We study biharmonic hypersurfaces in a generic Riemannian manifold. We first derive an invariant equation for such hypersurfaces generalizing the biharmonic hypersurface equation in space forms studied by Jiang, Chen, Caddeo, Montaldo, and Oniciuc. We then apply the equation to show that the generalized Chen conjecture is true for totally umbilical biharmonic hypersurfaces in an Einstein space, and construct a 2-parameter family of conformally flat metrics and a 4-parameter family of multiply warped product metrics, each of which turns the foliation of an upper-half space of ‫ޒ‬ m by parallel hyperplanes into a foliation with each leaf a proper biharmonic hypersurface. We also study the biharmonicity of Hopf cylinders of a Riemannian submersion. Biharmonic maps and submanifoldsAll manifolds, maps, and tensor fields that appear in this paper are assumed to be smooth unless stated otherwise.A biharmonic map is a map ϕ : (M, g) → (N , h) between Riemannian manifolds that is a critical point of the bienergy functionalfor every compact subset of M, where τ (ϕ) = Trace g ∇dϕ is the tension field of ϕ. The Euler-Lagrange equation of this functional gives the biharmonic map equation [Jiang 1986b] (which states that ϕ is biharmonic if and only if its bitension field τ 2 (ϕ) vanishes identically. In this equation we used R N to denote the curvature operator of (N , h) MSC2000: 53C12, 58E20, 53C42.

An important theorem about biharmonic submanifolds proved independently by Chen-Ishikawa [CI] and Jiang [Ji] states that an isometric immersion of a surface into 3-dimensional Euclidean space is biharmonic if and only if it is harmonic (i.e, minimal). In a later paper [CMO2], Cadeo-Monttaldo-Oniciuc shown that the theorem remains true if the target Euclidean space is replaced by a 3-dimensional hyperbolic space form. In this paper, we prove the dual results for Riemannian submersions, i.e., a Riemannian submersion from a 3-dimensional space form of non-positive curvature into a surface is biharmonic if and only if it is harmonic

The generalized Chen's conjecture on biharmonic submanifolds asserts that any biharmonic submanifold of a non-positively curved manifold is minimal.In this paper, we prove that this conjecture is false by constructing a foliation of proper biharmonic hyperplanes in a 5-dimensional conformally flat space with negative sectional curvature. Many examples of proper biharmonic submanifolds of non-positively curved spaces are also given.Date: 01/18/2011. 1991 Mathematics Subject Classification. 58E20, 53C12, 53C42.

The study of biharmonic submanifolds, initiated by B. Y. Chen [6,7] and G. Y. Jiang [12,13] independently, has received a great attention in the past 30 years with many important progress (see [17], [18] and the vast references therein). This note attempts to give a short survey on the study of biharmonic Riemannian submersions which are a dual concept of biharmonic submanifolds (i.e., biharmonic isometric immersions). Why biharmonic Riemannian submersions?A map ϕ : (M m , g) → (N n , h) between Riemannian manifolds is called a harmonic map if its tension field τ (ϕ) = Trace g ∇dϕ vanishes identically. Harmonic maps include many important objects studied in mathematics, such as harmonic functions, geodesics, minimal submanifolds, and Riemannian submersions with minimal fibers. For more on the related theory, applications and interesting links of harmonic maps see [8], [9], [10] and [26].2 M |τ (ϕ)| 2 dv g , where τ (ϕ) is the tension field of ϕ. The first variation of the bienergy (see [12]) gives the biharmonic map equation as:(1) τ 2 (ϕ) := Trace g (∇ ϕ ) 2 τ (ϕ) − R N (dϕ, τ (ϕ))dϕ = 0 where R N is the Riemannian curvature operator of the manifold (N, h).

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.