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Edit: Accoring to the comment of Asura Path I revise the question. Let $X$ be a vector field on a Riemannian manifold $(M,g)$. So we have a $1$-form $\beta$ with $\beta(Y)=\langle Y,X\...

Let $M$ be a compact manifold and $\varphi:M\rightarrow M$ a diffeomorphism. The invariant differential forms $$ \Omega^{k}_{inv}(M)=\{\alpha\in\Omega^{k}(M):\varphi^{*}\alpha=\alpha\} $$ form a ...