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Suppose we are given a vector bundle $E$ over a Riemannian manifold $(M,g)$ ,let $\nabla^{E}$ be a connection in $E$ and $\nabla^{g}$ be the Levi-Civita connection on $M$,

Is there a natural connection in $T^{*}M^{\otimes m}\otimes E$ associated to $\nabla^{g}$ and $\nabla^{E}$?

This relates to the definition of Sobolev spaces on manifolds which I encounterred in L.Nicolaescu's book "Lectures on the Geometry of Manifolds"pp.449.

Any answers,hints or references would be much appreciated!

C Weid
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    Induce a connection inductively. On $E \otimes T^*M$ act on sections $\sigma \otimes \omega$ using the Leibniz rule: $\nabla^E(\sigma) \otimes \omega + \sigma \otimes \nabla^g(\omega)$. –  Aug 06 '16 at 09:08
  • @MikeMiller so the uniform formula of $\nabla$ in $T^{*}M^{\otimes m}\otimes E$ is $$\nabla(\alpha_1\otimes...\otimes\alpha_m\otimes\sigma)=\nabla(\alpha_1)\otimes...\otimes\alpha_m\otimes\sigma+...+\alpha_1\otimes...\otimes\nabla(\alpha_m)otimes\sigma+\alpha_1\otimes...\otimes\alpha_m \otimes\nabla(\sigma)$$? – C Weid Aug 06 '16 at 09:17
  • Yeah, I just didn't want to write that. –  Aug 06 '16 at 09:19
  • @MikeMiller yeah it annoyed... – C Weid Aug 06 '16 at 09:22

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