# Explicit formula for the codifferential of a 2-form

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?

• 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. Nov 3, 2020 at 23:01
$$d^* \omega_j = -\nabla^l \omega_{lj} = -g^{kl} \nabla_{k}\omega_{lj} = -g^{kl}\left(\partial_{k}\omega_{lj}-\Gamma_{kl}^{s}\omega_{sj}-\Gamma^s_{kj}\omega_{ls}\right)$$