Biharmonic PNMC submanifolds in spheres

Balmus, Adina; Montaldo, Stefano; Oniciuc, Cezar

doi:10.1007/s11512-012-0169-5

Cited by 58 publications

(49 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…where p i , h i (i = 1, 2) are polynomials concerning function λ 1 and given by 4 1 + a 1 λ 2 1 + a 2 , h 2 = −54(n + 3)λ 3 1 + a 3 λ 1 , h 3 = 12(n − 4)λ 3 1 + a 4 λ 1 .…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

Biharmonic hypersurfaces with constant scalar curvature in space forms

Hong

2018

Pacific J. Math.

View full text Add to dashboard Cite

Let M n be a biharmonic hypersurface with constant scalar curvature in a space form M n+1 (c). We show that M n has constant mean curvature if c > 0 and M n is minimal if c ≤ 0, provided that the number of distinct principal curvatures is no more than 6. This partially confirms Chen's conjecture and Generalized Chen's conjecture. As a consequence, we prove that there exist no proper biharmonic hypersurfaces with constant scalar curvature in Euclidean space E n+1 or hyperbolic space H n+1 for n < 7.

show abstract

“…where p i , h i (i = 1, 2) are polynomials concerning function λ 1 and given by 4 1 + a 1 λ 2 1 + a 2 , h 2 = −54(n + 3)λ 3 1 + a 3 λ 1 , h 3 = 12(n − 4)λ 3 1 + a 4 λ 1 .…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

Biharmonic hypersurfaces with constant scalar curvature in space forms

Hong

2018

Pacific J. Math.

View full text Add to dashboard Cite

show abstract

“…It is clear from the definition that any biharmonic submanifold is biconservative. Now we are ready to prove the following theorem which generalizes the corresponding results in the Riemannian case given in [2]. Proof.…”

Section: )mentioning

confidence: 60%

Biharmonic submanifolds of pseudo-Riemannian manifolds

2017

Journal of Geometry and Physics

View full text Add to dashboard Cite

In this paper, we derived biharmonic equations for pseudo-Riemannian submanifolds of pseudo-Riemannian manifolds which includes the biharmonic equations for submanifolds of Riemannian manifolds as a special case. As applications, we proved that a pseudo-umbilical biharmonic pseudo-Riemannian submanifold of a pseudo-Riemannian manifold has constant mean curvature, we completed the classifications of biharmonic pseudo-Riemannian hypersurfaces with at most two distinct principal curvatures, which were used to give four construction methods to produce proper biharmonic pseudoRiemannian submanifolds from minimal submanifolds. We also made some comparison study between biharmonic hypersurfaces of Riemannian space forms and the space-like biharmonic hypersurfaces of pseudo-Riemannian space forms. Biharmonic maps between pseudo-Riemannian manifoldsAll manifolds, maps, tensor fields are assumed to be smooth in this paper. 1.1 Some basic concepts and notations from pseudo-Riemannian geometry Let (M m t , g) be a pseudo-Riemannian (or semi-Riemannian) manifold of dimension m with a nondegenerate metric with index t. Here, nondegeneracy means the only vector X ∈ T x M satisfying g x (X, Y ) = 0 for all Y ∈ T x M and all x ∈ M is X = 0. We use |X| = |g(X, X)| 1/2 to denote the norm of a vector X and Date: 12/02/2015. 1991 Mathematics Subject Classification. 58E20, 53C12.

show abstract

“…For classification of biharmonic submanifolds with parallel mean curvature vector and |A| 2 = constant in sphere see [BMO3].…”

Section: Space Form Of Constant Sectional Curvature C Is F -Biharmonimentioning

confidence: 99%

“…For some recent geometric study of general biharmonic maps see [BK], [BFO2], [BMO2], [MO], [NUG], [Ou1], [Ou4], [OL], [Oua] and the references therein. For some recent study of biharmonic submanifolds see [CI], [Ji2], [Ji3], [Di], [CMO1], [CMO2], [BMO1], [BMO3], [Ou3], [OT], [OW], [NU], [TO], * [CM], [AGR] and the references therein. For biharmonic conformal immersions and submersions see [Ou2], [Ou5], [BFO1], [LO], [WO] and the references therein.…”

mentioning

confidence: 99%

Onf-biharmonic maps andf-biharmonic submanifolds

2014

Pacific J. Math.

View full text Add to dashboard Cite

Abstractf -Biharmonic maps are the extrema of the f -bienergy functional. f -biharmonic submanifolds are submanifolds whose defining isometric immersions are fbiharmonic maps. In this paper, we prove that an f -biharmonic map from a compact Riemannian manifold into a non-positively curved manifold with constant f -bienergy density is a harmonic map; any f -biharmonic function on a compact manifold is constant, and that the inversions about S m for m ≥ 3 are proper f -biharmonic conformal diffeomorphisms. We derive f -biharmonic submanifolds equations and prove that a surface in a manifold (N n , h) is an fbiharmonic surface if and only it can be biharmonically conformally immersed into (N n , h). We also give a complete classification of f -biharmonic curves in 3-dimensional Euclidean space. Many examples of proper f -biharmonic maps and f -biharmonic surfaces and curves are given.

show abstract

Biharmonic PNMC submanifolds in spheres

Cited by 58 publications

References 25 publications

Biharmonic hypersurfaces with constant scalar curvature in space forms

Biharmonic hypersurfaces with constant scalar curvature in space forms

Biharmonic submanifolds of pseudo-Riemannian manifolds

Onf-biharmonic maps andf-biharmonic submanifolds

Contact Info

Product

Resources

About