  We classify biharmonic submanifolds with certain geometric properties in
Euclidean spheres. For codimension 1, we determine the biharmonic hypersurfaces
with at most two distinct principal curvatures and the conformally flat
biharmonic hypersurfaces. We obtain some rigidity results for pseudo-umbilical
biharmonic submanifolds of codimension 2 and for biharmonic surfaces with
parallel mean curvature vector field. We also study the type, in the sense of
B-Y. Chen, of compact proper biharmonic submanifolds with constant mean
curvature in spheres.

Differential Geometry