definition of \(C^\infty\) trivializing open subset and \(C^\infty\) local trivialization
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- 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 trivializing open subset and local trivialization.
- The reader knows a definition of diffeomorphism between arbitrary subsets of \(C^\infty\) manifolds with boundary.
- The reader knows a definition of Euclidean \(C^\infty\) manifold.
- The reader knows a definition of finite-product \(C^\infty\) manifold with boundary.
Target Context
- The reader will have a definition of \(C^\infty\) trivializing open subset and \(C^\infty\) local trivialization.
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:
\( M\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\( E\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\( k\): \(\in \mathbb{N} \setminus \{0\}\)
\( \pi\): \(: E \to M\), \(\in \{\text{ the } C^\infty \text{ maps }\}\)
\(*U\): \(\in \{\text{ the trivializing open subsets of } M\}\)
\(*\Phi\): \(: \pi^{-1} (U) \to U \times \mathbb{R}^k\), \(\in \{\text{ the local trivializations }\}\)
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Conditions:
\(\Phi \in \{\text{ the diffeomorphisms }\}\)
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\(\mathbb{R}^k\) and \(U \times \mathbb{R}^k\) are the Euclidean \(C^\infty\) manifold and the product \(C^\infty\) manifold with boundary.
