λ-biharmonic Riemannian submersions from manifolds with constant sectional curvature

Abstract

In this paper, we study λ-biharmonic Riemannian submersions, which generalize biharmonic Riemannian submersions. We prove non-existence results for λ-biharmonic Riemannian submersions from (n + 1)-dimensional Riemannian manifolds with constant sectional curvature c to n-dimensional Riemannian manifolds. Our results show that the critical value λ= 2(n - 1)c plays a decisive role. When λ\ne 2(n - 1)c, we prove a nonexistence theorem, although a dimensional assumption is needed in the positive curvature case. On the other hand, when λ= 2(n - 1)c, we prove a non-existence theorem in the nonnegative curvature case, whereas in the negative curvature case, we construct explicit examples. The only remaining local case is the positively curved case with λ\ne 2(n - 1)c and n \ge 5, while in the complete connected positive-curvature setting the theorem of Gromoll and Grove yields harmonicity in all dimensions.