404: Riemannian Bundle Has Compatible Connection

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A description/proof of that Riemannian bundle has compatible connection

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Topics

About: $C^{\mathrm{\infty }}$ manifold

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof

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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 $C^{\mathrm{\infty }}$ vectors bundle on any trivializing open set has an orthonormal frame.
• The reader admits the proposition that for any $C^{\mathrm{\infty }}$ vectors bundle over any $C^{\mathrm{\infty }}$ 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.

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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.

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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.

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Main Body

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1: Description

For any $C^{\mathrm{\infty }}$ manifold, $M$, and any Riemannian bundle, $\pi :E\to M$, there is a connection, $\mathrm{\nabla }$, on $E$ compatible with the metric.

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2: Proof

There is a trivializing open cover of $M$ for $E$, $\{U_{\alpha }|\alpha \in A\}$ where $A$ is a possibly uncountable indices set. For each ${\pi }^{-1}(U_{\alpha })$, there is an orthonormal $C^{\mathrm{\infty }}$ frame, $e_1,e_2,...,e_d$, by the proposition that the restriction of any $C^{\mathrm{\infty }}$ vectors bundle on any trivializing open set has an orthonormal frame. Any $C^{\mathrm{\infty }}$ sections, $s_1,s_2\in \mathrm{\Gamma }(E)$, are $s_j={s_j}^i{e}_i$ on ${\pi }^{-1}(U_{\alpha })$ where ${s_j}^i$ is a $C^{\mathrm{\infty }}$ function on $U_{\alpha }$. For any $C^{\mathrm{\infty }}$ vectors field, $V\in \mathrm{\Gamma }(TM)$, on $U_{\alpha }$, $V⟨s_1,s_2⟩=V(\sum _i{s_1}^i{s_2}^i)=\sum _i((V{s_1}^i){s_2}^i+{s_1}^{i}V{s_2}^i)$.

Let us define a connection on ${\pi }^{-1}(U_{\alpha })$, ${\mathrm{\nabla }}_{\alpha }$, as ${{\mathrm{\nabla }}_{\alpha }}_{V}s=V{s}^i{e}_i$. Let us confirm that ${\mathrm{\nabla }}_{\alpha }$ is really a connection. $V{s}^i{e}_i\in \mathrm{\Gamma }({\pi }^{-1}(U_{\alpha }))$, because $V{s}^i$ is a $C^{\mathrm{\infty }}$ function on $U_{\alpha }$. ${{\mathrm{\nabla }}_{\alpha }}_{V}s$ is $\mathbb{R}$ linear with respect to $s$, because ${{\mathrm{\nabla }}_{\alpha }}_V(rs)=V(r{s}^i)e_i=rV{s}^i{e}_i=r{{\mathrm{\nabla }}_{\alpha }}_{V}s$. ${{\mathrm{\nabla }}_{\alpha }}_{V}s$ is $C^{\mathrm{\infty }}(U_{\alpha })$ linear with respect to $V$, because ${{\mathrm{\nabla }}_{\alpha }}_{fV}s=fV{s}^i{e}_i=f{{\mathrm{\nabla }}_{\alpha }}_{V}s$. ${{\mathrm{\nabla }}_{\alpha }}_{V}s$ satisfies the Leibniz rule, ${{\mathrm{\nabla }}_{\alpha }}_V(fs)=(Vf)s+f{{\mathrm{\nabla }}_{\alpha }}_{V}s$, because ${{\mathrm{\nabla }}_{\alpha }}_V(fs)=V(f{s}^i)e_i=(Vf)s^i{e}_i+f(V{s}^i)e_i=(Vf)s+f{{\mathrm{\nabla }}_{\alpha }}_{V}s$. So, ${\mathrm{\nabla }}_{\alpha }$ is a connection.

Let us confirm that ${\mathrm{\nabla }}_{\alpha }$ is compatible with the metric on ${\pi }^{-1}(U_{\alpha })$. $V⟨s_1,s_2⟩=⟨{{\mathrm{\nabla }}_{\alpha }}_V{s}_1,s_2⟩+⟨s_1,{{\mathrm{\nabla }}_{\alpha }}_V{s}_2⟩$? $⟨{{\mathrm{\nabla }}_{\alpha }}_V{s}_1,s_2⟩+⟨s_1,{{\mathrm{\nabla }}_{\alpha }}_V{s}_2⟩=⟨V{s_1}^i{e}_i,{s_2}^i{e}_i⟩+⟨{s_1}^i{e}_i,V{s_2}^i{e}_i⟩=\sum _i((V{s_1}^i){s_2}^i+{s_1}^{i}V{s_2}^i)$, which equals $V⟨s_1,s_2⟩$, 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 $E$, $\mathrm{\nabla }$, as $\sum _{\alpha }{\rho }_{\alpha }{\mathrm{\nabla }}_{\alpha }$, which means that ${\mathrm{\nabla }}_{V}s=\sum _{\alpha }{\rho }_{\alpha }{{\mathrm{\nabla }}_{\alpha }}_{V}s$. $\mathrm{\nabla }$ is really a connection, by the proposition that for any $C^{\mathrm{\infty }}$ vectors bundle over any $C^{\mathrm{\infty }}$ 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 $\mathrm{\nabla }$ is compatible with the metric. $V⟨s_1,s_2⟩=⟨{{\mathrm{\nabla }}_{\alpha }}_V{s}_1,s_2⟩+⟨s_1,{{\mathrm{\nabla }}_{\alpha }}_V{s}_2⟩$ on each ${\pi }^{-1}(U_{\alpha })$. $\sum _{\alpha }({\rho }_{\alpha }V⟨s_1,s_2⟩)=V⟨s_1,s_2⟩=\sum _{\alpha }({\rho }_{\alpha }(⟨{{\mathrm{\nabla }}_{\alpha }}_V{s}_1,s_2⟩+⟨s_1,{{\mathrm{\nabla }}_{\alpha }}_V{s}_2⟩))=\sum _{\alpha }(⟨{\rho }_{\alpha }{{\mathrm{\nabla }}_{\alpha }}_V{s}_1,s_2⟩+⟨s_1,{\rho }_{\alpha }{{\mathrm{\nabla }}_{\alpha }}_V{s}_2⟩)=⟨\sum _{\alpha }({\rho }_{\alpha }{{\mathrm{\nabla }}_{\alpha }}_V{s}_1),s_2⟩+⟨s_1,\sum _{\alpha }({\rho }_{\alpha }{{\mathrm{\nabla }}_{\alpha }}_V{s}_2)⟩=⟨(\sum _{\alpha }{\rho }_{\alpha }{{\mathrm{\nabla }}_{\alpha }}_V)s_1,s_2⟩+⟨s_1,(\sum _{\alpha }{\rho }_{\alpha }{{\mathrm{\nabla }}_{\alpha }}_V)s_2⟩=⟨{\mathrm{\nabla }}_V{s}_1,s_2⟩+⟨s_1,{\mathrm{\nabla }}_V{s}_2⟩$.

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References

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