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 \(C^\infty\) 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.
- The reader admits the proposition that for any map between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at any point, where \(k\) includes \(\infty\), the restriction on any domain that contains the point is \(C^k\) at the point.
-
The reader admits
the proposition that any vectors field is \(C^\infty\) if and only if its operation result on any \(C^\infty\) function is \(C^\infty\) .
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) manifold with boundary and any \(C^\infty\) curve over any 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 with boundary 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: Structured Description
Here is the rules of Structured Description.
Entities:
\(M\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(\mathbb{R}\): \(= \text{ the Euclidean } C^\infty \text{ manifold }\)
\(J\): \(= (t_1, t_2), [t_1, t_2], (t_1, t_2], \text{ or } [t_1, t_2) \subseteq \mathbb{R}\) such that \(t_1 \lt t_2\), as the embedded submanifold with boundary of \(\mathbb{R}\)
\(\gamma\): \(: J \to M\), \(\in \{\text{ the curves }\} \cap \{\text{ the } C^\infty \text{ maps }\}\)
\(V\): \(: J \to \gamma (J) \to TM\), \(\in \{\text{ the vectors fields along } \gamma\}\)
Statements:
\(V \in \{\text{ the } C^\infty \text{ maps }\}\)
\(\iff\)
\(\forall f \in C^\infty (M) ((V f) \circ \gamma: J \to \mathbb{R} \in \{\text{ the } C^\infty \text{ maps }\} )\)
2: Proof
Whole Strategy: Step 1: suppose that \((V f) \circ \gamma\) is \(C^\infty\), and see that \(V\) is \(C^\infty\) by seeing its components function with respect to some appropriate charts; Step 2: suppose that \(V\) is \(C^\infty\), and see that \((V f) \circ \gamma\) is \(C^\infty\) by calculating it with respect to some appropriate charts.
Step 1:
Let us suppose that \((V f) \circ \gamma\) is \(C^\infty\).
For any point, \(j_0 \in J\), there is a chart, \((U'_{\gamma (j_0)} \subseteq M, \phi'_{\gamma (j_0)}: p \mapsto (x^1, x^2, ..., x^n))\), around \(\gamma (j_0)\).
\(x^k\) is a \(C^\infty\) function on \(U'_{\gamma (j_0)}\), 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, \((U_{\gamma (j_0)} \subseteq M, \phi_{\gamma (j_0)} = \phi'_{\gamma (j_0)}\vert_{U_{\gamma (j_0)}})\) such that \(U_{\gamma (j_0)} \subseteq U'_{\gamma (j_0)}\).
There is a chart, \((U_{j_0} \subseteq J, id_{j_0})\), such that \(\gamma (U_{j_0}) \subseteq U_{\gamma (j_0)}\), because \(\gamma\) is continuous.
Over \(U_{j_0}\), \(V (j) = V^l (j) \partial / \partial x^l\).
\(V (j) f_k = V^l (j) \partial / \partial x^l f_k = V^l (j) \partial / \partial x^l x^k = V^l (j) \delta^k_l = V^k (j)\), which really equals \((V f_k) \circ \gamma \vert_{U_{j_0}}\), which is \(C^\infty\) by the supposition: the proposition that for any map between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at any point, where \(k\) includes \(\infty\), the restriction on any domain that contains the point is \(C^k\) at the point.
There is the induced chart, \((\pi^{-1} (U_{\gamma (j_0)}) \subseteq TM, \widetilde{\phi_{\gamma (j_0)}})\).
The components function of \(V: J \to TM\) with respect \((U_{j_0} \subseteq J, id_{j_0})\) and \((\pi^{-1} (U_{\gamma (j_0)}) \subseteq TM, \widetilde{\phi_{\gamma (j_0)}})\) is \(: U_{j_0} \to \mathbb{R}^n \times \phi_{\gamma (j_0)} (U_{\gamma (j_0)}), j \mapsto (V^1 (j), ..., V^n (j), \gamma^1 (j), ..., \gamma^n (j))\), which is \(C^\infty\) because \(V^k\) is \(C^\infty\) and \(\gamma\) is \(C^\infty\).
So, \(V\) is \(C^\infty\).
Step 2:
Let us suppose that the map, \(V: J \to TM\), is \(C^\infty\).
For any point, \(j_0 \in J\), there are a chart, \((U_{\gamma (j_0)} \subseteq M, \phi_{\gamma (j_0)}: p \mapsto (x^1, x^2, ..., x^n))\), around \(\gamma (j_0)\), and a chart, \((U_{j_0} \subseteq J, id_{j_0})\), around \(j_0\) such that \(\gamma (U_{j_0}) \subseteq U_{\gamma (j_0)}\), because \(\gamma\) is continuous.
There is the induced chart, \((\pi^{-1} (U_{\gamma (j_0)}) \subseteq TM, \widetilde{\phi_{\gamma (j_0)}})\).
The components function of \(V: J \to TM\) with respect \((U_{j_0} \subseteq J, id_{j_0})\) and \((\pi^{-1} (U_{\gamma (j_0)}) \subseteq TM, \widetilde{\phi_{\gamma (j_0)}})\) is \(: U_{j_0} \to \mathbb{R}^n \times \phi_{\gamma (j_0)} (U_{\gamma (j_0)}), j \mapsto (V^1 (j), ..., V^n (j), \gamma^1 (j), ..., \gamma^n (j))\), which is \(C^\infty\) because \(V\) is \(C^\infty\), so, \(V^l (j)\) is \(C^\infty\).
\(V (j) = V^l (j) \partial / \partial x^l\).
For any \(C^\infty\) function, \(f: M \to \mathbb{R}\), over \(U_{j_0}\), \(V (j) f = V^l (j) \partial / \partial x^l f\), where \(\partial / \partial x^l f\) is a \(C^\infty\) function on \(U_{\gamma (j_0)}\), by
As \((V f) \circ \gamma\) is \(C^\infty\) on an open neighborhood of each point on \(J\), \((V f) \circ \gamma\) is \(C^\infty\) on \(J\).
