description/proof of that for map from subset of \(C^\infty\) manifold with boundary into subset of \(C^\infty\) manifold with boundary, map is local diffeomorphism iff for each domain point and its image, there are charts by which coordinates function is diffeomorphism
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
- 4: Note
Starting Context
- The reader knows a definition of map between arbitrary subsets of \(C^\infty\) manifolds with boundary locally diffeomorphic at point.
- The reader knows a definition of subspace topology of subset of topological space.
- 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.
Target Context
- The reader will have a description and a proof of the proposition that for any map from any subset of any \(C^\infty\) manifold with boundary into any subset of any \(C^\infty\) manifold with boundary, the map is a local diffeomorphism if and only if for each domain point and its image, there are some charts by which the coordinates function 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:
\(M_1\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(M_2\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(S_1\): \(\subseteq M_1\)
\(S_2\): \(\subseteq M_2\)
\(f\): \(: S_1 \to S_2\)
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Statements:
\(f \in \{\text{ the local diffeomorphisms }\}\)
\(\iff\)
\(\forall p \in S_1 (\exists (U_p \subseteq M_1, \phi_p) \in \{\text{ the charts of } M_1\}, \exists (U_{f (p)} \subseteq M_2, \phi_{f (p)}) \in \{\text{ the charts of } M_2\} (\phi_{f (p)} \circ f \circ {\phi_p}^{-1} \vert_{\phi_p (U_p \cap S_1)}: \phi_p (U_p \cap S_1) \to \phi_{f (p)} (U_{f (p)} \cap S_2) \in \{\text{ the diffeomorphisms }\}))\)
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2: Natural Language Description
For any \(C^\infty\) manifold with boundary, \(M_1\), any \(C^\infty\) manifold with boundary, \(M_2\), any subset, \(S_1 \subseteq M_1\), any subset, \(S_2 \subseteq M_2\), and any map, \(f: S_1 \to S_2\), \(f\) is a local diffeomorphisms if and only if for each \(p \in S_1\), there are some charts, \((U_p \subseteq M_1, \phi_p)\) and \((U_{f (p)} \subseteq M_2, \phi_{f (p)})\), such that the coordinates function, \(\phi_{f (p)} \circ f \circ {\phi_p}^{-1} \vert_{\phi_p (U_p \cap S_1)}: \phi_p (U_p \cap S_1) \to \phi_{f (p)} (U_{f (p)} \cap S_2)\) is a diffeomorphism.
3: Proof
Whole Strategy: Step 1: suppose that \(f\) is locally diffeomorphic; Step 2: for each \(p \in S_1\), take an open neighborhood of \(p\), \(U'_p \subseteq M_1\), and an open neighborhood of \(f (p)\), \(U'_{f (p)} \subseteq M_2\), that the definition of local diffeomorphism guarantees; Step 3: take a chart, \((U''_p \subseteq M_1, \phi''_p)\), and a chart, \((U''_{f (p)} \subseteq M_2, \phi''_{f (p)})\), that the definition of \(C^\infty\) map at \(p\) guarantees; Step 4: define the chart, \((U_p := U''_p \cap U'_p \subseteq M_1, \phi_p := \phi''_p \vert_{U_p})\) and see that \(f (U_p \cap U'_p \cap S_1) = U'''_{f (p)} \cap U''_{f (p)} \cap U'_{f (p)} \cap S_2\) for an open neighborhood of \(f (p)\) on \(M_2\), \(U'''_{f (p)}\); Step 5: define the chart, \((U_{f (p)} := U'''_{f (p)} \cap U''_{f (p)} \cap U'_{f (p)} \subseteq M_2, \phi_{f (p)} := \phi''_{f (p)} \vert_{U_{f (p)}})\); Step 6: see that \((U_p \subseteq M_1, \phi_p)\) and \((U_{f (p)} \subseteq M_2, \phi_{f (p)})\) satisfy the conditions; Step 7: suppose that there are some charts, \((U_p \subseteq M_1, \phi_p)\) and \((U_{f (p)} \subseteq M_2, \phi_{f (p)})\), that satisfy the conditions; Step 8: see that \(U_p\) and \(U_{f (p)}\) are some open neighborhoods that the definition of local diffeomorphism requires.
