Why the cohomologous harmonic functions are equal?

Question

As picture below, why the cohomologous harmonic functions are equal ? The cohomologous is respect to exterior operator on Riemannian manifold.

Answer 1

This has nothing to do with harmonic functions. By definition, $f_1,f_2 \in Z^0(M) = \ker (d \colon \Omega^0(M) \rightarrow \Omega^1(M))$ are cohomologous if they differ by an exact form in $B^0(M) = \operatorname{im}(d \colon \Omega^{-1}(M) \rightarrow \Omega^0(M)) = \{ 0 \}$. Since the only exact $0$-form is the zero function, two cohomologous functions must be equal.