Cross-post from [MSE](https://math.stackexchange.com/q/4064240/272127).




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Suppose $(M,g)$ be a closed Riemannian manifold. Because every parallel (nontrivial) $p$-form $\omega$ is harmonic so the $p$-th Betti number should be positive i.e. $b\_p\geq 1$. How to see this using De Rham cohomology? (of course without using properties of Hodge theory like $\delta$, 1-1 correspondence between De Rham and Harmonic forms, etc.)


It is well-known that parallel forms are closed. The more involved part is showing that it is not exact.


Such an easy argument based on harmonic forms I think there must be similar easy argument based on De Rham cohomology or it needs somewhat difficult and tricky argument similar to the [proof of Poincaré lemma](https://en.wikipedia.org/wiki/Closed_and_exact_differential_forms#Poincar%C3%A9_lemma)?



Existence non-trivial parallel $p$-form implies non-triviality of $p$-th cohomology group using De Rham cohomology
Here is an argument in the orientable case: If $\omega$ is a $g$-parallel $p$-form on an orientable Riemannian $n$-manifold $(M^n,g)$, then it is closed since $\mathrm{d}\omega = \pi\_{p+1}(\nabla^g\omega) = 0$, where $$\nabla^g:\Omega^p(M)=C^\infty(\Lambda^p(T^\*M))\to C^\infty(T^\*M\otimes\Lambda^p(T^\*M))$$ is the Levi-Civita connection and $\pi\_{p+1}:T^\*M\otimes\Lambda^p(T^\*M)\to\Lambda^{p+1}(T^\*M)$ is the obvious skew-symmetrization. Moreover, the $(n{-}p)$-form $\ast\_g\omega$ is also $g$-parallel, so it, too, is closed. Now
$$
\omega\wedge \ast\_g\omega = |\omega|^2\_g\,\,\,{\ast\_g}1,
$$
so, unless $\omega\equiv0$, we will have
$$
\int\_M \omega\wedge \ast\_g\omega >0.
$$
Thus, $\omega$ can't be exact, since $\omega = \mathrm{d}\eta$ would give
$$
\int\_M \omega\wedge \ast\_g\omega = \int\_M \mathrm{d}(\eta\wedge \ast\_g\omega)
= 0,
$$
by Stokes' Theorem. No Hodge theory required.


Finally, if $M$ is not orientable, pull $\omega$ up to the orientation double cover $\tilde M$, observe that it is not exact there, and conclude it couldn't have been exact on $M$ either.


