= 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.
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