On the harmonicity of normal almost contact metric structures

We consider normal almost contact structures on a Riemannian manifold and, through their associated sections of an ad-hoc twistor bundle, study their harmonicity, as sections or as maps. We rewrite these harmonicity equations in terms of the Riemann curvature tensor and find conditions relating the harmonicity of the almost contact and almost complex structures of the total and base spaces of the Morimoto fibration.Comment: 14 page

Biminimal immersions

We study biminimal immersions, that is immersions which are critical points of the bienergy for normal variations with fixed energy. We give a geometrical description of the Euler-Lagrange equation associated to biminimal immersions for: i) biminimal curves in a Riemannian manifold, with particular care to the case of curves in a space form ii) isometric immersions of codimension one in a Riemannian manifold, in particular for surfaces of a three-dimensional manifold. We describe two methods to construct families of biminimal surfaces using both Riemannian and horizontally homothetic submersions.Comment: Dedicated to Professor Renzo Caddeo on his 60th birthday. 2 figure

The biharmonic homotopy problem for unit vector fields on 2-tori

The bienergy of smooth maps between Riemannian manifolds, when restricted to unit vector fields, yields two different variational problems depending on whether one takes the full functional or just the vertical contribution. Their critical points, called biharmonic unit vector fields and biharmonic unit sections, form different sets. Working with surfaces, we first obtain general characterizations of biharmonic unit vector fields and biharmonic unit sections under conformal change of the metric. In the case of a 2-dimensional torus, this leads to a proof that biharmonic unit sections are always harmonic and a general existence theorem, in each homotopy class, for biharmonic unit vector fields

Generalized Cheeger-Gromoll Metrics and the Hopf map

We show, using two different approaches, that there exists a family of Riemannian metrics on the tangent bundle of a two-sphere, which induces metrics of constant curvature on its unit tangent bundle. In other words, given such a metric on the tangent bundle of a two-sphere, the Hopf map is identified with a Riemannian submersion from the universal covering space of the unit tangent bundle onto the two-sphere. A hyperbolic counterpart dealing with the tangent bundle of a hyperbolic plane is also presented.Comment: 17 pages, Dedicated to Professor Udo Simon on his seventieth birthda