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.     

Hypersurfaces in simply connected space forms

Let M be a hypersurface in a simply connected space form . We prove some rigidity results for M in terms of lower bounds on the Ricci curvature of the hypersurface M.     

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.      

Sasakian流形上的Schur引理

根据截面曲率给出维数≥5的Sasakian流形是Sasakian空间形式的一个充分必要条件,它是Schur引理的延伸.     

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.     

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.     

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.     

Integral curves of characteristic vector fields of real hypersurfaces in nonflat complex space forms

In this paper, we study real hypersurfaces all of whose integral curves of characteristic vector fields are plane curves in a nonflat complex space form.      

Ideal <Emphasis Type="Italic">C</Emphasis>-totally real submanifolds in Sasakian space forms

Roughly speaking, an ideal immersion of a Riemannian manifold into a space form is an isometric immersion which produces the least possible amount of tension from the ambient space at each point of the submanifold. Recently, B.-Y. Chen classified Lagrangian immersions in complex space forms, which are ideal. In the present paper, we investigate ideal C-totally real submanifolds in a Sasakian space form. Mathematics Subject Classification (2000) 53C40, 53C25     

Parallel mean curvature biconservative surfaces in complex space forms

In this paper, we consider PMC surfaces in complex space forms, and study the interaction between the notions of PMC, totally real and biconservative. We first consider PMC surfaces in a non-flat complex space form and prove that they are biconservative if and only if totally real. Then, we find a Simons-type formula for a well-chosen vector field constructed from the mean curvature vector field and use it to prove a rigidity result for CMC biconservative surfaces in two-dimensional complex space forms. We prove then a reduction codimension result for PMC biconservative surfaces in non-flat complex space forms. We conclude by constructing examples of CMC non-PMC biconservative submanifolds from the Segre embedding and discuss when they are proper-biharmonic.     

Convergence and the length spectrum

The author defines and analyzes the 1/k length spectra, L1/k(M), whose union, over all k∈N is the classical length spectrum. These new length spectra are shown to converge in the sense that limk→∞K1/k(Mi)⊂L1/k(M)∪{0} as Mi→M in the Gromov-Hausdorff sense. Energy methods are introduced to estimate the shortest element of L1/k, as well as a concept called the minimizing index which may be used to estimate the length of the shortest closed geodesic of a simply connected manifold in any dimension. A number of gap theorems are proven, including one for manifolds, Mn, with Ricci?(n−1) and volume close to Vol(Sn). Many results in this paper hold on compact length spaces in addition to Riemannian manifolds.     

Sasakian空间型的C-全实伪脐子流形

本文讨论了Ｓａｓａｋｉａｎ空间型的Ｃ-全实伪脐子流形，给出关于第二基本形式长度的一个Ｐｉｎｃｈｉｎｇ定理．     

The length of harmonic forms on a compact Riemannian manifold

We study -dimensional Riemannian manifolds with harmonic forms of constant length and first Betti number equal to showing that they are -step nilmanifolds with some special metrics. We also characterize, in terms of properties on the product of harmonic forms, the left-invariant metrics among them. This allows us to clarify the case of equality in the stable isosytolic inequalities in that setting. We also discuss other values of the Betti number.

     

The generalized Lichnerowicz problem: Uniformly quasiregular mappings and space forms

A uqr mapping of an -manifold is a mapping which is rational with respect to a bounded measurable conformal structure on . Remarkably, the only closed manifolds on which locally (but not globally) injective uqr mappings act are Euclidean space forms. We further characterize space forms admitting uniformly quasiregular self mappings and we show that the space forms admitting branched uqr maps are precisely the spherical space forms. We further show that every non-injective uqr map of a Euclidean space form is a quasiconformal conjugate of a conformal map. This is not true if the non-injective hypothesis is removed.

     

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空间形式中具有平行平均曲率向量的C-全实子流形

本文讨论了Sasakian空间形式中具有平行平均曲率向量的C-全实子流形,得到了一个Simons型公式并且改进了S.Yamaguchi等的一个结果.