description/proof of that for \(C^\infty\) vectors bundle, \(C^\infty\) section along closed subset of base space can be extended to over whole base space with support contained in any open neighborhood of subset
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
Starting Context
- The reader knows a definition of \(C^\infty\) vectors bundle.
- The reader knows a definition of section along subset of codomain of continuous surjection.
- The reader knows a definition of closed set.
- The reader knows a definition of neighborhood of subset.
- The reader knows a definition of support of map from topological space into field.
- The reader admits the proposition that for any \(C^\infty\) vectors bundle and any section from any subset of the base space \(C^k\) at any point where \(0 \lt k\), there is a \(C^k\) extension on an open-neighborhood-of-point domain.
- 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 on any domain that contains the point is \(C^k\) at the point.
- The reader admits the proposition that for any open cover of any \(C^\infty\) manifold, there is a \(C^\infty\) partition of unity subordinate to the cover.
- The reader admits the proposition that for any locally finite set of subsets of any topological space, the closure of the union of the subsets is the union of the closures of the subsets.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) vectors bundle, any \(C^\infty\) section along any closed subset of the base space can be extended to over the whole base space with the support contained in any open neighborhood of the subset.
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:
\((E, M, \pi)\): \(\in \{\text{ the } d \text{ -dimensional } C^\infty \text{ vectors bundles of rank } k\}\)
\(C\): \(\in \{\text{ the closed subsets of } M\}\)
\(U_C\): \(\in \{\text{ the open neighborhoods of } C \text{ on } M\}\)
\(s\): \(: C \subseteq M \to E\), \(\in \{\text{ the } C^\infty \text{ sections along } C\}\)
//
Statements:
\(\exists s': M \to E \in \{\text{ the } C^\infty \text{ sections of } \pi\} (s' \vert_C = s \land supp \text{ } s' \subseteq U_C)\)
//
2: Natural Language Description
For any \(d\)-dimensional \(C^\infty\) vectors bundle of rank \(k\), \((E, M, \pi)\), any closed subset, \(C \subseteq M\), and any open neighborhood of \(C\), \(U_C \subseteq M\), any \(C^\infty\) section along \(C\), \(s: C \to E\), can be extended to a \(C^\infty\) section of \(\pi\), \(s': M \to E\), such that \(s' \vert_C = s\) and \(supp \text{ } s' \subseteq U_C\).
3: Proof
Whole Strategy: Step 1: for each point, \(p \in C\), take an open neighborhood of \(p\), \(U_p \subseteq M\), and a \(C^\infty\) section, \(s'_p: U_p \to E\), such that \(U_p \subseteq U_C\) and \(s'_p \vert_{U_p \cap C} = s \vert_{C \cap U_p}\); Step 2: take the open cover of \(M\), \(\{U_p: p \in C\} \cup \{M \setminus C\}\), and a partition of unity subordinate to the cover, \(\{\rho_p \vert p \in C\} \cup \{\rho_0\}\); Step 3: define \(s''_p: M \to E\) as \(\rho_p s'_p\) on \(U_p\) and \(0\) on \(M \setminus U_p\); Step 4: define \(s' := \sum_{p \in C} s''_p\); Step 5: see that \(s'\) satisfies the conditions.
Step 1:
For each point, \(p \in C\), there are an open neighborhood of \(p\), \(U_p \subseteq M\), and a \(C^\infty\) section, \(s'_p: U_p \to E\), such that \(s'_p \vert_{U_p \cap C} = s \vert_{C \cap U_p}\), by the proposition that for any \(C^\infty\) vectors bundle and any section from any subset of the base space \(C^k\) at any point where \(0 \lt k\), there is a \(C^k\) extension on an open-neighborhood-of-point domain.
\(U_p\) can be taken to be such that \(U_p \subseteq U_C\): if not so, we can take \(U_p \cap U_C\) instead of \(U_p\), and \(C^\infty\)-ness is not harmed by it, 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 on any domain that contains the point is \(C^k\) at the point. So, let us suppose that \(U_p \subseteq U_C\).
Step 2:
\(\{U_p: p \in C\} \cup \{M \setminus C\}\) is an open cover of \(M\), and there is a partition of unity subordinate to the cover, \(\{\rho_p \vert p \in C\} \cup \{\rho_0\}\), by the proposition that for any open cover of any \(C^\infty\) manifold, there is a \(C^\infty\) partition of unity subordinate to the cover.
Step 3:
Let us define \(s''_p: M \to E\) as \(\rho_p s'_p\) on \(U_p\) and \(0\) on \(M \setminus U_p\).
\(s''_p\) is \(C^\infty\), because it is \(C^\infty\) on \(U_p\) and is \(0\) on open \(M \setminus supp \text{ } \rho_p\), so, \(C^\infty\) there, and \(M = U_p \cup (M \setminus supp \text{ } \rho_p)\) as \(supp \text{ } \rho_p \subseteq U_p\).
Step 4:
Let us define \(s' := \sum_{p \in C} s''_p\), which is well-defined, because at any point, \(p' \in M\), only some finite terms are nonzero, by the definition of partition of unity.
Step 5:
\(s'\) is \(C^\infty\), because for any point, \(p' \in M\), there is an open neighborhood, \(U'_{p'} \subseteq M\), on which the sum is a finite sum, which is a sum of some \(C^\infty\) sections.
For each point, \(p' \in C\), \(s' (p') = \sum_{p \in C'} \rho_p (p') s'_p (p')\) for a finite \(C' \subseteq C\), but \(s'_p (p') = s (p')\), so, \(= \sum_{p \in C'} \rho_p (p') s (p') = (\sum_{p \in C} \rho_p (p') + \rho_0 (p')) s (p') = s (p')\), because \(\rho_p (p')\) for \(p \in C \setminus C'\) and \(\rho_0 (p')\) are \(0\) anyway. So, \(s'\) is indeed a \(C^\infty\) extension of \(s\).
\(supp \text{ } s' \subseteq \overline{\cup_{p \in C} supp \text{ } \rho_p} = \cup_{p \in C} \overline{supp \text{ } \rho_p} = \cup_{p \in C} supp \text{ } \rho_p \subseteq \cup_{p \in C} U_p \subseteq U_C\), by the proposition that for any locally finite set of subsets of any topological space, the closure of the union of the subsets is the union of the closures of the subsets.
