A description/proof of that for \(C^\infty\) vectors bundle, global connection can be constructed with local connections over open cover, using partition of unity subordinate to open cover
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) vectors bundle.
- The reader knows a definition of connection on \(C^\infty\) vectors bundle.
- The reader knows a definition of restricted \(C^\infty\) vectors bundle.
- The reader knows a definition of partition of unity subordinate to open cover.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) vectors bundle over any \(C^\infty\) manifold, a global connection can be constructed with any local connections over any open cover using any partition of unity subordinate to the open cover.
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 \(C^\infty\) vectors bundle, \(\pi: E \rightarrow M\), any open cover of \(M\), \(\{U_\alpha\vert \alpha \in A\}\), where \(A\) is any possible uncountable indices set, any local connection, \(\nabla_\alpha\), on each restricted \(C^\infty\) vectors bundle, \(E\vert_{U_\alpha}\), and any partition of unity subordinate to the open cover, \(\{\rho_\alpha\vert \alpha \in A\}\), \(\nabla := \sum_\alpha \rho_\alpha \nabla_\alpha\), which means that \(\nabla_V s = \sum_\alpha \rho_\alpha {\nabla_\alpha}_V s\vert_{U_\alpha}\) is a connection on \(E\).
2: Proof
Around each point, \(p \in M\), there is an open neighborhood, \(U_p\), that intersects only finite elements of \(\{supp \rho_\alpha\vert \alpha \in A\}\), and on \(U_p\), \(\sum_\alpha \rho_\alpha {\nabla_\alpha}_V s\vert_{U_\alpha} = \sum_i \rho_i {\nabla_i}_V s\vert_{U_i}\) where \(i \in I\) where \(I\) is a finite indices set.
\(\nabla_V s\) is a \(C^\infty\) section on \(E\), because on each \(U_p\), \(\sum_i \rho_i {\nabla_i}_V s\vert_{U_i}\) is \(C^\infty\). \(\nabla_V s\) is \(\mathbb{R}\)-linear with respect to \(s\), because on each \(U_p\), \(\nabla_V (r s) = \sum_i \rho_i {\nabla_i}_V (r s\vert_{U_i}) = r \sum_i \rho_i {\nabla_i}_V s\vert_{U_i} = r \nabla_V s\). \(\nabla_V s\) is \(C^\infty (M)\)-linear with respect to \(V\), because on each \(U_p\), \(\nabla_{f V} s = \sum_i \rho_i {\nabla_i}_{f\vert_{U_i} V} s\vert_{U_i} = \sum_i \rho_i f\vert_{U_i} {\nabla_i}_V s\vert_{U_i} = f\vert_{U_i} \sum_i \rho_i {\nabla_i}_V s\vert_{U_i} = f \nabla_V s\). \(\nabla_V s\) satisfies the Leibniz rule, because on each \(U_p\), \(\nabla_V (f s) = \sum_i \rho_i {\nabla_i}_V (f\vert_{U_i} s\vert_{U_i}) = \sum_i \rho_i ((V f\vert_{U_i}) s\vert_{U_i} + f\vert_{U_i} {\nabla_i}_V s\vert_{U_i}) = (V f\vert_{U_p}) s\vert_{U_p} + f\vert_{U_p} \sum_i (\rho_i {\nabla_i}_V s\vert_{U_i}) = (V f) s + f \nabla_V s\).
To elaborate more on the evaluations of a construct like \(\sum_i \rho_i x_i\), let us evaluate it at each point, \(p' \in U_p\), as \(\sum_i \rho_i (p') x_i (p')\); let us factor \(x_i (p')\) as \(x (p') x'_i (p')\) where \(x (p)\) does not really depend on \(i\); \(\sum_i \rho_i (p') x_i (p') = x (p') \sum_i \rho_i (p') x'_i (p')\). In fact, in the previous \(\sum_i \rho_i (V f\vert_{U_i}) s\vert_{U_i}\), \(x_i (p') = (V f\vert_{U_i}) (p') s\vert_{U_i} (p') = x (p')\), which does not really depend on \(i\), so, \(\sum_i \rho_i (p') x_i (p') = x (p') \sum_i \rho_i (p') = x (p')\), so, \(\sum_i \rho_i (V f\vert_{U_i}) s\vert_{U_i} (p') = (V f\vert_{U_p}) s\vert_{U_p} (p')\). In the previous \(\sum_i \rho_i f\vert_{U_i} {\nabla_i}_V s\vert_{U_i}\), \(x_i (p') = f\vert_{U_i} (p') {\nabla_i}_V s\vert_{U_i} (p') = x (p') x'_i (p')\) where \(x (p') = f\vert_{U_i} (p')\), which does not really depend on \(i\), and \(x'_i (p') = {\nabla_i}_V s\vert_{U_i} (p')\), so, \(\sum_i \rho_i (p') x_i (p') = x (p') \sum_i \rho_i {\nabla_i}_V s\vert_{U_i} (p') = f\vert_{U_p} (p') \nabla\vert_V s (p')\), which is what we did.
3: Note
The open cover is usually a trivializing open cover (because any trivializing open cover allows a set of local connections), but is not necessarily so.
