Second variation and Legendrian stabilities of minimal Legendrian submanifolds in Sasakian manifolds

First, we derive a new second variation formula which holds for minimal Legendrian submanifolds in Sasakian manifolds. Using this, we prove that any minimal Legendrian submanifold in an η-Einstein Sasakian manifold with “nonpositive” η-Ricci constant is stable. Next we introduce the notion of the Legendrian stability of minimal Legendrian submanifolds in Sasakian manifolds. Using our second variation formula, we find a general criterion for the Legendrian stability of minimal Legendrian submanifolds in η-Einstein Sasakian manifolds with “positive” η-Ricci constant.     

Geometric classification of warped products isometrically immersed into Sasakian space forms

The main objective of this paper is to study the warped product pointwise semi‐slant submanifolds which are isometrically immersed into Sasakian manifolds. First, we prove some characterizations results in terms of the shape operator, under which influence a pointwise semi‐slant submanifold of a Sasakian manifold can be reduced to a warped product submanifold. Then, we determine a geometric inequality for the second fundamental form regarding to intrinsic invariant and extrinsic invariant using the Gauss equation instead of the Codazzi equation. Evenmore, we give some applications of this inequality into Sasakian space forms, and we will investigate the status of equalities in the inequality. As a particular case, we provide numerous applications of the Green lemma, the Laplacian of warped functions and some partial differential equations. Some triviality results for connected, compact warped product pointwise semi‐slant submanifolds of Sasakian space form by means of Hamiltonian and the kinetic energy of warped function involving boundary conditions are established.     

Biharmonic hypersurfaces in Sasakian space forms

We consider the Boothby–Wang fibration of a strictly regular Sasakian space form N and find the characterization of biharmonic Hopf cylinders over submanifolds of . Then, we determine all proper-biharmonic Hopf cylinders over homogeneous real hypersurfaces in complex projective spaces.     

Codimension two Kähler submanifolds of space forms

In this article we study isometric immersions from Kähler manifolds whose (1, 1) part of the second fundamental form is parallel, theppmc isometric immersions. When the domain is a Riemann surface these immersions are precisely those with parallel mean curvature. P. J. Ryan has classified the Kähler manifolds that admit isometric immersions, as real hypersurfaces, in space forms. We classify the codimension twoppmc isometric immersions into space forms.     

C-Totally real pseudo-parallel submanifolds of Sasakian space forms

Let be (2n + 1)-dimensional Sasakian space form of constant ϕ-sectional curvature (c) and M ⁿ be an ⁿ -dimensional C-totally real, minimal submanifold of . We prove that if M ⁿ is pseudo-parallel and , then M ⁿ is totally geodesic.     

Sasakian空间形式中子流形的全实截面曲率(英文)

By using the methods introduced by Chen[Chen Bang-yen,A series of Ka¨hlerian invarianrts and their applications to Khlerian geometry,Beitrge Algebra Geom,2001,42（1）：165-178],we establish some inequalities for invariant submanifolds in a Sasakian space form involving totally real sectional curvature and the scalar curvature.Moreover,we consider the case of equalities.     

On Ricci curvature ofC-totally real submanifolds in Sasakian space forms

LetM ⁿ be a Riemanniann-manifold. Denote byS(p) and Ric(p) the Ricci tensor and the maximum Ricci curvature onM _n, respectively. In this paper we prove that everyC-totally real submanifold of a Sasakian space formM ^2m+1(c) satisfies , whereH ² andg are the square mean curvature function and metric tensor onM ⁿ, respectively. The equality holds identically if and only if eitherM ⁿ is totally geodesic submanifold or n = 2 andM ⁿ is totally umbilical submanifold. Also we show that if aC-totally real submanifoldM ⁿ ofM ²ⁿ⁺¹ (c) satisfies identically, then it is minimal.     

On biconservative hypersurfaces in ‐dimensional Riemannian space forms

In this paper, we study biconservative hypersurfaces in and . Further, we obtain complete explicit classifications of biconservative hypersurfaces in 4‐dimensional Riemannian space forms with exactly three distinct principal curvatures.     

B.Y. Chen inequalities for slant submanifolds in Sasakian space forms

In the present paper, we establish Chen inequalities for slant submanifolds in Sasakian space forms, by using subspaces orthogonal to the Reeb vector field ξ.     

Minimal Legendrian submanifolds of S and absolutely area-minimizing cones

We use minimal Legendrian submanifolds in spheres to construct examples of absolutely area-minimizing cones and we prove a result about Legendrian 2-tori in S⁵.     

The third fundamental form metric for hypersurfaces in nonflat space forms

For hypersurfaces with regular Weingarten operator in nonflat space forms we study the relations between the intrinsic geometry of the third fundamental form metric and the extrinsic geometry of the hypersurface. We prove a theorema-egregium-type result for this metric and, in particular, give a local classification of hypersurfaces in case of an Einstein structure of this metric.Partially supported by the project 19701003 of NSFC.The geometry groops at TU Berlin and KU Leuven cooperate within the GA DGET program.     

An improved first Chen inequality for Legendrian submanifolds in Sasakian space forms

Legendrian submanifolds in Sasakian space forms play an important role in contact geometry. Defever et al. (Boll Unione Mat Ital B 7(11):365–374, 1997) established the first Chen inequality for C-totally real submanifolds in Sasakian space forms. In this article, we improve this first Chen inequality for Legendrian submanifolds in Sasakian space forms.     

Biharmonic curves in 3-dimensional Sasakian space forms

We show that every proper biharmonic curve in a 3-dimensional Sasakian space form of constant holomorphic sectional curvature H is a helix (both of whose geodesic curvature and geodesic torsion are constants). In particular, if H ≠  1, then it is a slant helix, that is, a helix which makes constant angle α with the Reeb vector field with the property . Moreover, we construct parametric equations of proper biharmonic herices in Bianchi–Cartan–Vranceanu model spaces of a Sasakian space form.      

Some explicit examples of Hamiltonian minimal Lagrangian submanifolds in complex space forms

In this paper, we find some new explicit examples of Hamiltonian minimal Lagrangian submanifolds among the Lagrangian isometric immersions of a real space form in a complex space form.     

Warped product structure of submanifolds with nonpositive extrinsic curvature in space forms

By means of a simple warped product construction we obtain examples of submanifolds with nonpositive extrinsic curvature and minimal index of relative nullity in any space form. We then use this to extend to arbitrary space forms four known splitting results for Euclidean submanifolds with nonpositive sectional curvature.     

Geometry of warped product pseudo-slant submanifolds of Kenmotsu manifolds

In this paper, we study pseudo-slant submanifolds and their warped products in Kenmotsu manifolds. We obtain the necessary conditions that a pseudoslant submanifold is locally a warped product and establish an inequality for the squared norm of the second fundamental form in terms of the warping function. The equality case is also considered.