description/proof of that \(C^\infty\) local section of \(C^\infty\) vectors bundle is \(C^\infty\) embedding
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) vectors bundle of rank \(k\).
- The reader knows a definition of open submanifold with boundary of \(C^\infty\) manifold with boundary.
- The reader knows a definition of section of continuous surjection.
- The reader knows a definition of \(C^\infty\) embedding.
- 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 for any \(C^\infty\) map between any \(C^\infty\) manifolds with boundary and any corresponding charts, the components function of the differential of the map with respect to the standard bases is this.
- The reader admits the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
Target Context
- The reader will have a description and a proof of the proposition that any \(C^\infty\) local section of any \(C^\infty\) vectors bundle is a \(C^\infty\) embedding.
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:
\(k\): \(\in \mathbb{N} \setminus \{0\}\)
\((E, M, \pi)\): \(\in \{\text{ the } C^\infty \text{ vectors bundles of rank } k\}\)
\(U\): \(\in \{\text{ the open embedded submanifolds with boundary of } M\}\)
\(s\): \(: U \to E\), \(\in \{\text{ the } C^\infty \text{ local sections of } \pi\}\)
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Statements:
\(s \in \{\text{ the } C^\infty \text{ embeddings }\}\)
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2: Proof
Whole Strategy: Step 1: see that \(s\) is a \(C^\infty\) injective immersion; Step 2: see that the codomain restriction of \(s\) is a homeomorphism; Step 3: conclude the proposition.
Step 1:
\(s\) is an injection, because for each \(u_1, u_2 \in U\) such that \(u_1 \neq u_2\), \(s (u_1) \neq s (u_2)\), because if \(s (u_1) = s (u_2)\), \(u_1 = \pi \circ s (u_1) = \pi \circ s (u_2) = u_2\), a contradiction against \(u_1 \neq u_2\).
Let us see that \(s\) is a \(C^\infty\) immersion.
\(s\) is \(C^\infty\), by the supposition.
Let \(u \in U\) be any.
Let us take any trivializing chart, \((U_u \subseteq U, \phi_u)\), with a trivialization, \(\Phi: \pi^{-1} (U_u) \to U_u \times \mathbb{R}^k\), 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: take a trivializing open neighborhood of \(u\), \(U'_u \subseteq M\), such that \(U'_u \subseteq U\), and take \(U_u\) such that \(U_u \subseteq U'_u\).
By the proposition that for any \(C^\infty\) vectors bundle, the trivialization of any chart trivializing open subset induces the canonical chart map, there is the canonical chart, \((\pi^{-1} (U_u) \subseteq E, \widetilde{\phi_u})\), where \(\widetilde{\phi_u}: \pi^{-1} (U_u) \to U_u \times \mathbb{R}^k \to \mathbb{R}^{d + k} \text{ or } \mathbb{H}^{d + k}, v \mapsto (\pi_2 (\Phi (v)), \phi_u (\pi (v)))\), where \(\pi_2: U_u \times \mathbb{R}^k \to \mathbb{R}^k\) is the projection.
Let us think of the differential of \(s\) at \(u\), \(d s_u: T_uU \to T_{s (u)}E\).
With the standard bases for \(T_uU\) and \(T_{s (u)}E\) with respect to the charts, the components function of \(d s_u\) is \((v^j) \mapsto (\partial_l \hat{s}^j v^l)\) where \(\hat{s} = \widetilde{\phi_u} \circ s \circ {\phi_u}^{-1}: \phi_u (U_u) \subseteq \mathbb{R}^d \to \widetilde{\phi_u} (\pi^{-1} (U_u)) \subseteq \mathbb{R}^{d + k} \text{ or } \mathbb{H}^{d + k}\) is the components function of \(s\), by the proposition that for any \(C^\infty\) map between any \(C^\infty\) manifolds with boundary and any corresponding charts, the components function of the differential of the map with respect to the standard bases is this.
For each \(j \in \{k + 1, ..., k + d\}\), \(\hat{s}^j: (x^1, ..., x^d) \mapsto x^{j - k}\), so, \(\partial_l \hat{s}^j = \delta^{j - k}_l\), so, \(\partial_l \hat{s}^j v^l = \delta^{j - k}_l v^l = v^{j - k}\).
So, for each \(v, v' \in T_uU\) such that \(v \neq v'\), \(v^l \neq v'^l\) for an \(l \in \{1, ..., d\}\), so, \(\partial_l \hat{s}^j v^l = v^{j - k} \neq v'^{j - k} = \partial_l \hat{s}^j v'^l\) for \(j = l + k\).
So, \(d s_u (v) \neq d s_u (v')\).
So, \(d s_u\) is injective, so, \(s\) is a \(C^\infty\) immersion.
Step 2:
Let us see that the codomain restriction of \(s\), \(s': U \to s (U)\), is a homeomorphism.
\(s\) is continuous, because \(s\) is \(C^\infty\), and \(s'\) is continuous, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
As \(s\) is injective, there is the inverse, \(s'^{-1}: s (U) \to U\).
\(s'^{-1}\) is nothing but the restriction of \(\pi\), \(\pi \vert_{s (U)}: s (U) \to U\).
\(\pi\) is continuous, and \(\pi \vert_{s (U)}\) is continuous, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
So, \(s'^{-1}\) is continuous.
So, \(s'\) is a homeomorphism.
Step 3:
So, \(s\) is a \(C^\infty\) embedding.
