scholarly journals Hypersurfaces of a Sasakian Manifold

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 877 ◽  
Author(s):  
Haila Alodan ◽  
Sharief Deshmukh ◽  
Nasser Bin Turki ◽  
Gabriel-Eduard Vîlcu
Keyword(s):  
Vector Field ◽  
Laplace Operator ◽  
Upper Bound ◽  
Mean Curvature ◽  
Sasakian Manifold ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Isometric To A Sphere ◽  
The Laplace Operator ◽  
Orientable Hypersurface

We extend the study of orientable hypersurfaces in a Sasakian manifold initiated by Watanabe. The Reeb vector field ξ of the Sasakian manifold induces a vector field ξ T on the hypersurface, namely the tangential component of ξ to hypersurface, and it also gives a smooth function ρ on the hypersurface, which is the projection of the Reeb vector field on the unit normal. First, we find volume estimates for a compact orientable hypersurface and then we use them to find an upper bound of the first nonzero eigenvalue of the Laplace operator on the hypersurface, showing that if the equality holds then the hypersurface is isometric to a certain sphere. Also, we use a bound on the energy of the vector field ∇ ρ on a compact orientable hypersurface in a Sasakian manifold in order to find another geometric condition (in terms of mean curvature and integral curves of ξ T ) under which the hypersurface is isometric to a sphere. Finally, we study compact orientable hypersurfaces with constant mean curvature in a Sasakian manifold and find a sharp upper bound on the first nonzero eigenvalue of the Laplace operator on the hypersurface. In particular, we show that this upper bound is attained if and only if the hypersurface is isometric to a sphere, provided that the Ricci curvature of the hypersurface along ∇ ρ has a certain lower bound.

scholarly journals Hypersurfaces of a Sasakian manifold - revisited

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sharief Deshmukh ◽  
Olga Belova ◽  
Nasser Bin Turki ◽  
Gabriel-Eduard Vîlcu
Keyword(s):  
Vector Field ◽  
Laplace Operator ◽  
Sasakian Manifold ◽  
Necessary Condition ◽  
Dimensional Sphere ◽  
Normal Vector ◽  
Normal Vector Field ◽  
Isometric To A Sphere ◽  
The Laplace Operator ◽  
Orientable Hypersurface

AbstractWe study orientable hypersurfaces in a Sasakian manifold. The structure vector field ξ of a Sasakian manifold determines a vector field v on a hypersurface that is the component of the Reeb vector field ξ tangential to the hypersurface, and it also gives rise to a smooth function σ on the hypersurface, namely the projection of ξ on the unit normal vector field N. Moreover, we have a second vector field tangent to the hypersurface, given by $\mathbf{u}=-\varphi (N)$ u = − φ ( N ) . In this paper, we first find a necessary and sufficient condition for a compact orientable hypersurface to be totally umbilical. Then, with the assumption that the vector field u is an eigenvector of the Laplace operator, we find a necessary condition for a compact orientable hypersurface to be isometric to a sphere. It is shown that the converse of this result holds, provided that the Sasakian manifold is the odd dimensional sphere $\mathbf{S}^{2n+1}$ S 2 n + 1 . Similar results are obtained for the vector field v under the hypothesis that this is an eigenvector of the Laplace operator. Also, we use a bound on the integral of the Ricci curvature $Ric ( \mathbf{u},\mathbf{u} ) $ R i c ( u , u ) of the compact hypersurface to find a necessary condition for the hypersurface to be isometric to a sphere. We show that this condition is also sufficient if the Sasakian manifold is $\mathbf{S}^{2n+1}$ S 2 n + 1 .

scholarly journals B.-Y. Chen inequalities for semislant submanifolds in Sasakian space forms

2003 ◽  
Vol 2003 (27) ◽  
pp. 1731-1738 ◽  
Author(s):  
Dragoş Cioroboiu
Keyword(s):  
Vector Field ◽  
Mean Curvature ◽  
Riemannian Space ◽  
Sasakian Manifold ◽  
Sharp Inequality ◽  
Space Forms ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Sasakian Space Forms ◽  
Riemannian Space Forms

Chen (1993) established a sharp inequality for the sectional curvature of a submanifold in Riemannian space forms in terms of the scalar curvature and squared mean curvature. The notion of a semislant submanifold of a Sasakian manifold was introduced by J. L. Cabrerizo, A. Carriazo, L. M. Fernandez, and M. Fernandez (1999). In the present paper, we establish Chen inequalities for semislant submanifolds in Sasakian space forms by using subspaces orthogonal to the Reeb vector fieldξ.

