description/proof of that for 2 \(C^\infty\) vectors bundles over same \(C^\infty\) manifold with boundary, bijective \(C^\infty\) vectors bundle homomorphism is '\(C^\infty\) vectors bundles - \(C^\infty\) vectors bundle homomorphisms' isomorphism
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 bijection.
- The reader knows a definition of %structure kind name% homomorphism.
- The reader knows a definition of %category name% isomorphism.
- 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 open submanifold with boundary of \(C^\infty\) manifold with boundary.
- The reader admits the proposition that any open subset of any \(C^\infty\) trivializing open subset is a \(C^\infty\) trivializing open subset.
- The reader admits the proposition that for any \(C^\infty\) manifold with boundary and any 2 real finite-dimensional vectors spaces turned \(C^\infty\) manifolds, any \(C^\infty\) bijection from the product of the manifold with boundary and the former vectors space onto the product of the manifold with boundary and the latter vectors space that is 1st-factor-preserving and 1st-factor-fixed-linear is a diffeomorphism.
- The reader admits the proposition that for any map between any embedded submanifolds with boundary of any \(C^\infty\) manifolds with boundary, \(C^k\)-ness does not change when the domain or the codomain is regarded to be the subset.
- The reader admits the proposition that for any maps between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at corresponding points, where \(k\) includes \(\infty\), the composition is \(C^k\) at the point.
- The reader admits the proposition that any bijective linear map is a 'vectors spaces - linear morphisms' isomorphism.
Target Context
- The reader will have a description and a proof of the proposition that for any 2 \(C^\infty\) vectors bundles over any same \(C^\infty\) manifold with boundary, any bijective \(C^\infty\) vectors bundle homomorphism is a '\(C^\infty\) vectors bundles - \(C^\infty\) vectors bundle homomorphisms' isomorphism.
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:
\(M\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\((E, M, \pi)\): \(\in \{\text{ the } C^\infty \text{ vectors bundles } \}\)
\((E', M, \pi')\): \(\in \{\text{ the } C^\infty \text{ vectors bundles } \}\)
\(f\): \(: E \to E'\), \(\in \{\text{ the bijections }\}\), such that \((id: M \to M, f) \in \{\text{ the } C^\infty \text{ vectors bundle homomorphisms }\}\)
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Statements:
\(f \in \{\text{ the ' } C^\infty \text{ vectors bundles - } C^\infty \text{ vectors bundle homomorphisms' isomorphisms }\}\)
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2: Proof
Whole Strategy: see that \((id^{-1}, f^{-1})\) is a \(C^\infty\) vectors bundle homomorphism; Step 1: see that \(\pi \circ f^{-1} = id^{-1} \circ \pi'\); Step 2: see that \(f^{-1}\) is \(C^\infty\); Step 3: for each \(m \in M\), see that \(f^{-1} \vert_{\pi'^{-1} (m)}: \pi'^{-1} (m) \to \pi^{-1} (m)\) is linear; Step 4: conclude the proposition.
Step 1:
As \(f\) is bijective, there is \(f^{-1}\).
Let us see that \(\pi \circ f^{-1} = id^{-1} \circ \pi'\).
\(\pi' \circ f = id \circ \pi\), because \((id: M \to M, f)\) is a \(C^\infty\) vectors bundle homomorphism.
\(id^{-1} \circ \pi' = id^{-1} \circ \pi' \circ f \circ f^{-1} = id^{-1} \circ id \circ \pi \circ f^{-1} = \pi \circ f^{-1}\).
Step 2:
Step 2 Strategy: Step 2-1: around each \(m \in M\), take a trivializing open subset, \(U_m \subseteq M\), a trivialization, \(\Phi_m: \pi^{-1} (U_m) \to U_m \times \mathbb{R}^k\), and a trivialization, \(\Phi'_m: \pi'^{-1} (U_m) \to U_m \times \mathbb{R}^k\); Step 2-2: think of \(\Phi'_m \circ f \circ {\Phi_m}^{-1}\), and see that it is a \(C^\infty\) bijection that is 1st-factor-preserving and 1st-factor-fixed linear; Step 2-3: conclude that \(\Phi'_m \circ f \circ {\Phi_m}^{-1}\) is diffeomorphic; Step 2-4: conclude that \(f\) is diffeomorphic; Step 3: conclude the proposition.
Step 2-1:
Around each \(m \in M\), let us take a trivializing open subset, \(U_m \subseteq M\), a trivialization, \(\Phi_m: \pi^{-1} (U_m) \to U_m \times \mathbb{R}^k\), and a trivialization, \(\Phi'_m: \pi'^{-1} (U_m) \to U_m \times \mathbb{R}^k\), which is possible because while there are a trivializing open subset, \(V_m\), for \(E\) and a trivializing open subset, \(V'_m\), for \(E'\), \(U_m := V_m \cap V'_m\) will do, by the proposition that any open subset of any \(C^\infty\) trivializing open subset is a \(C^\infty\) trivializing open subset.
Step 2-2:
Let us think of \(\Phi'_m \circ f \circ {\Phi_m}^{-1}: U_m \times \mathbb{R}^k \to U_m \times \mathbb{R}^k\).
