I tried to compute the explicit expression for $d^*d\omega$ where $\omega=\omega_jdx^j$ is a 1-form on a manifold $M^n$. My hope was to find an expression that doesn't involve the Christoffel symbols. I tried using the fact that $$ \int d^*d\omega\cdot\eta=\int d\omega\cdot d\eta $$ where both $\omega,\eta$ are 1-forms.
In what follows we denote $\omega_{ij}=\partial_i\omega_j-\partial_j\omega_i$ and similarly for $\eta_{kt}$. In this way $d\omega=\omega_{ij}dx^i\wedge dx^j$ (sum for $i<j$). With this notation we compute (until specified differently, sums are taken for $i<j$ and $k<t$)
\begin{align*} \int d^{*}d\omega\cdot\eta&=\int d\omega\cdot d\eta\\ &=\int\omega_{ij}\eta_{kt}g\left(dx^{i}\wedge dx^{j},dx^{k}\wedge dx^{t}\right)\sqrt{g}dx\\ &=\int\omega_{ij}\left(\partial_{k}\eta_{t}-\partial_{t}\eta_{k}\right)\left(g^{ik}g^{jt}-g^{it}g^{jk}\right)\sqrt{g}dx\\ &=\int\omega_{ij}\partial_{k}\eta_{t}\left(g^{ik}g^{jt}-g^{it}g^{jk}\right)\sqrt{g}dx-\int\omega_{ij}\partial_{k}\eta_{t}\left(g^{it}g^{jk}-g^{ik}g^{jt}\right)\sqrt{g}dx\\ (k,t=1,\dots,n)&=\int\omega_{ij}\partial_{k}\eta_{t}\left(g^{ik}g^{jt}-g^{it}g^{jk}\right)\sqrt{g}dx\\ &=-\int\frac{1}{\sqrt{g}}\partial_{k}\left[\omega_{ij}\left(g^{ik}g^{jt}-g^{it}g^{jk}\right)\sqrt{g}\right]\eta_{t}\sqrt{g}dx\\ (i,j=1,\dots,n)&=-\int\frac{1}{\sqrt{g}}\partial_{k}\left[\omega_{ij}g^{ik}g^{jt}\sqrt{g}\right]\eta_{t}\sqrt{g}dx\\ &=-\int\frac{1}{\sqrt{g}}g_{st}\partial_{k}\left[\omega_{ij}g^{ik}g^{jt}\sqrt{g}\right]g^{sl}\eta_{l}\sqrt{g}dx \end{align*} So that \begin{align*} d^{*}d\omega&=-\frac{1}{\sqrt{g}}g_{st}\partial_{k}\left[\omega_{ij}g^{ik}g^{jt}\sqrt{g}\right]dx^{s}\\ &=-g^{ik}\partial_{k}\omega_{ij}dx^{j}-\frac{1}{\sqrt{g}}g_{st}\partial_{k}\left[g^{ik}g^{jt}\sqrt{g}\right]\omega_{ij}dx^{s}\\ &=-\left(g^{ik}\partial_{k}\omega_{ij}+\frac{1}{\sqrt{g}}\partial_{k}\left[g^{ik}\sqrt{g}\right]\omega_{ij}\right)dx^{j}-g^{ik}g_{st}\partial_{k}\left[g^{jt}\right]\omega_{ij}dx^{s}. \end{align*} At this point, the first part of the last expression reminds me strongly of the form of the operator $d^*d=-\Delta_M$ on functions $$ -\Delta_Mf=g^{ik}\partial_{k}\partial_{i}f+\frac{1}{\sqrt{g}}\partial_{k}\left[g^{ik}\sqrt{g}\right]\partial_{i}f $$ so I would expect something similar on 1-forms. I can't find a sense for the second term $$-g^{ik}g_{st}\partial_{k}\left[g^{jt}\right]\omega_{ij}dx^{s}$$ so I started thinking that I did something wrong in my calculations. Could you please confirm (or disprove) the formula that I found?
Any help is very appreciated.