On the geometry of biharmonic submanifolds in Sasakian space forms
D. Fetcu, C. Oniciuc
备注：
10 pages; Proposition 3.10 was improved; Contribution to the Proceedings of the 10-th International Conference on Geometry, Integrability and Quantization, Varna 2008, Bulgaria
We classify all proper-biharmonic Legendre curves in a Sasakian space form and point out some of their geometric properties. Then we provide a method for constructing anti-invariant proper-biharmonic submanifolds in Sasakian space forms. Finally, using the Boothby-Wang fibration, we determine all proper-biharmonic Hopf cylinders over homogeneous real hypersurfaces in complex projective spaces.
In this article we consider the concept of biharmonicity about the hypersurfaces in the Sasakian space form which is equipped with the Tanaka-Webster connection.
We classify the biharmonic Legendre curves in a Sasakian space form, and obtain their explicit parametric equations in the $(2n+1)$-dimensional unit sphere endowed with the canonical and deformed Sasakian structures defined by Tanno. Then, composing with the flow of the Reeb vector field, we transform a biharmonic integral submanifold into a biharmonic anti-invariant submanifold. Using this method we obtain new examples of biharmonic submanifolds in spheres and, in particular, in $\mathbb{S}^{7}$.
In [5], D. Fetcu and C. Oniciuc presented the classification result for biharmonic $\mathcal{C}$-parallel Legendrian submanifolds in $7$-dimensional Sasakian space forms. However, it is incomplete. In this paper, all such submanifolds are explicitly determined.
Sasakian manifolds provide explicit formulae of some Jacobi operators which describe the biharmonic equation of curves in Riemannian manifolds. In this paper we characterize non-geodesic biharmonic curves in Sasakian manifolds which are either tangent or normal to the Reeb vector field. In the three-dimensional case, we prove that such curves are some helixes whose geodesic curvature and geodesic torsion satisfy a given relation.
In this paper we give necessary and sufficient conditions for spacelike and timelike curves in a conformally flat, quasi conformally flat and conformally symmetric 4-dimensional \textit{LP}-Sasakian manifold to be proper biharmonic. Also, we investigate proper biharmonic curves in the Lorentzian sphere $S^{4}_{1}$.
We find the characterization of maximum dimensional proper-biharmonic integral $\mathcal{C}$-parallel submanifolds of a Sasakian space form and then classify such submanifolds in a 7-dimensional Sasakian space form. Working in the sphere $\mathbb{S}^7$ we explicitly find all 3-dimensional proper-biharmonic integral $C$-parallel submanifolds. We also determine the proper-biharmonic parallel Lagrangian submanifolds of $\mathbb{C}P^3$.
We classify the biharmonic non-Legendre curves in a Sasakian space form for which the angle between the tangent vector field and the characteristic vector field is constant and obtain explicit examples of such curves in $\mathbb{R}^{2n+1}(-3)$.
We characterize biharmonic anti-invariant surfaces in $3$-dimensional generalized $(κ, μ)$-manifolds with non-zero constant mean curvature by means of the scalar curvature of the ambient space and the mean curvature. In addition, we give a method for constructing infinity many examples of biharmonic submanifolds in a certain $3$-dimensional generalized $(κ, μ)$-manifold. Moreover, we determine $3$-dimensional generalized $(κ, μ)$-manifolds which admit a certain kind of proper biharmonic foliation.
The object of this paper is to study the invariant submanifolds of Sasakian generalized-Sasakian-space-form. Here, we obtain some equivalent conditions for an invariant submanifold of a Sasakian generalized-Sasakian-space-forms to be totally geodesic.
In the present paper first, we define the conformal Sasakian manifolds and then we study geometry of invariant, anti-invariant and CR-submanifolds of conformal Sasakian manifolds.
We consider biharmonic submanifolds in both generalized complex and Sasakian space forms. After giving the biharmonicity conditions for submanifolds in these spaces, we study different particular cases for which we obtain curvature estimates. We consider curves, complex and Lagrangian surfaces and hypersurfaces for the generalized complex space form as well as hypersurfaces, invariant and anti-invariant submanifolds in case of generalized Sasakian space form.
