description/proof of that map from open subset of \(C^\infty\) manifold with boundary onto open subset of Euclidean \(C^\infty\) manifold or closed upper half Euclidean \(C^\infty\) manifold with boundary is chart map iff it is diffeomorphism
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) manifold with boundary.
- The reader knows a definition of diffeomorphism between arbitrary subsets of \(C^\infty\) manifolds with boundary.
- The reader admits the proposition that for any map between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at any point, where \(k\) excludes \(0\) and includes \(\infty\), any pair of domain chart around the point and codomain chart around the corresponding point such that the intersection of the domain chart and the domain is mapped into the codomain chart satisfies the condition of the definition.
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The reader admits
the proposition that for any map between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at any point, where \(k\) includes \(\infty\), the restriction or expansion on any codomain that contains the range is \(C^k\) at the point . - The reader admits the proposition that for any \(C^\infty\) manifold with boundary and its any chart, the restriction of the chart on any open subset domain is a chart.
- The reader admits the proposition that the identity map from any subset of any Euclidean \(C^\infty\) manifold or any closed upper half Euclidean \(C^\infty\) manifold with Boundary into any subset of Euclidean \(C^\infty\) manifold or any closed upper half Euclidean \(C^\infty\) manifold with boundary is \(C^\infty\).
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) manifold with boundary, any map from any open subset of the manifold with boundary onto any open subset of the corresponding-dimensional Euclidean \(C^\infty\) manifold or closed upper half Euclidean \(C^\infty\) manifold with boundary is a chart map if and only if it is a diffeomorphism.
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:
\(M\): \(\in \{\text{ the } d \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\(\mathbb{R}^d\): \(= \text{ the Euclidean } C^\infty \text{ manifold }\)
\(\mathbb{H}^d\): \(= \text{ the closed upper half Euclidean } C^\infty \text{ manifold with boundary }\)
\(U\): \(\in \{\text{ the open subsets of } M\}\)
\(\overline{U}\): \(\in \{\text{ the open subsets of } \mathbb{R}^d \text{ or } \mathbb{H}^d\}\)
\(f\): \(: U \to \overline{U}\)
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Statements:
\(f \in \{\text{ the diffeomorphisms }\}\)
\(\iff\)
\((U \subseteq M, f) \in \{\text{ the charts for } M\}\)
//
2: Proof
Whole Strategy: Step 1: suppose that \(f\) is diffeomorphic, take any chart, \((U' \subseteq M, \phi)\), such that \(U' \cap U \neq \emptyset\), and see that \((U \subseteq M, f)\) and \((U' \subseteq M, \phi)\) are \(C^\infty\) compatible; Step 2: suppose that \((U \subseteq M, f)\) is a chart, and take the chart, \((\overline{U} \subseteq \mathbb{R}^d \text{ or } \mathbb{H}^d, id)\), and see the components functions of \(f\) and \(f^{-1}\) with respect to the charts are \(C^\infty\).
Step 1:
Let us suppose that \(f\) is a diffeomorphism.
\(f\) is a homeomorphism.
Let us take any chart, \((U' \subseteq M, \phi)\), such that \(U' \cap U \neq \emptyset\).
Let us take the chart, \((\overline{U} \subseteq \mathbb{R}^d \text{ or } \mathbb{H}^d, id)\).
\(f (U' \cap U) \subseteq \overline{U}\).
By the proposition that for any map between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at any point, where \(k\) excludes \(0\) and includes \(\infty\), any pair of domain chart around the point and codomain chart around the corresponding point such that the intersection of the domain chart and the domain is mapped into the codomain chart satisfies the condition of the definition, the coordinates function of \(f\), \(id \circ f \circ \phi^{-1} \vert_{\phi (U' \cap U)}: \phi (U' \cap U) \to \overline{U}\), is \(C^\infty\).
But that equals \(f \circ {\phi \vert_{U' \cap U}}^{-1}: \phi (U' \cap U) \to \overline{U}\), whose codomain restriction, \(f \circ {\phi \vert_{U' \cap U}}^{-1}: \phi (U' \cap U) \to f (U' \cap U)\), is the transition map, which is \(C^\infty\), by the proposition that for any map between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at any point, where \(k\) includes \(\infty\), the restriction or expansion on any codomain that contains the range is \(C^k\) at the point.
\(f (U'\cap U) \subseteq f (U)\) is an open subset of \(\overline{U'} = f (U)\) because \(f\) is homeomorphic, and so, \((f (U'\cap U) \subseteq \mathbb{R}^d \text{ or } \mathbb{H}^d, id)\) is a chart for \(\mathbb{R}^d\) or \(\mathbb{H}^d\), by the proposition that for any \(C^\infty\) manifold with boundary and its any chart, the restriction of the chart on any open subset domain is a chart.
\(f^{-1} (f (U'\cap U)) \subseteq U'\).
By the proposition that for any map between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at any point, where \(k\) excludes \(0\) and includes \(\infty\), any pair of domain chart around the point and codomain chart around the corresponding point such that the intersection of the domain chart and the domain is mapped into the codomain chart satisfies the condition of the definition, the coordinates function of \(f^{-1}\), \(\phi \circ f^{-1} \circ id^{-1}: id (f (U'\cap U)) \to \phi (U')\), is \(C^\infty\).
But that equals \(\phi \circ f^{-1}: f (U'\cap U) \to \phi (U')\), whose codomain restriction, \(\phi \circ f^{-1}: f (U'\cap U) \to \phi (U' \cap U)\), is the transition map, which is \(C^\infty\), by the proposition that for any map between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at any point, where \(k\) includes \(\infty\), the restriction or expansion on any codomain that contains the range is \(C^k\) at the point.
That means that \((U \subseteq M, f)\) is \(C^\infty\) compatible with \((U' \subseteq M, \phi)\).
So, \((U \subseteq M, f)\) is in the maximal atlas.
Step 2:
Let us suppose that \((U \subseteq M, f)\) is a chart.
Let us take the chart, \((\overline{U} \subseteq \mathbb{R}^d \text{ or } \mathbb{H}^d, id)\).
The components function of \(f\), \(id \circ f \circ f^{-1}: f (U) \to id (\overline{U})\), is \(C^\infty\), because it is \(id\), by the proposition that the identity map from any subset of any Euclidean \(C^\infty\) manifold or any closed upper half Euclidean \(C^\infty\) manifold with Boundary into any subset of Euclidean \(C^\infty\) manifold or any closed upper half Euclidean \(C^\infty\) manifold with boundary is \(C^\infty\).
The components function of \(f^{-1}\), \(f \circ f^{-1} \circ id^{-1}: id (\overline{U}) \to f (U)\), is \(C^\infty\), because it is \(id\), by the proposition that the identity map from any subset of any Euclidean \(C^\infty\) manifold or any closed upper half Euclidean \(C^\infty\) manifold with Boundary into any subset of Euclidean \(C^\infty\) manifold or any closed upper half Euclidean \(C^\infty\) manifold with boundary is \(C^\infty\).
So, \(f\) is diffeomorphic.
