A description/proof of that Riemannian bundle has compatible connection
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of Riemannian bundle.
- The reader knows a definition of connection compatible with metric.
- The reader admits the proposition that the restriction of any \(C^\infty\) vectors bundle on any trivializing open set has an orthonormal frame.
- The reader admits 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.
Target Context
- The reader will have a description and a proof of the proposition that on any Riemannian bundle, there is a connection compatible with the metric.
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, \(M\), and any Riemannian bundle, \(\pi: E \to M\), there is a connection, \(\nabla\), on \(E\) compatible with the metric.
2: Proof
There is a trivializing open cover of \(M\) for \(E\), \(\{U_\alpha\vert \alpha \in A\}\) where \(A\) is a possibly uncountable indices set. For each \(\pi^{-1} (U_\alpha)\), there is an orthonormal \(C^\infty\) frame, \(e_1, e_2, ..., e_d\), by the proposition that the restriction of any \(C^\infty\) vectors bundle on any trivializing open set has an orthonormal frame. Any \(C^\infty\) sections, \(s_1, s_2 \in \Gamma (E)\), are \(s_j = {s_j}^i e_i\) on \(\pi^{-1} (U_\alpha)\) where \({s_j}^i\) is a \(C^\infty\) function on \(U_\alpha\). For any \(C^\infty\) vectors field, \(V \in \Gamma (TM)\), on \(U_\alpha\), \(V \langle s_1, s_2 \rangle = V (\sum_i {s_1}^i {s_2}^i) = \sum_i ((V {s_1}^i) {s_2}^i + {s_1}^i V {s_2}^i)\).
Let us define a connection on \(\pi^{-1} (U_\alpha)\), \(\nabla_\alpha\), as \({\nabla_\alpha}_V s = V s^i e_i\). Let us confirm that \(\nabla_\alpha\) is really a connection. \(V s^i e_i \in \Gamma (\pi^{-1} (U_\alpha))\), because \(V s^i\) is a \(C^\infty\) function on \(U_\alpha\). \({\nabla_\alpha}_V s\) is \(\mathbb{R}\) linear with respect to \(s\), because \({\nabla_\alpha}_V (r s) = V (r s^i) e_i = r V s^i e_i = r {\nabla_\alpha}_V s\). \({\nabla_\alpha}_V s\) is \(C^\infty (U_\alpha)\) linear with respect to \(V\), because \({\nabla_\alpha}_{f V} s = f V s^i e_i = f {\nabla_\alpha}_V s\). \({\nabla_\alpha}_V s\) satisfies the Leibniz rule, \({\nabla_\alpha}_V (fs) = (V f) s + f {\nabla_\alpha}_V s\), because \({\nabla_\alpha}_V (fs) = V (f s^i) e_i = (V f) s^i e_i + f (V s^i) e_i = (V f) s + f {\nabla_\alpha}_V s\). So, \(\nabla_\alpha\) is a connection.
Let us confirm that \(\nabla_\alpha\) is compatible with the metric on \(\pi^{-1} (U_\alpha)\). \(V \langle s_1, s_2 \rangle = \langle {\nabla_\alpha}_V s_1, s_2 \rangle + \langle s_1, {\nabla_\alpha}_V s_2\rangle\)? \(\langle {\nabla_\alpha}_V s_1, s_2 \rangle + \langle s_1, {\nabla_\alpha}_V s_2\rangle = \langle V {s_1}^i e_i, {s_2}^i e_i \rangle + \langle {s_1}^i e_i, V {s_2}^i e_i\rangle = \sum_i ((V {s_1}^i) {s_2}^i + {s_1}^i V {s_2}^i)\), which equals \(V \langle s_1, s_2 \rangle\), which has been expanded in the 1st paragraph.
Let us take a partition of unity, \(\rho_\alpha\), subordinate to the trivializing open cover, \(\{U_\alpha\}\). Let us define the connection on \(E\), \(\nabla\), as \(\sum_\alpha \rho_\alpha \nabla_\alpha\), which means that \(\nabla_V s = \sum_\alpha \rho_\alpha {\nabla_\alpha}_V s\). \(\nabla\) is really a connection, by 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.
Let us confirm that \(\nabla\) is compatible with the metric. \(V \langle s_1, s_2 \rangle = \langle {\nabla_\alpha}_V s_1, s_2 \rangle + \langle s_1, {\nabla_\alpha}_V s_2\rangle\) on each \(\pi^{-1} (U_\alpha)\). \(\sum_\alpha (\rho_\alpha V \langle s_1, s_2 \rangle) = V \langle s_1, s_2 \rangle = \sum_\alpha (\rho_\alpha (\langle {\nabla_\alpha}_V s_1, s_2 \rangle + \langle s_1, {\nabla_\alpha}_V s_2\rangle)) = \sum_\alpha ( \langle \rho_\alpha {\nabla_\alpha}_V s_1, s_2 \rangle + \langle s_1, \rho_\alpha {\nabla_\alpha}_V s_2\rangle) = \langle \sum_\alpha (\rho_\alpha {\nabla_\alpha}_V s_1), s_2 \rangle + \langle s_1, \sum_\alpha (\rho_\alpha {\nabla_\alpha}_V s_2) \rangle = \langle (\sum_\alpha \rho_\alpha {\nabla_\alpha}_V) s_1, s_2 \rangle + \langle s_1, (\sum_\alpha \rho_\alpha {\nabla_\alpha}_V) s_2 \rangle = \langle \nabla_V s_1, s_2 \rangle + \langle s_1, \nabla_V s_2 \rangle\).