We introduce a weaker notion of generic submanifold of a Sasakian manifold and we prove a Frankel type theorem for this kind of submanifolds under suitable hypotesis on the index of the scalar Levi forms determined by normal directions. It concerns the intersection between a generic and an invariant submanifold and the intersection between two generic submanifolds. From this theorem we derive some topological information about generic submanifolds of Sasakian space forms.
Warped product CR-submanifolds in Kaehlerian manifolds were intensively studied only since 2001 after the impulse given by B.Y. Chen. Immediately after, another line of research, similar to that concerning Sasakian geometry as the odd dimensional version of Kaehlerian geometry, was developed, namely warped product contact CR-submanifolds in Sasakian manifolds. In this note we proved that there exists no proper doubly warped product contact CR-submanifolds in trans-Sasakian manifolds.
In this paper we provide a second variation formula for L-minimal Lagrangian submanifolds in a pseudo-Sasakian manifold. We apply it to the case of Lorentzian-Sasakian manifolds and relate the L-stability of L-minimal Legendrians in a Sasakian M to their L-stability in an associated Lorentzian-Sasakian structure on M.
We give necessary and sufficient conditions for a Lagrangian submanifold of a Kähler manifold to be biharmonic. Furthermore, we classify biharmonic PNMC Lagrangian submanifolds in the complex space forms.
In this note, we give a brief survey on some recent developments of biharmonic submanifolds. After reviewing some recent progress on Chen's biharmonic conjecture, the Generalized Chen's conjecture on biharmonic submanifolds of non-positively curved manifolds, and some classifications of biharmonic submanifolds of spheres, we give a short list of some open problems in this topic.
The present paper deals with the study of invariant submanifolds of generalized Sasakian-space-forms with respect to Levi-Civita connection as well as semi-symmetric metric connection. We provide some examples of such submanifolds and obtain many new results including, the necessary and sufficient conditions under which the submanifolds are totally geodesic. The Ricci solitons of such submanifolds are also studied.
In this paper, we introduce and study a new class of CR-lightlike submanifold of an indefinite nearly Sasakian manifold, called Quasi Generalized Cauchy-Riemann (QGCR) lightlike submanifold. We give some characterization theorems for the existence of QGCR-lightlike submanifolds and finally derive necessary and sufficient conditions for some distributions to be integrable.
The present paper deals with some results of submanifolds of generalized Sasakian-space-forms in \cite{ALEGRE3} with respect to semisymmetric metric connection, semisymmetric non-metric connection, Schouten-van Kampen connection and Tanaka-webster connection.
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 pseudo-Riemannian submanifol…
A structure on an almost contact metric manifold is defined as a generalization of well-known cases: Sasakian, quasi-Sasakian, Kenmotsu and cosymplectic. Then we consider a semi-invariant $ξ^{\bot}$-submanifold of a manifold endowed with such a structure and two topics are studied: the integrability of distributions defined by this submanifold and characterizations for the totally umbilical case. In particular we recover results of Kenmotsu, Eum and Papaghiuc.
We construct biharmonic real hypersurfaces and Lagrangian submanifolds of Clifford torus type in $CP^n$ via the Hopf fibration; and get new examples of biharmonic submanifolds in $S^{2n+1}$ as byproducts .
The present paper deals with the study of Chaki-pseudo parallel and Deszcz-pseudo parallel invariant submanifolds of SQ-Sasakian manifolds with respect to Levi-Civita connection and semisymmetric metric connection and obtain that these two classes are equivalent with a certain condition. Also the invariant and anti-invariant submanifolds of SQ-Sasakian manifolds with respect to Levi-Civita connection as well as semisymmetric metric connection whose metrics are Ricci solitons are studied.
We classify biharmonic submanifolds with certain geometric properties in Euclidean spheres. For codimension 1, we determine the biharmonic hypersurfaces with at most two distinct principal curvatures and the conformally flat biharmonic hypersurfaces. We obtain some rigidity results for pseudo-umbilical biharmonic submanifolds of codimension 2 and for biharmonic surfaces with parallel mean curvature vector field. We also study the type, in the sense of B-Y. Chen, of compact proper biharmonic submanifolds with constant mean curvature in spheres.
In this paper we construct proper biharmonic submanifolds into various types of ellipsoids. We also prove, in this context, some useful composition properties which can be used to produce large families of new proper biharmonic immersions.
