A description/proof of that vectors field along \(C^\infty\) curve is \(C^\infty\) iff 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 \(C^\infty\) vectors field along curve.
- 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 for any \(C^\infty\) manifold and any \(C^\infty\) curve over any open interval, any vectors field along the curve is \(C^\infty\) if and only if its operation result on any \(C^\infty\) function on the manifold 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 open interval, \(I \subseteq \mathbb{R}\), where \(I\) is given the canonical manifold structure, and any \(C^\infty\) curve, \(c: I \to M\), any vectors field along \(c\), \(V: I \to c (I) \to TM\), is \(C^\infty\) if and only if for any \(C^\infty\) function, \(f: M \to \mathbb{R}\), \((V f) \circ c\) is \(C^\infty\) as the map, \(I \to \mathbb{R}\).
2: Proof
Let us suppose that \((V f) \circ c\) is \(C^\infty\).
For any point, \(i_0 \in I\), there is a chart, \((U'_{c (i_0)} \subseteq M, \phi'_{c (i_0)}: p \mapsto (x^1, x^2, ..., x^n))\), around \(c (i_0)\). \(x^k\) is a \(C^\infty\) function on \(U'_{c (i_0)}\), and there is a \(C^\infty\) function on \(M\), \(f_k\), that equals \(x^k\) on a possibly smaller open neighborhood, by the proposition that for any \(C^\infty\) function on any point neighborhood of any \(C^\infty\) manifold, exists a \(C^\infty\) function on the whole manifold that equals the original function on a possibly smaller neighborhood of the point. Let us take the possibly smaller chart, \((U_{c (i_0)} \subseteq U'_{c (i_0)}, \phi_{c (i_0)} = \phi'_{c (i_0)}\vert_{U_{c (i_0)}})\).
There is a chart, \((U_{i_0} \subseteq I, \phi_{i_0})\), such that \(c (U_{i_0}) \subseteq U_{c (i_0)}\), because \(c\) is continuous. Over \(U_{i_0}\), \(V (i) = V^j (i) \partial_j\). \(V (i) f = V^j (i) \partial_j f\). \(f\) can be taken to be \(f_k\), and \(V (i) f_k = V^j (i) \partial_j x^k = V^k (i)\), which really equals \((V f_k) \circ c\), so, is \(C^\infty\) by the supposition. As the components function of \(V: I \to TM\) is \(i \mapsto (c^1 (i), c^2 (i), ..., c^n (i), V^1 (i), V^2 (i), ..., V^n (i))\), which is \(C^\infty\), so, the map is \(C^\infty\).
Let us suppose that the map, \(V: I \to TM\), is \(C^\infty\).
For any point, \(i_0 \in I\), there are a chart, \((U_{c (i_0)} \subseteq M, \phi_{c (i_0)}: p \mapsto (x^1, x^2, ..., x^n))\), around \(c (i_0)\), and a chart, \((U_{i_0} \subseteq I, \phi_{i_0})\), around \(i_0\), and \(V (i) = V^j (i) \partial_j\) where \(V^j (i)\) is \(C^\infty\). For any \(C^\infty\) function, \(f: M \to \mathbb{R}\), over \(U_{i_0}\), \(V (i) f = V^j (i) \partial_j f\) where \(\partial_j f\) is a \(C^\infty\) function on \(M\), so, \((V f) \circ c\) is a \(C^\infty\) function over \(I\).
