A short note on biharmonic submanifolds in non-Sasakian contact metric 3-manifolds

TL;DR

This paper characterizes biharmonic anti-invariant surfaces in 3D generalized $(ppa,rac12)$-manifolds with constant mean curvature, providing construction methods and identifying manifolds with specific biharmonic foliations.

Contribution

It offers a characterization of biharmonic surfaces in non-Sasakian contact metric 3-manifolds and introduces a method to construct many examples, advancing understanding of biharmonic submanifolds.

Findings

Characterization of biharmonic anti-invariant surfaces with constant mean curvature.

Method for constructing numerous biharmonic submanifolds.

Identification of manifolds admitting proper biharmonic foliations.

Abstract

We characterize biharmonic anti-invariant surfaces in $3$-dimensional generalized $(\kappa, \mu)$-manifolds with non-zero constant mean curvature by means of the scalar curvature of the ambient space and the mean curvature. In addition, we give a method for constructing infinity many examples of biharmonic submanifolds in a certain $3$-dimensional generalized $(\kappa, \mu)$-manifold. Moreover, we determine $3$-dimensional generalized $(\kappa, \mu)$-manifolds which admit a certain kind of proper biharmonic foliation.