A description and a proof of that any vector bundle connection depends only on the section values on any vector curve
Topics
About: differential geometry
About: vector bundle connection
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) vector bundle.
- The reader knows a definition of vector bundle connection.
- The reader knows a definition of trivializing neighborhood of point with respect to vector bundle. The reader knows a definition of \(C^\infty\) section.
- The reader knows a definition of local operator.
- The reader knows a definition of \(C^\infty\) frame.
- The reader admits the proposition that any vector bundle connection (\(\Gamma (TM) \times \Gamma (E) \rightarrow \Gamma (E)\)) is a local operator with respect to its any argument.
- The reader admits the proposition that there is a \(C^\infty\) frame over any trivializing neighborhood of point with respect to any vector bundle.
The reader knows a definition of \(C^\infty\) tangent vector field.
Target Context
- The reader will have a description and a proof of the proposition that any vector bundle connection depends only on the section values on any vector curve.
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 \(C^\infty\) vector bundle, \((E, M, \pi)\), any \(C^\infty\) tangent vector field on M, \(X \in \Gamma (TM)\), any \(C^\infty\) section on E, \(s \in \Gamma (E)\), any \(C^\infty\) section on E, \(t \in \Gamma (E)\), any connection on E, \(\nabla\), and any point on M, \(p \in M\), if there are any neighborhood, \(U_p\), of p on M and any curve on \(U_p\) through p that (the curve) represents X such that \(s = t\) there, then \((\nabla_X s)_p = (\nabla_X t)_p\).
2: Proof
As any connection is a local operator with respect to any argument, its result can be evaluated on a trivializing neighborhood, \(U_p\), of p. There is a \(C^\infty\) frame, {\(e_1, \ldots, e_r\)}, over the trivializing neighborhood of p. Over the trivializing neighborhood of p, \(s = s^ie_i\) and \(t = t^ie_i\), where \(s^i, t^i \in C^\infty\). \((\nabla_Xs)_p = ( (X{s_i})e_i) (p) + (s_i\nabla_Xe_i) (p)\) and \((\nabla_Xt)_p = ( (X{t_i})e_i) (p) + (t_i\nabla_Xe_i) (p)\) by the Leibniz rule, but as X is a directional derivative, \(Xs_i\) or \(Xt_i\) depends only on the values of \(s_i\) or \(t_i\) on any curve that represents X, but as \(s_i = t_i\) on the curve, \(Xs_i = Xt_i\) at p. As also \(s_i (p) = t_i (p)\), \((\nabla_X s)_p = (\nabla_X t)_p\).
