A description/proof of that \(C^\infty\) vectors field on regular submanifold is \(C^\infty\) as vectors field along regular submanifold on supermanifold
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) vectors field.
- The reader knows a definition of \(C^\infty\) vectors field along regular submanifold on super \(C^\infty\) manifold.
- The reader admits the proposition that any vectors field is \(C^\infty\) if and only if its components function on any chart is \(C^\infty\).
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) manifold and its any regular submanifold, any \(C^\infty\) vectors field on the regular submanifold is \(C^\infty\) as a vectors field along the regular submanifold on the supermanifold.
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, \(M_1\), and any regular submanifold, \(M_2 \subseteq M_1\), any \(C^\infty\) vectors field, \(V \in \mathfrak{X} (M_2)\), is a \(C^\infty\) vectors field along \(M_2\) on \(M_1\), which is \(\pi_* V \in \Gamma (T M_1\vert M_2)\) where \(\pi: M_2 \rightarrow M_1\) is the inclusion.
2: Proof
There is a adopted chart on \(M_1\) and the corresponding adopting chart on \(M_2\). On the adopting chart, \(V = V^i \partial_i\) where \(0 \leq i \leq d_2\) where \(d_2\) is the dimension of \(M_2\) and \(V^i\) is a \(C^\infty\) function, by the proposition that any vectors field is \(C^\infty\) if and only if its components function on any chart is \(C^\infty\). The vectors field along \(M_2\) on \(M_1\) is the push-forward, \(\pi_* V\), and \(\pi_* V = V^i \partial_i + \sum_j 0 \partial_j\) by the adopted chart where \(d_2 + 1 \leq j \leq d_1\) where \(d_1\) is the dimension of \(M_1\). For any \(C^\infty\) function, \(f \in C^\infty (M_1)\), \((\pi_* V) (f) = V^i \partial_i f\), which is a \(C^\infty\) function on \(M_2\), which means that \(\pi_* V\) is a \(C^\infty\) vectors field along \(M_2\) on \(M_1\) by the definition.