Step 1:
Let us suppose that \(f\) is locally diffeomorphic.
Step 2:
For each \(p \in S_1\), let us take an open neighborhood of \(p\), \(U'_p \subseteq M_1\), and an open neighborhood of \(f (p)\), \(U'_{f (p)} \subseteq M_2\), that the definition of local diffeomorphism guarantees. That means that \(f \vert_{U'_p \cap S_1}: U'_p \cap S_1 \to U'_{f (p)} \cap S_2\) is diffeomorphic.
Step 3:
Especially, \(f \vert_{U'_p \cap S_1}\) is \(C^\infty\) at \(p\), so, let us take a chart, \((U''_p \subseteq M_1, \phi''_p)\), and a chart, \((U''_{f (p)} \subseteq M_2, \phi''_{f (p)})\), that the definition of map \(C^\infty\) at \(p\) guarantees. That mean that \(f \vert_{U'_p \cap S_1} (U''_p \cap U'_p \cap S_1) \subseteq U''_{f (p)}\) and \(\phi''_{f (p)} \circ f \vert_{U'_p \cap S_1} \circ {\phi''_p}^{-1} \vert_{\phi''_p (U''_p \cap U'_p \cap S_1)}: \phi''_p (U''_p \cap U'_p \cap S_1) \to \phi''_{f (p)} (U''_{f (p)})\) is \(C ^\infty\) at \(\phi''_p (p)\).
Step 4:
Let us define the chart, \((U_p := U''_p \cap U'_p \subseteq M_1, \phi_p := \phi''_p \vert_{U_p})\).
\(U_p \cap U'_p \cap S_1\) is an open subset of \(U'_p \cap S_1\) by the definition of subspace topology, and as \(f \vert_{U'_p \cap S_1}\) is homeomorphic, \(f \vert_{U'_p \cap S_1} (U_p \cap U'_p \cap S_1)\) is an open subset of \(U'_{f (p)} \cap S_2\), so, \(f \vert_{U'_p \cap S_1} (U_p \cap U'_p \cap S_1) = U'''_{f (p)} \cap U'_{f (p)} \cap S_2\) for an open neighborhood of \(f (p)\), \(U'''_{f (p)} \subseteq M_2\), by the definition of subspace topology.
\(U'''_{f (p)} \cap U'_{f (p)} \cap S_2 = U'''_{f (p)} \cap U''_{f (p)} \cap U'_{f (p)} \cap S_2\), because \(f \vert_{U'_p \cap S_1} (U_p \cap U'_p \cap S_1) = f \vert_{U'_p \cap S_1} (U''_p \cap U'_p \cap U'_p \cap S_1) = f \vert_{U'_p \cap S_1} (U''_p \cap U'_p \cap S_1) \subseteq U''_{f (p)}\), so, \(U'''_{f (p)} \cap U'_{f (p)} \cap S_2 = f \vert_{U'_p \cap S_1} (U_p \cap U'_p \cap S_1) = f \vert_{U'_p \cap S_1} (U_p \cap U'_p \cap S_1) \cap U''_{f (p)} = U'''_{f (p)} \cap U'_{f (p)} \cap S_2 \cap U''_{f (p)} = U'''_{f (p)} \cap U''_{f (p)} \cap U'_{f (p)} \cap S_2\).
Step 5:
Let us define the chart, \((U_{f (p)} := U'''_{f (p)} \cap U''_{f (p)} \cap U'_{f (p)} \subseteq M_2, \phi_{f (p)} := \phi''_{f (p)} \vert_{U_{f (p)}})\).
Step 6:
Let us see that \((U_p \subseteq M_1, \phi_p)\) and \((U_{f (p)} \subseteq M_2, \phi_{f (p)})\) satisfy the conditions of this proposition.
