Explicit formulas for biharmonic submanifolds in Sasakian space forms

Fetcu, Dorel; Oniciuc, Cezar

doi:10.2140/pjm.2009.240.85

Cited by 39 publications

(56 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The survey article [Montaldo and Oniciuc 2006] contains an account of the study of biharmonic curves in various models. See [Arslan et al 2005;Inoguchi 2004;Fetcu and Oniciuc 2009a;2009b;Sasahara 2005; for special biharmonic submanifolds in contact manifolds or Sasakian space forms.…”

Section: Biharmonic Maps and Submanifoldsmentioning

confidence: 99%

Biharmonic hypersurfaces in Riemannian manifolds

Ou¹

2010

Pacific J. Math.

129

117

View full text Add to dashboard Cite

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.

show abstract

Section: Biharmonic Maps and Submanifoldsmentioning

confidence: 99%

Biharmonic hypersurfaces in Riemannian manifolds

Ou¹

2010

Pacific J. Math.

129

117

View full text Add to dashboard Cite

show abstract

“…After space forms, a series of non-constant sectional curvature spaces were considered as ambient spaces for the study of proper-biharmonic submanifolds (see, for example, [4,23,30,31,35,36,54,58,61]). In this respect, one of the research directions was the study of proper-biharmonic submanifolds in Sasakian space forms.…”

Section: Conjecture ([8]) Any Proper-biharmonic Submanifold In S N Hmentioning

confidence: 99%

“…Then all proper-biharmonic Legendre curves in arbitrary dimensional Sasakian spaces were classified. It was proved that they are helices (see [30]), that certain angles involving their Frenet frame field are constant and, by using the (2n + 1)-dimensional unit sphere endowed with its canonical and deformed Sasakian structures (see [63]), the explicit parametric equations of such curves were given.…”

Section: Conjecture ([8]) Any Proper-biharmonic Submanifold In S N Hmentioning

confidence: 99%

Harmonic and Biharmonic Maps at Iaşi

Balmus¹,

Fetcu²,

Oniciuc³

2010

Annals of the Alexandru Ioan Cuza University - Mathematics

View full text Add to dashboard Cite

Abstract. We report on the achievements of the geometers from Iaşi in the field of harmonic and biharmonic maps.Mathematics Subject Classification 2000: 53C07, 53C40, 53C42, 53C43, 58E20. Key words: harmonic and biharmonic maps, minimal and biharmonic submanifolds.Nowadays, the theory of harmonic maps between Riemannian manifolds is a very important field of Riemannian geometry. The harmonic maps ϕ : (M, g) → (N, h) are critical points of the energy functional E which is defined on the infinit dimensional manifold of the smooth maps from M to N ,Here dϕ is the differential of the map ϕ and ∥·∥ denotes the Hilbert-Schmidt norm. The Euler-Lagrange equation associated to the energy E is given by the vanishing of the tension field τ (ϕ) = trace ∇dϕ and it is a second order elliptic equation, as one can see from its local expressionwhere Γ denotes the Christoffel symbols and ∆ is the Laplacian acting on functions. From here raises the deep link established by the harmonic maps *

show abstract

“…Since any harmonic maps is biharmonic, we are interested in proper biharmonic maps, that is non-harmonic biharmonic maps. Biharmonic maps have been studied intensively in the last decade (see [7], [8], [9], [24], [25], [26], [2], [19], [20], [21]). In the study of almost contact manifolds, Legendre curves play an important role, e. g., a diffeomorphism of a contact manifold is a contact transformation if and only if it maps Legendre curve to Legendre curve.…”

Section: Introductionmentioning

confidence: 99%