We study biharmonic hypersurfaces and biharmonic submanifolds in a Riemannian manifold. One of interesting problems in this direction is Chen's conjecture which says that any biharmonic submanifold in a Euclidean space is minimal. From the invariant equation for biharmonic submanifolds, we derive a fundamental identity involving the mean curvature vector field, and using this, we prove Chen's conjecture on biharmonic submanifolds in a Euclidean space. More generally, it is proved that any biharmonic submanifold in a space form of nonpositively sectional curvature is minimal. Furthermore we pro…
f-Biharmonic maps are the extrema of the f-bienergy functional. f-biharmonic submanifolds are submanifolds whose defining isometric immersions are f-biharmonic 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 $m\ge 3$ are proper f-biharmonic conformal diffeomorphisms. We derive f-biharmonic submanifolds equations and prove that a surface in a manifold $(N^n, h)$ is …
We generalize the Ruh-Vilms problem by characterizing the submanifolds in Euclidean spaces with proper biharmonic Gauss map and we construct examples of such hypersurfaces.
We find some integral formulas of Simons and Bochner type and use them to study biharmonic and biconservative submanifolds in space forms. We obtain rigidity results that in the biharmonic case represent partial answers to two well-known conjectures on such submanifolds in spheres.
We present some results on the boundedness of the mean curvature of proper biharmonic submanifolds in spheres. A partial classification result for proper biharmonic submanifolds with parallel mean curvature vector field in spheres is obtained. Then, we completely classify the proper biharmonic submanifolds in spheres with parallel mean curvature vector field and parallel Weingarten operator associated to the mean curvature vector field.
We continue our study [Ou4] of f-biharmonic maps and f-biharmonic submanifolds by exploring the applications of f-biharmonic maps and the relationships among biharmonicity, f-biharmonicity and conformality of maps between Riemannian manifolds. We are able to characterize harmonic maps and minimal submanifolds by using the concept of f-biharmonic maps and prove that the set of all f-biharmonic maps from 2-dimensional domain is invariant under the conformal change of the metric on the domain. We give an improved equation for f-biharmonic hypersurfaces and use it to prove some rigidity theorems a…
In the present paper we survey the most recent classification results for proper biharmonic submanifolds in unit Euclidean spheres. We also obtain some new results concerning geometric properties of proper biharmonic constant mean curvature submanifolds in spheres.
We obtain several rigidity results for biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector field. We classify biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector field and with at most two distinct principal curvatures. In particular, we determine all biharmonic surfaces with parallel normalized mean curvature vector field in $\mathbb{S}^n$. Then we investigate, for (not necessarily compact) proper biharmonic submanifolds in $\mathbb{S}^n$, their type in the sense of B-Y. Chen. We prove: (i) a proper biharmonic subma…
The generalized Chen's conjecture on biharmonic submanifolds asserts that any biharmonic submanifold of a non-positively curved manifold is minimal (see e.g., [CMO1], [MO], [BMO1], [BMO2], [BMO3], [Ba1], [Ba2], [Ou1], [Ou2], [IIU]). In this paper, we prove that this conjecture is false by constructing foliations 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.
For a Legendrian submanifold $M$ of a Sasaki manifold $N$, we study harmonicity and biharmonicity of the corresponding Lagrangian cone submanifold C(M) of a Kaehler manifold C(N). We show that, if $C(M)$ is biharmonic in C(N), then it is harmonic; and $M$ is proper biharmonic in $N$ if and only if C(M) has a non-zero eigen-section of the Jacobi operator with the eigenvalue $m=dim M$.
In this note, we classify biharmonic submanifolds in a sphere defined by bi-eigenmaps ($Δ^2 ϕ=λϕ$) or buckling eigenmaps ($Δ^2 ϕ=-μΔϕ$). We then classify biharmonic bi-eigenmaps and buckling eigenmaps into spheres with constant energy density. The results can be viewed as generalizations of Takahashi's characterization of minimal submanifolds in a sphere by eigenmaps.
We find a Simons type formula for submanifolds with parallel mean curvature vector (pmc submanifolds) in product spaces $M^n(c)\times\mathbb{R}$, where $M^n(c)$ is a space form with constant sectional curvature $c$, and then we use it to prove a gap theorem for the mean curvature of certain complete proper-biharmonic pmc submanifolds, and classify proper-biharmonic pmc surfaces in $\mathbb{S}^n(c)\times\mathbb{R}$.
