A description/proof of that chart on regular submanifold is extension of adapting chart
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) manifold.
- The reader knows a definition of regular submanifold.
- The reader admits the proposition that for any map between \(C^\infty\) manifolds, its continuousness in the topological sense equals its continuousness in the norm sense for the coordinates functions.
- The reader admits the invariance of domain theorem: any injective continuous map from any open set on any Euclidean topological space to any Euclidean topological space is an open map.
Target Context
- The reader will have a description and a proof of the proposition that any chart on any regular submanifold of any \(C^\infty\) manifold is an extension of an adapting chart.
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: Description
For any \(C^\infty\) manifold, M, of dimension \(n\), and any regular submanifold, \(M_1 \subseteq M\), of dimension \(n_1\), any chart around any point, \(p \in M_1\), on \(M_1\), \((U_p, \phi)\) where \(U_p \subseteq M_1\), is an extension of an adopting chart, \((U'_p \cap M_1, \pi \circ \phi'|_{U'_p \cap M_1})\) where \((U'_p, \phi')\) is the adopted chart such that \(U'_p \subseteq M\) and \(\pi\) is the vanishing of the always '0', \(n_1 + 1\) ~ \(n\) coordinates.
2: Proof
Around \(p\), there is an adopting chart, \((U''_p \cap M_1, \pi \circ \phi''|_{U''_p \cap M_1})\) where \((U''_p, \phi'')\) is the adopted chart such that \(U''_p \subseteq M\). As \(U_p\) is open in the subspace topology, \(U_p = U'''_p \cap M_1\) where \(U'''_p \subseteq M\) is open on \(M\). So, \(U_p \cap U''_p \cap M_1 = U'''_p \cap M_1 \cap U''_p \cap M_1 = U''_p \cap U'''_p \cap M_1\), and \((U''_p \cap U'''_p \cap M_1, \pi \circ \phi''|_{U''_p \cap U'''_p \cap M_1})\) is an adopting chart with \((U''_p \cap U'''_p, \phi''|_{U''_p \cap U'''_p})\) as the adopted chart. On the other hand, \((U_p \cap U''_p \cap M_1 = U''_p \cap U'''_p \cap M_1, \phi|_{U''_p \cap U'''_p \cap M_1})\) is a chart on \(M_1\).
There is a diffeomorphism \(f: \pi \circ \phi'' (U''_p \cap U'''_p \cap M_1) \rightarrow \phi (U''_p \cap U'''_p \cap M_1)\), because the 2 charts belong to the same atlas, and we define an injective map, \(f': \phi'' (U''_p \cap U'''_p) \rightarrow \mathbb{R}^n\) , such that the 1 ~ \(n_1\) components are mapped by \(f\) and the remaining components are mapped identically. By the proposition that for any map between \(C^\infty\) manifolds, its continuousness in the topological sense equals its continuousness in the norm sense for the coordinates functions, \(f'\) is continuous, and by the invariance of domain theorem: any injective continuous map from any open set on any Euclidean topological space to any Euclidean topological space is an open map, \(f'\) is an open map, so, \(f' (\phi'' (U''_p \cap U'''_p))\) is open, and \(f': \phi'' (U''_p \cap U'''_p) \rightarrow f' (\phi'' (U''_p \cap U'''_p))\) is a homeomorphism, in fact, obviously, a diffeomorphism. So, \((U''_p \cap U'''_p, f' \circ \phi'')\) is a chart, and \(f' \circ \phi'' (U''_p \cap U'''_p \cap M_1)\) equals \(f' \circ \phi'' (U''_p \cap U'''_p)\) whose \(n_1 + 1\) ~ \(n\) components vanished, because any point on \(f' \circ \phi'' (U''_p \cap U'''_p)\) whose one of the \(n_1 + 1\) ~ \(n\) coordinates is not 0 is not on \(f' \circ \phi'' (U''_p \cap U'''_p \cap M_1)\) and any point on \(f' \circ \phi'' (U''_p \cap U'''_p)\) whose all the \(n_1 + 1\) ~ \(n\) coordinates are 0 is on \(f' \circ \phi'' (U''_p \cap U'''_p \cap M_1)\). So, \((U''_p \cap U'''_p \cap M_1, \pi \circ f' \circ \phi''|_{U''_p \cap U'''_p \cap M_1})\) is the adopting char of the adopted chart, \((U''_p \cap U'''_p, f' \circ \phi''|_{U''_p \cap U'''_p}\), but \(\pi \circ f' \circ \phi''|_{U''_p \cap U'''_p \cap M_1} = \phi|_{U''_p \cap U'''_p \cap M_1} = \phi|_{U_p \cap U''_p \cap M_1}\), of which the chart, \((U_p, \phi)\), is an extension.
