Biharmonic PNMC submanifolds in spheres

We obtain several rigidity results for biharmonic submanifolds in with parallel normalized mean curvature vector fields. We classify biharmonic submanifolds in with parallel normalized mean curvature vector fields and with at most two distinct principal curvatures. In particular, we determine all biharmonic surfaces with parallel normalized mean curvature vector fields in . Then we investigate, for (not necessarily compact) proper-biharmonic submanifolds in , their type in the sense of B.-Y. Chen. We prove that (i) a proper-biharmonic submanifold in is of 1-type or 2-type if and only if it has constant mean curvature f=1 or fa(0,1), respectively; and (ii) there are no proper-biharmonic 3-type submanifolds with parallel normalized mean curvature vector fields in .