\(f \vert_{U_p \cap S_1}: U_p \cap S_1 \to U_{f (p)} \cap S_2\) is bijective, because while \(f \vert_{U_p \cap S_1}\) is injective as the restriction of the bijective \(f \vert_{U'_p \cap S_1}\), \(f \vert_{U_p \cap S_1} (U_p \cap S_1) = f \vert_{U'_p \cap S_1} (U_p \cap U'_p \cap S_1) = U'''_{f (p)} \cap U''_{f (p)} \cap U'_{f (p)} \cap S_2 = U_{f (p)} \cap S_2\).
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 \((U_p \subseteq M_1, \phi_p)\) and \((U_{f (p)} \subseteq M_2, \phi_{f (p)})\) pair satisfies the condition of the definition of \(C^\infty\)-ness for \(f \vert_{U'_p \cap S_1}\).
So, \(\phi_{f (p)} \circ f \vert_{U'_p \cap S_1} \circ {\phi_p}^{-1} \vert_{\phi_p (U_p \cap S_1)}: \phi_p (U_p \cap S_1) \to \phi_{f (p)} (U_{f (p)}) = \phi_{f (p)} \circ f \vert_{U_p \cap S_1} \circ {\phi_p}^{-1} \vert_{\phi_p (U_p \cap S_1)}\) is \(C^\infty\).
Likewise, the \((U_{f (p)} \subseteq M_2, \phi_{f (p)})\) and \((U_p \subseteq M_1, \phi_p)\) pair satisfies the condition of the definition of \(C^\infty\)-ness for \({f \vert_{U'_p \cap S_1}}^{-1}\).
So, \(\phi_p \circ {f \vert_{U'_p \cap S_1}}^{-1} \circ {\phi_{f (p)}}^{-1} \vert_{\phi_{f (p)} (U_{f (p)} \cap S_2)}: \phi_{f (p)} (U_{f (p)} \cap S_2) \to \phi_p (U_p) = \phi_p \circ {f \vert_{U_p \cap S_1}}^{-1} \circ {\phi_{f (p)}}^{-1} \vert_{\phi_{f (p)} (U_{f (p)} \cap S_2)}\) is \(C^\infty\).
\(\phi_p \circ {f \vert_{U'_p \cap S_1}}^{-1} \circ {\phi_{f (p)}}^{-1} \vert_{\phi_{f (p)} (U_{f (p)} \cap S_2)} = (\phi_{f (p)} \circ f \vert_{U'_p \cap S_1} \circ {\phi_p}^{-1} \vert_{\phi_p (U_p \cap S_1)})^{-1}\).
So, \(\phi_{f (p)} \circ f \circ {\phi_p}^{-1} \vert_{\phi_p (U_p \cap S_1)} = \phi_{f (p)} \circ f \vert_{U'_p \cap S_1} \circ {\phi_p}^{-1} \vert_{\phi_p (U_p \cap S_1)}\) is a diffeomorphism.
Step 7:
Let us suppose that there are some charts, \((U_p \subseteq M_1, \phi_p)\) and \((U_{f (p)} \subseteq M_2, \phi_{f (p)})\), that satisfy the conditions of this proposition.
Step 8:
\(f \vert_{U_p \cap S_1}: U_p \cap S_1 \to U_{f (p)} \cap S_2\) is a diffeomorphism, because it is bijective and for each \(p' \in U_p \cap S_1\), there are the charts pair, \((U_p \subseteq M_1, \phi_p)\) and \((U_{f (p)} \subseteq M_2, \phi_{f (p)})\), which are some charts around \(p'\) and \(f (p')\), that satisfies the conditions for \(f \vert_{U_p \cap S_1}\)' s being \(C^\infty\) at \(p'\) and for \({f \vert_{U_p \cap S_1}}^{-1}\)'s being \(C^\infty\) at \(f (p')\).
So, \(f\) is locally diffeomorphic.
4: Note
\(\phi_{f (p)} \circ f \circ {\phi_p}^{-1} \vert_{\phi_p (U_p \cap S_1)}\)'s being diffeomorphic does not necessarily mean that it has a diffeomorphic extension.
