Let $(M^3, g)$ be a (closed) Riemannian manifold and let $u: M \to S$ be a harmonic function, where $S$ is a closed orientable surface. If $\omega$ is a $2$-form on $S$, what are sufficient conditions on $\omega$ in order $u^* \omega$ to be a harmonic $2$-form on $M$?

The concrete case I am analyzing is the following: we have $u_1, u_2 : M \to \mathbb{S}^1$ harmonic functions, $u = (u_1, u_2):M \to \mathbb{S}^1 \times \mathbb{S}^1$ and $\omega = d\theta_1 \wedge d\theta_2$, where $d\theta_i$ denotes the volume form on each $\mathbb{S}^1$.

Let $(M^3, g)$ be a (closed) Riemannian manifold and let $u: M \to S$ be a harmonic function, where $S$ is a closed orientable surface. If $\omega$ is a $2$-form on $S$, what are sufficient conditions on $\omega$ in order $u^* \omega$ to be a harmonic $2$-form on $M$?

The concrete case I am analyzing is the following: we have $u_1, u_2 : M \to \mathbb{S}^1$ harmonic functions, $u = (u_1, u_2):M \to \mathbb{S}^1 \times \mathbb{S}^1$ and $\omega = d\theta_1 \wedge d\theta_2$, where $d\theta_i$ denotes the volume form on each $\mathbb{S}^1$.

I’d be glad to have a solution in either case.