Given a diffeomorphism between two surfaces, is there an expression for the pullback of the covariant derivative of a vector field?

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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