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 .

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We give some general results on proper-biharmonic submanifolds of a complex space form and, in particular, of the complex projective space. These results are mainly concerned with submanifolds with constant mean curvature or parallel mean curvature vector field. We find the relation between the bitension field of the inclusion of a submanifoldM in CP n and the bitension field of the inclusion of the corresponding Hopf-tube in S 2n+1 . Using this relation we produce new families of proper-biharmonic submanifolds of CP n . We study the geometry of biharmonic curves of CP n and we characterize the proper-biharmonic curves in terms of their curvatures and complex torsions.

We consider biconservative surfaces M 2 , g in a space form N 3 (c), with mean curvature function f satisfying f > 0 and ∇f = 0 at any point, and determine a certain Riemannian metric gr on M such that M 2 , gr is a Ricci surface in N 3 (c). We also obtain an intrinsic characterization of these biconservative surfaces.2010 Mathematics Subject Classification. 53A10, 53C42.

Abstract. We consider surfaces with parallel mean curvature vector (pmc surfaces) in CP n ×R and CH n ×R, and, more generally, in cosymplectic space forms. We introduce a holomorphic quadratic differential on such surfaces. This is then used in order to show that the anti-invariant pmc 2-spheres of a 5-dimensional non-flat cosymplectic space form of product type are actually the embedded rotational spheres S 2 H ⊂M 2 × R of Hsiang and Pedrosa, whereM 2 is a complete simply-connected surface with constant curvature. When the ambient space is a cosymplectic space form of product type and its dimension is greater than 5, we prove that an immersed non-minimal non-pseudo-umbilical anti-invariant 2-sphere lies in a product spaceM 4 × R, whereM 4 is a space form. We also provide a reduction of codimension theorem for the pmc surfaces of a non-flat cosymplectic space form.

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