Biharmonic PNMC Submanifolds in Spheres

We obtain several rigidity results for biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector field. We classify biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector field and with at most two distinct principal curvatures. In particular, we determine all biharmonic surfaces with parallel normalized mean curvature vector field in $\mathbb{S}^n$. Then we investigate, for (not necessarily compact) proper biharmonic submanifolds in $\mathbb{S}^n$, their type in the sense of B-Y. Chen. We prove: (i) a proper biharmonic submanifold in $\mathbb{S}^n$ is of 1-type or 2-type if and only if it has constant mean curvature {\mcf}=1 or {\mcf}\in(0,1), respectively; (ii) there are no proper biharmonic 3-type submanifolds with parallel normalized mean curvature vector field in $\mathbb{S}^n$.Comment: 17 page