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Hodge theory for elliptic complexes $E$ identifies the space of harmonic sections with cohomology $$\mathcal{H}(E) \simeq H(E).$$ A classical example with differential forms ($E = (\Omega,d)$) ...

Consider the Hopf fibrations $S^1\to S^{2n+1}\to CP^n$ and $S^3\to S^{4n+3}\to HP^n$. These are Riemannian submersions with totally geodesic fibers. Consider now their canonical variations (the so-...