We consider a complete biharmonic immersed submanifold $M$ in an Euclidean space $\mathbb{E}^N$. Assume that the immersion is proper, that is, the preimage of every compact set in $\mathbb{E}^N$ is also compact in $M$. Then, we prove that $M$ is minimal. It is considered as an affirmative answer to the global version of Chen's conjecture for biharmonic submanifolds.
We give some general results on proper-biharmonic submanifolds of a complex space form and, in particular, of the complex projective space. These results are mainly concerned with submanifolds with constant mean curvature or parallel mean curvature vector field. We find the relation between the bitension field of the inclusion of a submanifold $\bar{M}$ in $\mathbb{C}P^n$ and the bitension field of the inclusion of the corresponding Hopf-tube in $\mathbb{S}^{2n+1}$. Using this relation we produce new families of proper-biharmonic submanifolds of $\mathbb{C}P^n$. We study the geometry of biharm…
In this paper, we study warped products of contact skew-CR submanifolds, called contact skew CR-warped products. We establish an inequality for the squared norm of the second fundamental form in terms of the warping function and the slant angle. The equality case in the statement of the inequality is investigated and some applications of derived inequality are given. Furthermore, we provide a non-trivial example of such submanifolds.
We study f-biharmonic and bi-f-harmonic submanifolds in both generalized complex and Sasakian space forms. We prove necessary and sufficient condition for f-biharmonicity and bi-f-harmonicity in the general case and many particular cases. Some non-existence results are also obtained.
We prove some new rigidity results for proper biharmonic immersions in ${\mathbb S}^n$ of the following types: Dupin hypersurfaces; hypersurfaces, both compact and non-compact, with bounded norm of the second fundamental form; hypersurfaces satisfying intrinsic properties; PMC submanifolds; parallel submanifolds.
First, we prove that indefinite Sasakian manifolds do not admit any screen conformal $r$-null submanifolds, tangent to the structure vector field. We, therefore, define a special class of null submanifolds, called; {\it contact screen conformal} $r$-null submanifold of indefinite Sasakian manifolds. Several characterization results, on the above class of null submanifolds, are proved. In particular, we prove that such null submanifolds exist in indefinite Sasakian space forms of constant holomorphic sectional curvatures of $-3$.
We obtain the parametric equations of all biharmonic Legendre curves and Hopf cylinders in the 3-dimensional unit sphere endowed with the modified Sasakian structure defined by Tanno.
A submanifold $M$ of a Euclidean $m$-space is said to be biharmonic if $Δ\overrightarrow H=0$ holds identically, where $\overrightarrow H$ is the mean curvature vector field and $Δ$ is the Laplacian on $M$. In 1991, the author conjectured that every biharmonic submanifold of a Euclidean space is minimal. The study of biharmonic submanifolds is nowadays a very active research subject. In particular, since 2000 biharmonic submanifolds have been receiving a growing attention and have become a popular subject of study with many progresses. In this article, we provide a brief survey on recent devel…
In this paper, we solve affirmatively B.-Y. Chen's conjecture for hypersurfaces in the Euclidean space, under a generic condition. More precisely, every biharmonic hypersurface of the Euclidean space must be minimal if their principal curvatures are simple, and the associated frame field is irreducible.
We obtain a basic inequality involving the Laplacian of the warping function and the squared mean curvature of any warped product isometrically immersed in a Riemannian manifold without assuming any restriction on the Riemann curvature tensor of the ambient manifold. Applying this general theory, we obtain basic inequalities involving the Laplacian of the warping function and the squared mean curvature of $C$-totally real warped product submanifolds of $(κ,μ) $-space forms, Sasakian space forms and non-Sasakian $(κ,μ) $-manifolds. Then we obtain obstructions to the existence of minimal isometr…
In this paper, we study warped product submanifolds of nearly trans-Sasakian manifolds. The non-existence of the warped product semi-slant submanifolds of the type $N_θ\times{_{f}N_T}$ is shown, whereas some characterization and new geometric obstructions are obtained for the warped products of the type $N_T\times{_{f}N_θ}$. We establish two general inequalities for the squared norm of the second fundamental form. The first inequality generalizes derived inequalities for some contact metric manifolds [16, 18, 19, 24], while by a new technique, the second inequality is constructed to express th…
