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Suppose that we have two smooth manifolds $M$ and $N$ and a smooth mapping $\phi : M \rightarrow N$. The Differential of that smooth mapping induces a bundle map $D\phi : TM \rightarrow TM$ between the tangential bundles.

We then have the pullback bundle $\phi^\star TN$ over the manifold $M$.

Let's now introduce Riemannian structures. Suppose that $M$ and $N$ are Riemannian manifolds with metrics $g^M$ and $g^N$ and the corresponding Levi-Civita connections $\nabla^M$ and $\nabla^N$.

  1. Does the pullback connection $\phi^\star \nabla^N$ act as a connection on the pullback bundle $TN$?

  2. Does the original connection $\nabla^M$ over $M$ act on the pullback bundle in any discernable way?

  3. What is a natural connection on "mixed" $TM \otimes \phi^\star TN$ over $M$?

I believe the first question has a positive answer but I am in the dark on the second and third one. My motivation is the third one in particular.

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  • $\begingroup$ Here’s a discussion about this. mathoverflow.net/q/49272/613 $\endgroup$
    – Deane Yang
    Commented Dec 27, 2020 at 16:35
  • $\begingroup$ @DeaneYang: thanks. I reformulated my posting to clarify what I am looking for. I don't think this is standard textbook material. $\endgroup$
    – shuhalo
    Commented Dec 27, 2020 at 17:00
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    $\begingroup$ I agree this is not standard textbook material, even though it should be. My answer on the page I cited is a terse summary of the situation for the pullback bundle and connection. This shows that the connection on $M$ plays no role in the definition of the connection on $\phi^*TN$. As for the tensor product, this is a separate independent step. If you have two vector bundles with connections, then the tensor product has a natural connection given by the product rule: $\nabla_X (\sigma\otimes\tau)=(\nabla_X \sigma)\otimes\tau+\sigma\otimes(\nabla_X\tau)$. $\endgroup$
    – Deane Yang
    Commented Dec 27, 2020 at 17:56
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    $\begingroup$ Yes, the former for $M$ and the latter for $N$. They could be any connections on $TM\to M$ and $TN\to N$, it does not matter if they are metric-compatible or torsion-free. $\endgroup$
    – Quarto Bendir
    Commented Dec 30, 2020 at 14:59
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    $\begingroup$ I don't know of anywhere where I find it comprehensible. It is essentially inside of volume 1 of Kobayashi and Nomizu's "Foundations of Differential Geometry". You could also check Eells and Sampson's article "Harmonic mappings of Riemannian manifolds" and Eells and Lemaire "A report on harmonic maps" and "Selected topics in harmonic maps", maybe Schoen and Yau "Lectures on Harmonic Maps" $\endgroup$
    – Quarto Bendir
    Commented Dec 30, 2020 at 18:21

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