A description/proof of that vectors field is \(C^\infty\) if and only if operation result on any \(C^\infty\) function is \(C^\infty\)
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 chart.
- The reader knows a definition of \(C^\infty\) vectors field.
- The reader admits the proposition that for any \(C^\infty\) function on any point open neighborhood of any \(C^\infty\) manifold, there exists a \(C^\infty\) function on the whole manifold that equals the original function on a possibly smaller neighborhood of the point.
Target Context
- The reader will have a description and a proof of the proposition that any vectors field is \(C^\infty\) if and only if its operation result on any \(C^\infty\) function is \(C^\infty\).
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 vectors field, \(V\), is \(C^\infty\) if and only if its operation result function on any \(C^\infty\) function is \(C^\infty\).
2: Proof
Let us suppose that its operation result function on any \(C^\infty\) function is \(C^\infty\). At any point, \(p \in M\), there is a chart, \((U_p, x^1, . . ., x^n )\). There, \(V = V^j \frac{\partial}{\partial x^j}\). Each coordinate function, \(x^j: U_p \to \mathbb{R}\), is a \(C^\infty\) function on \(U_p\), and there is a \(C^\infty\) function, \(\tilde{x^j}: M \to \mathbb{R}\), on \(M\) that equals \(x^j\) on a possibly smaller open neighborhood, \(U'_p \subseteq U_p\), of \(p\), by the proposition that for any \(C^\infty\) function on any point open neighborhood of any \(C^\infty\) manifold, there exists a \(C^\infty\) function on the whole manifold that equals the original function on a possibly smaller neighborhood of the point. On \(U'_p\), \(V \tilde{x^j} = V^i \frac{\partial \tilde{x^j}}{\partial x^i} = V^j\), \(C^\infty\) by the supposition, and \(V\) is \(C^\infty\) on \(U'_p\). As \(V\) is \(C^\infty\) on a neighborhood of any point on \(M\), \(V\) is \(C^\infty\) on \(M\).
Let us suppose that \(V\) is \(C^\infty\). At any point, \(p \in M\), there is a chart, \((U_p, x^1, . . ., x^n )\). There, \(V = V^j \frac{\partial}{\partial x^j}\) where \(\{V^i\}\) are \(C^\infty\). For any \(C^\infty\) function, \(f\), on \(M\), on \(U_p\), \(V f = V^j \frac{\partial f}{\partial x^j}\) is \(C^\infty\). As \(V f\) is \(C^\infty\) on a neighborhood of any point on \(M\), it is \(C^\infty\) on \(M\).
