A description/proof of that vectors field on restricted tangent vectors bundle is \(C^\infty\) iff operation result on any \(C^\infty\) function on super manifold is \(C^\infty\) on regular submanifold
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of restricted vectors bundle.
- The reader knows a definition of \(C^\infty\) section.
- 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 on any restricted tangent vectors bundle is \(C^\infty\) iff the operation result on any \(C^\infty\) function on the super manifold is \(C^\infty\) on the regular submanifold.
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 regular submanifold, \(M \subseteq M'\), and any vectors field, \(V: M \to TM'\vert_{M}\), \(V\) is \(C^\infty\) if and only if for any \(C^\infty\) function, \(f \in C^\infty (M')\), \(V f\) is \(C^\infty\) on \(M\).
2: Proof
For any point, \(p \in M\), there is an adopted chart, \((U_p \subseteq M', \phi)\), of \(p\). The corresponding adopting chart is \((U_p \cap M \subseteq M, \pi\circ \phi \vert_{U_p \cap M})\) where \(\pi\) is the projection into the first \(n\) components where \(n\) is the dimension of \(M\).
Let us suppose that \(V f\) is a \(C^\infty\) function. 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 \(U'_p \cap M\), \(V \tilde{x^j} = V^i \frac{\partial \tilde{x^j}}{\partial x^i} = V^j\), \(C^\infty\) on \(U'_p \cap M\) by the supposition, and \(V\) is \(C^\infty\) on \(U'_p \cap M\). 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 \(C^\infty\). \(V = V^j \frac{\partial}{\partial x^j}\) on \(U_p \cap M\) where \(V^j\) is a \(C^\infty\) function on \(U_p \cap M\). \(V f = V^j \frac{\partial f}{\partial x^j}\) is a \(C^\infty\) function on \(U_p \cap M\): \(\frac{\partial f}{\partial x^j}\) is a \(C^\infty\) function on \(U_p\), but on \(U_p \cap M\), \((\frac{\partial f}{\partial x^j}) (x^1, x^2, ..., x^n, x^{n + 1}, x^{n + 2}, ..., x^{n + m}) = (\frac{\partial f}{\partial x^j}) (x^1, x^2, ..., x^n, 0, 0, ..., 0)\), which is \(C^\infty\) on \(U_p \cap M\). As \(V f\) is \(C^\infty\) on a neighborhood of any point on \(M\), it is \(C^\infty\) on \(M\).
3: Note
\(V\) does not operate on any function on \(M\), because \(V (p) \in TpM'\) not in \(TpM\).
