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I was looking for an explicit formula for the codifferential of a 2-form $\omega=\omega_{ij}dx^i\wedge dx^j$ but I found two expressions that look different. One from Jost's book Riemannian Geometry and Geometric Analysis according to which $$ (d^{*}\omega)_{j}=-g^{kl}\left(\partial_{l}\omega_{kj}-\Gamma_{kl}^{s}\omega_{sj}\right) $$ and the other here in Stackexchange reading $$ (d^{*}\omega)_{j}=-g^{kl}\nabla_k\omega_{lj} $$ Being $\nabla_k$ the covariant derivative. Explicitly $$ (d^{*}\omega)_{j}=-g^{kl}\left(\partial_{l}\omega_{kj}-\Gamma_{kl}^{s}\omega_{sj}-\Gamma^s_{kj}\omega_{ls}\right) $$ I tried but I can't tell which of the two formulas is correct. Can anybody point out a reference where the validity of the correct expression is proved?

Thanks in advance.

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    $\begingroup$ Without looking at the two books, I'm guessing this is simply a difference of convention for the summation on the indices of the 2-form. Afterall, $dx^i\wedge dx^j=-dx^j\wedge dx^i$, meaning there could be two different notations for the components based on how the summation works (i.e. sum over $i,j$ independently or sum over $i<j$).Though I haven't checked the formulas closely so perhaps I'm off on this. But it's worth checking. $\endgroup$
    – TK-421
    Nov 3, 2020 at 23:01

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"Riemannian Geometry" by Peter Peterson might be useful. Check Theorem 9.4.1.

$$d^* \omega_j = -\nabla^l \omega_{lj} = -g^{kl} \nabla_{k}\omega_{lj} = -g^{kl}\left(\partial_{k}\omega_{lj}-\Gamma_{kl}^{s}\omega_{sj}-\Gamma^s_{kj}\omega_{ls}\right)$$

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