We are going to see that it is diffeomorphic, using the proposition that for any \(C^\infty\) manifold with boundary and any 2 real finite-dimensional vectors spaces turned \(C^\infty\) manifolds, any \(C^\infty\) bijection from the product of the manifold with boundary and the former vectors space onto the product of the manifold with boundary and the latter vectors space that is 1st-factor-preserving and 1st-factor-fixed-linear is a diffeomorphism. Step 2-2 is about seeing that \(\Phi'_m \circ f \circ {\Phi_m}^{-1}\) satisfies the requirements for the proposition. "1st-factor-preserving" and "1st-factor-fixed-linear" hereafter means what are meant in the proposition.
\(U_m\) is a \(C^\infty\) manifold with boundary, by Note for the definition of open submanifold with boundary of \(C^\infty\) manifold with boundary.
Note the proposition that for any map between any embedded submanifolds with boundary of any \(C^\infty\) manifolds with boundary, \(C^k\)-ness does not change when the domain or the codomain is regarded to be the subset.
\(\Phi'_m \circ f \circ {\Phi_m}^{-1}\) is bijective, because \({\Phi_m}^{-1}: U_m \times \mathbb{R}^k \to \pi^{-1} (U_m)\) is bijective, \(f \vert_{\pi^{-1} (U_m)}: \pi^{-1} (U_m) \to \pi^{-1} (U_m)\) is bijective: \(f\) is fiber-preserving and is 'vectors spaces - linear morphisms' isomorphic (so, bijective) on each fiber, and \(\Phi'_m: \pi^{-1} (U_m) \to U_m \times \mathbb{R}^k\) is bijective.
\(\Phi'_m \circ f \circ {\Phi_m}^{-1}\) is \(C^\infty\), by the proposition that for any maps between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at corresponding points, where \(k\) includes \(\infty\), the composition is \(C^k\) at the point.
\(\Phi'_m \circ f \circ {\Phi_m}^{-1}\) is 1st-factor-preserving, because \({\Phi_m}^{-1}\) maps \(\{p\} \times \mathbb{R}^k\) into \(\pi^{-1} (p)\), \(f\) maps \(\pi^{-1} (p)\) into \(\pi'^{-1} (p)\), and \(\Phi'_m\) maps \(\pi'^{-1} (p)\) into \(\{p\} \times \mathbb{R}^k\).
\(\Phi'_m \circ f \circ {\Phi_m}^{-1}\) is 1st-factor-fixed linear, because \({\Phi_m}^{-1} \vert_{\{p\} \times \mathbb{R}^k}\) is linear into \(\pi^{-1} (p)\), \(f \vert_{\pi^{-1} (p)}\) is linear into \(\pi'^{-1} (p)\), and \(\Phi'_m \vert_{\pi'^{-1} (p)}\) is linear into \(\{p\} \times \mathbb{R}^k\).
Step 2-3:
So, by the proposition that for any \(C^\infty\) manifold with boundary and any 2 real finite-dimensional vectors spaces turned \(C^\infty\) manifolds, any \(C^\infty\) bijection from the product of the manifold with boundary and the former vectors space onto the product of the manifold with boundary and the latter vectors space that is 1st-factor-preserving and 1st-factor-fixed-linear is a diffeomorphism, \(\Phi'_m \circ f \circ {\Phi_m}^{-1}\) is diffeomorphic.
So, \((\Phi'_m \circ f \circ {\Phi_m}^{-1})^{-1} = \Phi_m \circ f^{-1} \circ {\Phi'_m}^{-1}\) is \(C^\infty\).
Step 2-4:
Then, \(f^{-1} = {\Phi_m}^{-1} \circ \Phi_m \circ f^{-1} \circ {\Phi'_m}^{-1} \circ \Phi'_m\) is \(C^\infty\), by the proposition that for any maps between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at corresponding points, where \(k\) includes \(\infty\), the composition is \(C^k\) at the point.
So, \(f\) is a diffeomorphism.
Step 3:
For each \(m \in M\), let us see that \(f^{-1} \vert_{\pi'^{-1} (m)}: \pi'^{-1} (m) \to \pi^{-1} (m)\) is linear.
As \(f\) is fiber-preserving and bijective, \(f^{-1} \vert_{\pi'^{-1} (m)}\) is the inverse of \(f \vert_{\pi^{-1} (m)}: \pi^{-1} (m) \to \pi'^{-1} (m)\), which is linear by the definition of \(C^\infty\) vectors bundle homomorphism. By the proposition that any bijective linear map is a 'vectors spaces - linear morphisms' isomorphism, \(f \vert_{\pi^{-1} (m)}\) is a 'vectors spaces - linear morphisms' isomorphism, and so, \(f^{-1} \vert_{\pi'^{-1} (m)}\) is linear.
Step 4:
So, \((id^{-1}, f^{-1})\) is a \(C^\infty\) vectors bundle homomorphism.
So, \((id, f)\) is a '\(C^\infty\) vectors bundles - \(C^\infty\) vectors bundle homomorphisms' isomorphism.
