Manifold maps commuting with the Laplacian

The commutative algebra Q)(G) of isometry-invariant differential operators on a Riemannian symmetric space always contains the Laplace-Beltrami operator Δ.In fact, Δ is the generator of Q)(G) exactly when G/H is of rank one.Therefore it is natural to ask which manifold maps φ: G 1 /H 1 -+ G 2 /H 2 commute with the Laplacian on C°° functions of G 2 /H 2 , i.e., φ*Δ 2 f = Δ^f for all / e C°°(G 2 /H 2 ).Helgason [3, p. 387] showed for a general pseudo-Riemannian manifold M that the only diffeomorphisms Φ: M -> M which commute with Δ are the isometries.Recalling the powerful de Rham-Hodge theorem (classical real pth cohomology group = pth de Rham cohomology group = space of harmonic p-forms) on compact Riemannian manifolds, the above question should be: which surjective maps ψ: M -> N commute with Δ on differential p-forms for compact M and NΊ Our main results are:(1) Every such Laplacian-commuting map is a Riemannian submersion, and therefore is a locally trivial differentiable Riemannian fibre space.(2) If there exists such a map ψ: M -> N commuting with Δ on p-ίorms for compact M and fixed p, then b p (N) < b p (M).(3) For compact M,ψ: M -> N commutes with the Laplacian on functions if and only if ψ is a harmonic Riemannian submersion.An analogous question "which compact fibre space mappings π: E-^B commute with the codifferential operator δ on forms of all degrees simultaneously" has been answered in certain specific cases [5], but our result is more general.