A description/proof of that for map \(C^\infty\) at point, coordinates function with any charts is \(C^\infty\) at point image
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of map \(C^\infty\) at point.
Target Context
- The reader will have a description and a proof of the proposition that for any map \(C^\infty\) at point between any \(C^\infty\) manifolds, the coordinates function with any charts is \(C^\infty\) at the point image.
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 \(C^\infty\) manifolds, \(M_1, M_2\), any point, \(p \in M_1\), any map \(C^\infty\) at \(p\), \(f: M_1 \to M_2\), and any charts, \((U_1 \subseteq M_1, \phi_1)\) and \((U_2 \subseteq M_2, \phi_2)\), such that \(f (U_1) \subseteq U_2\), the coordinates function with the charts, \(\tilde{f} = \phi_2 \circ f \circ {\phi_1}^{-1}: \phi_1 (U_1) \to \phi_2 (U_2)\) is \(C^\infty\) at \(\phi_1 (p)\).
2: Note
The definition of map \(C^\infty\) at point dictates only the existence of a chart around the point on the domain and a chart around the corresponding point on the codomain for which the coordinates function is \(C^\infty\) at the point image (not necessarily on the whole chart range), and it is not about any charts on the domain and the codomain.
3: Proof
There are a chart, \((U_p \subseteq M_1, \phi_p)\), around \(p\) and a chart, \((U_{f (p)} \subseteq M_2, \phi_{f (p)})\), around \(f (p)\) such that \(f (U_p) \subseteq U_{f (p)}\) and \(\phi_{f (p)} \circ f \circ {\phi_p}^{-1}: \phi_p (U_p) \to \phi_{f (p)} (U_{f (p)})\) is \(C^\infty\) at \(\phi (p)\).
\(\tilde{f} \vert_{\phi_1 (U_1 \cap U_p)} = \phi_2 \circ f \circ {\phi_1}^{-1} \vert_{\phi_1 (U_1 \cap U_p)} = \phi_2 \circ {\phi_{f (p)}}^{-1} \circ \phi_{f (p)} \circ f \circ {\phi_p}^{-1} \circ \phi_p \circ {\phi_1}^{-1} \vert_{\phi_1 (U_1 \cap U_p)}\) is \(C^\infty\) at \(\phi_1 (p)\), because \(\phi_p \circ {\phi_1}^{-1}\) is \(C^\infty\) at \(\phi_1 (p)\), \(\phi_{f (p)} \circ f \circ {\phi_p}^{-1}\) is \(C^\infty\) at \(\phi_p (p)\), and \(\phi_2 \circ {\phi_{f (p)}}^{-1}\) is \(C^\infty\) at \(\phi_{f (p)} (f (p))\).
So, \(\tilde{f}\) is \(C^\infty\) at \(\phi_1 (p) \in \phi_1 (U_1)\).
