**In a nutshell:**

Given a metric on the torus $\mathbb{T}^n$, can we extend any element $\sigma \in \bigwedge^k T_p^*\mathbb{T}^n$ to a global harmonic form?

Let $\mathbb{T}^n$ be the $n$-Torus. Fix $1<k<n$. Given a Riemannian metric $g$ on $\mathbb{T}^n$, we denote by $H^k_g$ the space of $g$-harmonic forms of degree $k$. Suppose that $g$ is **flat** on some open neighbourhood $U \subseteq \mathbb{T}^n$.

Let $p \in U$, and consider the evaluation map at $p$, $ev_p:H^k_g \to \bigwedge^k T_p^*\mathbb{T}^n$, given by $ev_p(\omega)=\omega_p$.

Is it true that $ev_p$ is an isomorphism? (Note that the dimensions of both spaces are equal).

Of course, this holds for the standard flat metric on $\mathbb{T}^n$ induced by the Euclidean metric on $\mathbb{R}^n$. (There we essentially have $dx^{i_1} \wedge \dots \wedge dx^{i_k}$ as a global basis for $H^k_g$).

Since $g$ is flat around $p$, we can take isometric coordinates, where $g_{ij}=\delta_{ij}$. So, *locally* we have all the $dx^{i_1} \wedge \dots \wedge dx^{i_k}$ as harmonic forms. However, I am not sure that all these local forms can be extended into global harmonic forms on $\mathbb{T}^n$.
If such an extension is possible, then of course the evaluation map would be surjective.

Is there any reason to believe such an extension is always possible?