A description/proof of that \(C^\infty\) vectors field is uniquely defined by its \(C^\infty\) metric value functions with all \(C^\infty\) vectors fields
Topics
About: Riemannian manifold
The table of contents of this article
Starting Context
- The reader knows a definition of metric.
- The reader admits the proposition that around any point on any Riemannian manifold, there is a chart whose canonical frame is orthonormal.
- The reader admits the proposition that for any \(C^\infty\) vectors field on any point neighborhood of any \(C^\infty\) manifold, exists a \(C^\infty\) vectors field on the whole manifold that equals the original vectors field 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 Riemannian manifold, the set of any \(C^\infty\) metric value functions with respect to all the vectors fields defines a unique \(C^\infty\) vectors field.
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 Riemannian manifold, \(M\), any \(C^\infty (M)\) linear map, \(f: \mathfrak{X} (M) \rightarrow C^\infty (M)\) defines the unique \(C^\infty\) vectors field, \(V \in \mathfrak{X} (M)\), such that \(\langle V, V' \rangle = f (V')\).
2: Proof
Around any point, \(p \in M\), let us take a chart, \((U_p, \phi_p)\), such that \((\partial_1, \partial_2, . . ., \partial_n)\) is an orthonormal frame, which is possible by the proposition that around any point on any Riemannian manifold, there is a chart whose canonical frame is orthonormal. Let us define \(V^i := f (\partial_i)\) and \(V := V^i \partial_i\). \(V' = V'^i \partial_i\). \(\langle V, V' \rangle = \sum_i V^i V'^i = \sum_i f (\partial_i) V'^i = f (\sum_i V'^i \partial_i) = f (V')\). So, there is such a \(V\) on \(U_p\) that satisfies the condition.
For any \(p' \in U_p\), supposing that there is another \(V\) by another chart that contains \(p'\), taking \(V'\) such that \(V'^i = 1\) and \(V'^j = 0\) for \(j \neq i\) at \(p'\) (there is \(V''\) such that \(V''^i = 1\) and \(V''^j = 0\) for \(j \neq i\) on \(U_p\), and there is such a \(V'\) on \(M\), by the proposition that for any \(C^\infty\) vectors field on any point neighborhood of any \(C^\infty\) manifold, exists a \(C^\infty\) vectors field on the whole manifold that equals the original vectors field on a possibly smaller neighborhood of the point), \(\langle V, V' \rangle \vert_{p'} = V^i (p') = f (V') (p') = f (\partial_i) (p')\), so, that another \(V\) coincides with \(V\) in the previous paragraph on the intersection. So, \(V\) is well-defined on whole \(M\).
As \(V\) is locally a \(C^\infty\) vectors field around any point on \(M\), \(V\) is \(C^\infty\) on whole \(M\).
\(V\) is unique, because in whatever way it is constructed, it has to satisfy \(V^i = f (\partial_i)\) on \(U_p\), so, that \(V\) equals our \(V\) on any \(U_p\), so, on whole \(M\).
