definition of subset of \(C^\infty\) manifold with boundary that satisfies local-slice condition
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of J-slice of chart domain with respect to point.
Target Context
- The reader will have a definition of subset of \(C^\infty\) manifold with boundary that satisfies local-slice condition.
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:
\( M\): \(\in \{\text{ the } d' \text{ -dimensional } C^\infty \text{ manifolds with boundary } \}\)
\(*S\): \(\subseteq M\)
\( d\): \(\in \mathbb{N}\) such that \(d \le d'\)
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Conditions:
\(\forall s \in S (\exists (U_s \subseteq M, \phi_s) \in \{\text{ the charts around } s \text{ for } M\}, \exists J \subseteq \{1, ..., d'\} = (j_1, ..., j_d), \exists u \in U_s (U_s \cap S = S_{J, u} (U_s)))\)
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2: Note
\((U_s \subseteq M, \phi_s)\) is called "adopted chart".
\((U_s \cap S \subseteq S, \pi_J \circ \phi_s \vert_{U_s \cap S})\) is called "corresponding adopting chart".
\(\{(U_s \cap S \subseteq S, \pi_J \circ \phi_s \vert_{U_s \cap S}) \vert s \in S\}\) is called "adopting atlas for \(S\)".
\(S\) becomes an embedded submanifold with boundary of \(M\) with the subspace topology and the adopting atlas, by the proposition that any subset of any \(C^\infty\) manifold with boundary that satisfies the local-slice condition is an embedded submanifold with boundary of the manifold with boundary with the subspace topology and the adopting atlas.
