Let $u: V \rightarrow M$, where $(V,g)$ is a Lorentzian manifold and $(M,h)$ is a Riemannian manifold.
The wave equation is defined as $g. \nabla^2 u$.
As far as I see, $\nabla^2 u \in \Gamma(T^{*}V \otimes T^{*}V \otimes TM)$, since $\nabla_{\partial_{\alpha}} u= \partial_{\alpha} u \in \Gamma(T^{*}V \otimes TM)$. Is that correct?
But then, how can you apply $g$ on this?
This mapping is a harmonic map, if the mapping $u$ satisifes $g \cdot \nabla^2 u=0$ then in local coordinates \begin{align} g^{\alpha \beta}\nabla_\alpha \partial_{\beta}u^A &= g^{\alpha \beta}(\partial^2_{\alpha \beta}u^A - \Gamma^\lambda_{\alpha \beta}\partial_{\lambda} u^A+ \Gamma^A_{BC}\partial_{\alpha}u^B \partial_{\beta}u^C)\\ &=0 \end{align} And this is how you apply $g$, as you say.