..., k, k' + 1, ..., k' + d\}\) and \(u = e\): \(u \in \pi'^{-1} (V_m)\), because \(e \in E_m \subseteq \pi'^{-1} (V_m)\).
\(\widetilde{\phi_m} (\pi'^{-1} (V_m) \cap E) = \{(v^1, ..., v^k, 0, ..., 0, \phi_m (m')) \vert v^1, ..., v^k \in \mathbb{R}, m' \in V_m\}\).
\(\widetilde{\phi_m} (\pi'^{-1} (V_m)) = \{(v^1, ..., v^k, v^{k + 1}, ..., v^{k'}, \phi_m (m')) \vert v^1, ..., v^{k'} \in \mathbb{R}, m' \in V_m\}\).
\(\widetilde{\phi_m} (u) = (\widetilde{\phi_m} (u)^1, ..., \widetilde{\phi_m} (u)^k, 0, ..., 0, \phi_m (m))\).
\(S_{J, \widetilde{\phi_m} (u)} (\mathbb{R}^{k' + d}) = \{(v^1, ..., v^k, 0, ..., 0, x^1, ..., x^d) \in \mathbb{R}^{k' + d} \vert v^1, ..., v^k \in \mathbb{R}, x^1, ..., x^d \in \mathbb{R}\}\).
So, \(\widetilde{\phi_m} (\pi'^{-1} (V_m)) \cap S_{J, \widetilde{\phi_m} (u)} (\mathbb{R}^{k' + d}) = \{(v^1, ..., v^k, 0, ..., 0, \phi_m (m')) \vert v^1, ..., v^k \in \mathbb{R}, m' \in V_m\}\).
So, \(\widetilde{\phi_m} (\pi'^{-1} (V_m) \cap E) = \widetilde{\phi_m} (\pi'^{-1} (V_m)) \cap S_{J, \widetilde{\phi_m} (u)} (\mathbb{R}^{k' + d})\).
So, \(\pi'^{-1} (V_m) \cap E = S_{J, u} (\pi'^{-1} (V_m))\).
So, \(E\) satisfies the local-slice condition, so, \(E\) with the subspace topology and the adopting atlas, \(\{(\pi'^{-1} (V_m) \cap E = \pi^{-1} (V_m) \subseteq E, \widetilde{\phi_m} = \pi_J \circ \widetilde{\phi_m}' \vert_{\pi^{-1} (V_m)}) \vert m \in M\}\), is an embedded submanifold with boundary of \(E'\), by the proposition that any subset of any \(C^\infty\) manifold with boundary that satisfies the local-slice condition is an embedded submanifold with boundary of the manifold with boundary with the subspace topology and the adopting atlas.
Step 3:
Let us see that \((E, M, \pi)\) is a \(C^\infty\) vectors bundle.
\(\pi\) is \(C^\infty\), because for each \(e \in E\), denoting \(m = \pi (e)\), there are a chart, \((U_m \subseteq M, \phi_m)\), and the corresponding chart, \((\pi^{-1} (V_m) \subseteq E, \widetilde{\phi_m})\), and the components function is \(: (v^1, ..., v^k, \phi_m (m')) \mapsto \phi_m (m')\), which is \(C^\infty\).
For each \(m \in M\), the chart, \((V_m \subseteq M, \phi_m)\), taken in Step 1 is a \(C^\infty\) trivializing open neighborhood of \(m\), because \(: \pi^{-1} (V_m) \to V_m \times \mathbb{R}^k, v \mapsto (\pi (v), \pi_2 (\widetilde{\phi_m} (v)))\) is a \(C^\infty\) trivialization: it is a restriction of the \(C^\infty\) trivialization, \(\Phi_m: \pi'^{-1} (V_m) \to V_m \times \mathbb{R}^{k'}\).
Step 4:
So, \(E\) is an embedded submanifold with boundary of \(E'\), \((E, M, \pi)\) is a \(C^\infty\) vectors bundle, and for each \(m \in M\), \(\pi^{-1} (m) = E_m\) is a \(k\)-dimensional vectors subspace of \(\pi'^{-1} (m)\), so, \((E, M, \pi)\) is a \(C^\infty\) vectors subbundle of \((E', M, \pi')\).
