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We study the total lifts of curves by means of a submersion π: M 3 s → B 2 r that satisfy the condition ΔH = λH analyzing, in particular, the cases in which the submersion has totally geodesic fibres or integrable horizontal distribution. We also consider in detail the case A = 0 (biharmonic lifts). Moreover, we obtain a biharmonic lift in R 3 by means of a Riemannian submersion that has non-constant mean curvature, getting so a counterexample to the Chen conjecture for R 3 with a non-flat Riemannian metric.

Biharmonic lifts by means of pseudo-Riemannian submersions in dimension three