\lambda-biharmonic Riemannian submersions from manifolds with constant sectional curvature

TL;DR

This paper investigates iharmonic Riemannian submersions from manifolds with constant sectional curvature, establishing non-existence results for various curvature and parameter conditions, and constructing explicit examples in certain cases.

Contribution

It generalizes biharmonic submersions to iharmonic ones, proving non-existence theorems under specific curvature and parameter conditions, and providing explicit examples when possible.

Findings

Non-existence results for iharmonic submersions when   2(n - 1)c

Explicit examples of iharmonic submersions in negative curvature cases

Critical parameter  = 2(n - 1)c determines existence or non-existence

Abstract

In this paper, we study \lambda-biharmonic Riemannian submersions, which generalize biharmonic Riemannian submersions. We prove non-existence results for \lambda-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 \lambda = 2(n - 1)c plays a decisive role. When \lambda \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 \lambda = 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 \lambda \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.