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

Dedicated to Professor Renzo Caddeo on his 60th birthday.Abstract 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 with biminimal immersions for both biminimal curves in a Riemannian manifold, with particular attention given to the case of curves in a space form, and isometric immersions of codimension 1 in a Riemannian manifold, in particular for surfaces of a three-dimensional manifold. We describe two methods of constructing families of biminimal surfaces using both Riemannian and horizontally homothetic submersions.

We study harmonic sections of a Riemannian vector bundle E → M whose total space is equipped with a 2-parameter family of metrics h p,q which includes both the Sasaki and Cheeger-Gromoll metrics. The restrictions of the h p,q to the total space of any sphere subbundle SE(k) of E (where k > 0 is the radius) are essentially the same for all (p, q), and it is shown that for every k there exists a unique p such that the harmonic sections of SE(k) are harmonic sections of E with respect to h p,q for all q. In both the compact and non-compact cases Bernstein regions of the (p, q)-plane are identified, where the only harmonic sections of E with respect to h p,q are parallel. Examples are constructed of compact vector fields which are harmonic sections of E = T M in the case where M has non-zero Euler characteristic.1991 Mathematics Subject Classification. 53C43 (53C07, 53C24, 58E15, 58E20, 58G30).

Inspired by the all-important conformal invariance of harmonic maps on two-dimensional domains, this article studies the relationship between biharmonicity and conformality. We first give a characterization of biharmonic morphisms, analogues of harmonic morphisms investigated by Fuglede and Ishihara, which, in particular, explicits the conditions required for a conformal map in dimension four to preserve biharmonicity and helps producing the first example of a biharmonic morphism which is not a special type of harmonic morphism. Then, we compute the bitension field of horizontally weakly conformal maps, which include conformal mappings. This leads to several examples of proper (i.e. nonharmonic) biharmonic conformal maps, in which dimension four plays a pivotal role. We also construct a family of Riemannian submersions which are proper biharmonic maps.

We define, on smooth manifolds, the notions of almost twistorial structure and twistorial map, thus providing a unified framework for all known examples of twistor spaces. The condition of being harmonic morphisms naturally appears among the geometric properties of submersive twistorial maps between low-dimensional Weyl spaces endowed with a nonintegrable almost twistorial structure due to Eells and Salamon. This leads to the twistorial characterisation of harmonic morphisms between Weyl spaces of dimensions four and three. Also, we give a thorough description of the twistorial maps with one-dimensional fibres from four-dimensional Weyl spaces endowed with the almost twistorial structure of Eells and Salamon.

Abstract. We study a geometrical condition (PHWC) which is weaker than horizontal weak conformality. In particular, we show that harmonic maps satisfying this condition, which will be called pseudo harmonic morphisms, include harmonic morphisms and can be described as pulling back certain germs to certain other germs. Finally, we construct a canonical f-structure associated to every map satisfying (PHWC) and find conditions on this fstructure to ensure the harmonicity of the map.

Abstract. Biharmonic maps are the critical points of the bienergy functional and, from this point of view, generalize harmonic maps. We consider the Hopf map ψ : S 3 → S 2 and modify it into a nonharmonic biharmonic map φ : S 3 → S 3 . We show φ to be unstable and estimate its biharmonic index and nullity. Resolving the spectrum of the vertical Laplacian associated to the Hopf map, we recover Urakawa's determination of its harmonic index and nullity.

We study the geometry of the tangent bundle equipped with a two-parameter family of metrics, deforming the Sasaki and Cheeger-Gromoll metrics. After deriving the expression for the Levi-Civita connection, we compute the Riemann curvature tensor and the sectional, Ricci and scalar curvatures. We identify all metrics whose restriction to the fibres has positive sectional curvature. When the base manifold is a space form, we characterise metrics with non-negative sectional curvature and show that one can always find parameters ensuring positive scalar curvature. This extends to compact manifolds and, under some curvature conditions, to general manifolds.

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