We explicitly classify the nonharmonic biharmonic submanifolds of the unit threedimensional sphere S 3 .

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We study biharmonic submanifolds of the Euclidean sphere that satisfy certain geometric properties. We classify: (i) the biharmonic hypersur- faces with at most two distinct principal curvatures; (ii) 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

We introduce the notion of biconservative hypersurfaces, that is hypersurfaces with conservative stress-energy tensor with respect to the bienergy. We give the (local) classification of biconservative surfaces in three-dimensional space forms

We investigate proper biharmonic hypersurfaces with at most three distinct principal curvatures in space forms. We obtain the full classification of proper biharmonic hypersurfaces in 4-dimensional space forms

Using Hilbert’s criterion, we consider the stress-energy tensor associated to the bienergy functional. We show that it derives from a variational problem on metrics and exhibit the peculiarity of dimension four. First, we use this tensor to construct new exam- ples of biharmonic maps, then classify maps with vanishing or parallel stress-energy tensor and Riemannian immersions whose stress-energy tensor is proportional to the metric, thus obtaining a weaker but high-dimensional version of the Hopf Theorem on compact constant mean curvature immersions. We also relate the stress-energy tensor of the inclusion of a submanifold in Euclidean space with the harmonic stress-energy tensor of its Gauss map

We classify all biharmonic Legendre curves in a Sasakian space form and obtain their explicit parametric equations in the (2n + 1)-dimensional unit sphere endowed with the canonical and deformed Sasakian structures defined by Tanno. We also show that, under the flow-action of the characteristic vector field, a biharmonic integral submanifold becomes a biharmonic anti-invariant submanifold. Then, we obtain new examples of biharmonic submanifolds in the Euclidean sphere ‫ޓ‬ 7 .

We obtain several rigidity results for biharmonic submanifolds in S n with parallel normalized mean curvature vector field. We classify biharmonic submanifolds in 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 S n .Then we investigate, for (not necessarily compact) proper biharmonic submanifolds in S n , their type in the sense of B-Y. Chen. We prove: (i) a proper biharmonic submanifold in S n is of 1-type or 2-type if and only if it has constant mean curvature f = 1 or f ∈ (0, 1), respectively; (ii) there are no proper biharmonic 3-type submanifolds with parallel normalized mean curvature vector field in S n .where τ (ϕ) = trace ∇dϕ is the tension field. The functional E 2 is called the bienergy functional. In the particular case when ϕ : (M, g) → (N, h) is a Riemannian immersion, the tension field has the expression τ (ϕ) = mH and equation (1.1) is equivalent to ϕ being a critical point of E 2 . Obviously, any minimal submanifold (H = 0) is biharmonic. The non-harmonic biharmonic submanifolds are called proper biharmonic.The study of proper biharmonic submanifolds is nowadays becoming a very active subject and its popularity initiated with the challenging conjecture of B-Y. Chen: any biharmonic submanifold in an Euclidean space is minimal.