Harmonic forms are fixed by a connected isometric group action

On a closed oriented Riemannian manifold, a smooth isometric action of a connected Lie group fixes every harmonic differential form. Each group element is connected to the identity through a smooth path, so its action is smoothly homotopic to the identity and fixes de Rham cohomology. Isometries preserve the Hodge Laplacian; uniqueness of the harmonic representative, from the Hodge decomposition theorem, then makes each form itself fixed.