I am a physics student and I work on General Relativity in this day and age. I have a little mathematical understanding in Differential geometry.
However, when I have to counter d'Alembert operator in my work, I do not find much information about this object.
As I know, we can write: $$\mathop\Box\equiv g_{\mu\nu}\frac{\partial}{\partial x^{\mu}}\frac{\partial}{\partial x^{\nu}}\equiv g_{\mu\nu}\partial_{\mu}\partial_{\nu}\equiv \partial^{\nu}\partial_{\nu}\equiv\partial_{\nu}\partial^{\nu}$$ Now, it acts on a tensor-like $h^{\gamma\lambda}$ with $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ for $h_{\mu\nu}<<1$: $$\mathop\Box h^{\gamma\lambda}=\partial_{\nu}\partial^{\nu}h^{\gamma\lambda}$$ Here the question, if I multiply both sides by an metric $\eta_{\alpha\beta}$, is $\eta_{\alpha\beta}$ able to contract $h^{\gamma\lambda}$ like this $$\eta_{\alpha\beta}\partial_{\nu}\partial^{\nu}h^{\gamma\lambda}=\partial_{\nu}\partial^{\nu}h^{\lambda}_{\beta} \quad\mathrm{if}\quad\alpha=\gamma$$ I think it depend on specific $\nu$ compare with $\alpha$, $\beta$, $\gamma$ and $\lambda$, right?
Suppose that $h^{\mu\nu}$ is symmetrical.