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We study bifurcation for the constant scalar curvature equation along aone-parameter family of Riemannian metrics on the total space of a harmonicRiemannian submersion. We provide an existence theorem for bifurcation pointsand a criterion to see that the conformal factors corresponding to thebifurcated metrics must be indeed constant along the fibers. In the case of thecanonical variation of a Riemannian submersion with totally geodesic fibers, wecharacterize discreteness of the set of all degeneracy points along the familyand give a sufficient condition to guarantee that bifurcation necessarilyoccurs at every point where the linearized equation has a nontrivial solution.In the model case of quaternionic Hopf fibrations, we show thatsymmetry-breaking bifurcation does not occur except at the round metric.

Bifurcation for the constant scalar curvature equation and harmonic Riemannian submersions