Bulletin of the
Korean Mathematical Society BKMS

We characterize proper biharmonic anti-invariant surfaces in $3$-dimensional generalized $(\kappa, \mu)$-manifolds with 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 proper 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.

Keywords: biharmonic submanifolds, Legendre curves, anti-invariant surfaces, generalized $(\kappa, \mu)$-manifolds

MSC numbers: Primary 53C42; Secondary 53B25