We obtain several rigidity results for biharmonic submanifolds in
Sn with parallel normalized mean curvature vector field. We
classify biharmonic submanifolds in Sn 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 Sn.
Then we investigate, for (not necessarily compact) proper biharmonic
submanifolds in Sn, their type in the sense of B-Y. Chen. We prove:
(i) a proper biharmonic submanifold in Sn 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 Sn.Comment: 17 page