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Let $A$ and $B$ be two surfaces (smooth enough) in an affine space $M$ with metric $g$. Let $g^A$, $g^B$ be the metric tensors on the two surfaces induced by $g$, and $\nabla^A$, $\nabla^B$ the Levi-Civita connections on the two surfaces. Let $f:A\rightarrow B$ be a diffeomorphism.

I'm wondering if, given a vector field $X^B$ on $B$, there is an expression for $f^* (\nabla^{B} X^B)$. I think it should be something including $\nabla^A(f^*X^B)$, but I can't find it.

Thanks for your help.

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Differentials pull back, vector fields push forward. So no, unless you use $f^{-1}$ (which is perfectly defined, and smooth). – geodude Apr 22 '14 at 15:27
Yes, I'm using $f^{-1}$, in order to extend the pullback definition to a (1,1) tensor field. – Gaff Apr 22 '14 at 15:32
You could express it as $\nabla^{f^*B} F^* X^B$, i.e. the covariant derivative with respect to the pulled-back connection. – Ryan Budney Apr 22 '14 at 16:02
Actually I was thinking to something including the connection $\nabla^A$ already defined, maybe using the fact that both $g^A$ and $g^B$ are induced by the same metric. – Gaff Apr 22 '14 at 18:18
@Gaff: The fact that they are both induced by the same ambient metric is irrelevant: Any two Riemannian manifolds embed isometrically in some Euclidean space. – studiosus Apr 23 '14 at 0:07

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