A description/proof of that map from open subset of \(C^\infty\) manifold onto open subset of Euclidean \(C^\infty\) manifold is chart map iff it is diffeomorphism
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 diffeomorphism.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) manifold, any map from any open subset of the manifold onto any open subset of the corresponding-dimensional Euclidean \(C^\infty\) manifold is a chart map if and only if it 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: Description
For any \(C^\infty\) manifold, \(M\), any open subset, \(U \subseteq M\), any open subset of the corresponding-dimensional Euclidean \(C^\infty\) manifold, \(U' \subseteq \mathbb{R}^d\), and any map, \(f: U \to U'\), \((U, f)\) is a chart if and only if \(f\) is a diffeomorphism.
2: Proof
Let us suppose that \((U, f)\) is a chart. \(f\) is a homeomorphism, by the definition of chart. Around any point, \(p \in U\), there is the chart, \((U, f)\), and around \(f (p) \in U'\), there is the chart on the Euclidean \(C^\infty\) manifold, \((U', id)\) where \(id\) is the identity map, \(id: U' \to U'\). \(id \circ f \circ f^{-1}\) is \(C^\infty\), because it is \(id\), so, \(f\) is \(C^\infty\). \(f \circ f^{-1} \circ id^{-1}\) is \(C^\infty\), because it is \(id\), so, \(f^{-1}\) is \(C^\infty\).
Let us suppose that \(f\) is a diffeomorphism. \(f\) is a homeomorphism. For any chart, \((U'', \phi)\), such that \(U'' \cap U \neq \emptyset\), \(f \circ \phi^{-1}\vert_{\phi (U'' \cap U)}\) is \(C^\infty\) as a composition of \(C^\infty\) maps, and \(\phi \circ f^{-1}\vert_{f (U'' \cap U)}\) is \(C^\infty\) as a composition of \(C^\infty\) maps. So, \((U, f)\) is compatible with \((U'', \phi)\), and \((U, f)\) is included in the maximal atlas.
3: Note
Talking of "corresponding-dimensional", in fact, there cannot be any diffeomorphism onto any open subset of any different-dimensional Euclidean \(C^\infty\) manifold, by the \(C^\infty\) invariance of dimension theorem, so, if there is a diffeomorphism onto an open subset of the Euclidean \(C^\infty\) manifold of a dimension, that dimension is nothing but the corresponding dimension.
