Miho Shito | Searcharxiv

arXiv2026

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

In this paper, we study \lambda-biharmonic Riemannian submersions, which generalize biharmonic Riemannian submersions. We prove nonexistence 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. We prove that if c \ge 0, or if c < 0 and \lambda \ne 2(n - 1)c, then any \lambda-biharmonic Riemannian submersion must be harmonic. On the other hand, when \lambda = 2(n - 1)c with c < 0, we construct explicit examples of \lambda-biharmonic Riemannian submersions.

math.DG↗

arXiv2025

Classification of biharmonic Riemannian submersions from manifolds with constant sectional curvature

In 2011, Wang and Ou (Math. Z. {\bf 269}:917-925, 2011) showed that any biharmonic Riemannian submersion from a 3-dimensional Riemannian manifold with constant sectional curvature to a surface is harmonic. In this paper, we generalize the 3-dimensional setting to arbitrary dimensions. By constructing an adapted orthonormal frame, we simplify the biharmonic equation for Riemannian submersions and analyze the curvature properties of Riemannian manifolds with constant sectional curvature. As a result, we prove that a Riemannian submersion from an $(n+1)$-dimensional Riemannian manifold with constant sectional curvature to an $n$-dimensional Riemannian manifold is biharmonic if and only if it is harmonic. This result may also be viewed as an affirmative codimension-one Riemannian submersion analogue of Chen's conjecture, the generalized Chen's conjecture, and the BMO conjecture.

math.DG↗