Totally Biharmonic Hypersurfaces in Space Forms and 3-Dimensional BCV Spaces

2019/11/06 by Montaldo, Stefano, Pampano, Alvaro
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1911.02625

Abstract

A hypersurface is said to be totally biharmonic if all its geodesics are biharmonic curves in the ambient space. We prove that a totally biharmonic hypersurface into a space form is an isoparametric biharmonic hypersurface, which allows us to give the full classification of totally biharmonic hypersurfaces in these spaces. Moreover, restricting ourselves to the 3-dimensional case, we show that totally biharmonic surfaces into Bianchi-Cartan-Vranceanu spaces are isoparametric surfaces and we give their full classification. In particular, we show that, leaving aside surfaces in the 3-dimensional sphere, the only non-trivial example of a totally biharmonic surface appears in the product space \mathbbS²(ρ)×ℝ.

Related

• Biharmonic hypersurfaces in a product space L^m× ℝ
• Biharmonic space-like hypersurfaces in pseudo-Riemannian space
• f-Biharmonic hypersurfaces into a conformally flat space
• Biharmonic Riemannian submersions from a 3-dimensional BCV space
• Some classifications of biharmonic hypersurfaces with constant scalar curvature