A contact metric manifold whose characteristic vector field is a harmonic vector field is called an H-contact metric manifold. We introduce the notion of (κ, µ, ν)-contact metric manifolds in terms of a specific curvature condition. Then, we prove that a contact metric 3-manifold M is an H-contact metric manifold if and only if it is a (κ, µ, ν)-contact metric manifold on an everywhere open and dense subset of M. Also, we prove that, for dimensions greater than three, such manifolds are reduced to (κ, µ)-contact metric manifolds whereas, in three dimensions, (κ, µ, ν)-contact metric manifolds exist.

In this paper, we characterize biharmonic Legendre curves in 3-dimensional ðk; m; nÞcontact metric manifolds. Moreover, we give examples of Legendre geodesics in these spaces. We also give a geometric interpretation of 3-dimensional generalized ðk; mÞcontact metric manifolds in terms of its Legendre curves. Furthermore, we study biharmonic anti-invariant surfaces of 3-dimensional generalized ðk; mÞ-contact metric manifolds with constant norm of the mean curvature vector field. Finally, we give examples of anti-invariant surfaces with constant norm of the mean curvature vector field immersed in these spaces.

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In this paper, we study minimal maps between euclidean spheres. The Hopf fibrations provide explicit examples of such minimal maps. Moreover, their corresponding graphs have second fundamental form of constant norm. We prove that a minimal submersion from S 3 to S 2 whose Gauss map satisfies a suitable pinching condition must be weakly conformal with totally geodesic fibers. As a consequence, we obtain that an equivariant minimal submersion from S 3 to S 2 coincides with the Hopf fibration. Furthermore, we prove that a minimal map f : U ⊂ S 3 → S 2 with constant singular values and constant norm of the second fundamental form is either constant or, up to isometries, coincides with the Hopf fibration.