Lagrangian submersions from normal almost contact manifolds

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3885-3895 ◽  
Author(s):  
Hakan Taştan
Keyword(s):  
Vector Field ◽  
Riemannian Manifolds ◽  
Sasakian Manifold ◽  
Horizontal Distribution ◽  
Contact Manifolds ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Kenmotsu Manifolds ◽  
The One ◽  
Lagrangian Submersion

We study Lagrangian submersions from Sasakian and Kenmotsu manifolds onto Riemannian manifolds. We prove that the horizontal distribution of a Lagrangian submersion from a Sasakian manifold onto a Riemannian manifold admitting vertical Reeb vector field is integrable, but the one admitting horizontal Reeb vector field is not. We also show that the horizontal distribution of a such submersion is integrable when the total manifold is Kenmotsu. Moreover, we give some applications of these results.

Some characterizations of property of trans-Sasakian 3-manifolds

2021 ◽  
Vol 71 (2) ◽  
pp. 383-390
Author(s):  
Yan Zhao
Keyword(s):  
Vector Field ◽  
Sasakian Manifold ◽  
Reeb Vector ◽  
Reeb Vector Field

Abstract In this paper, we obtain some characterizations on the Reeb vector field for a trans-Sasakian manifold to be proper.

scholarly journals Singularities of plane complex curves and limits of Kähler metrics with cone singularities. I: Tangent Cones

2017 ◽  
Vol 4 (1) ◽  
pp. 43-72 ◽  
Author(s):  
Martin de Borbon
Keyword(s):  
Vector Field ◽  
Einstein Metrics ◽  
Projective Line ◽  
Tangent Cones ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Complex Curves ◽  
Complex Dimensions ◽  
Irregular Case ◽  
Kähler Einstein Metrics

Abstract The goal of this article is to provide a construction and classification, in the case of two complex dimensions, of the possible tangent cones at points of limit spaces of non-collapsed sequences of Kähler-Einstein metrics with cone singularities. The proofs and constructions are completely elementary, nevertheless they have an intrinsic beauty. In a few words; tangent cones correspond to spherical metrics with cone singularities in the projective line by means of the Kähler quotient construction with respect to the S1-action generated by the Reeb vector field, except in the irregular case ℂβ₁×ℂβ₂ with β₂/ β₁ ∉ Q.

scholarly journals Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow

2020 ◽  
Vol 18 (1) ◽  
pp. 1518-1530
Author(s):  
Xuesen Qi ◽  
Ximin Liu
Keyword(s):  
Laplace Operator ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
Coefficient Function ◽  
First Eigenvalue ◽  
Initial Hypersurface ◽  
The Mean ◽  
The Laplace Operator

Abstract In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.

Yamabe solitons on 3-dimensional contact metric manifolds with Qφ = φQ

2019 ◽  
Vol 16 (03) ◽  
pp. 1950039 ◽  
Author(s):  
V. Venkatesha ◽  
Devaraja Mallesha Naik
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Sasakian Manifold ◽  
Contact Metric Manifold ◽  
Reeb Vector Field ◽  
3 Dimensional ◽  
Flow Vector ◽  
Contact Metric Manifolds ◽  
Yamabe Soliton

If [Formula: see text] is a 3-dimensional contact metric manifold such that [Formula: see text] which admits a Yamabe soliton [Formula: see text] with the flow vector field [Formula: see text] pointwise collinear with the Reeb vector field [Formula: see text], then we show that the scalar curvature is constant and the manifold is Sasakian. Moreover, we prove that if [Formula: see text] is endowed with a Yamabe soliton [Formula: see text], then either [Formula: see text] is flat or it has constant scalar curvature and the flow vector field [Formula: see text] is Killing. Furthermore, we show that if [Formula: see text] is non-flat, then either [Formula: see text] is a Sasakian manifold of constant curvature [Formula: see text] or [Formula: see text] is an infinitesimal automorphism of the contact metric structure on [Formula: see text].

scholarly journals Lie Derivatives and Ricci Tensor on Real Hypersurfaces in Complex Two-plane Grassmannians

2018 ◽  
Vol 61 (3) ◽  
pp. 543-552
Author(s):  
Imsoon Jeong ◽  
Juan de Dios Pérez ◽  
Young Jin Suh ◽  
Changhwa Woo
Keyword(s):  
Vector Field ◽  
Differential Operator ◽  
Ricci Tensor ◽  
Real Hypersurface ◽  
Real Hypersurfaces ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Lie Derivatives ◽  
Lie Derivation

AbstractOn a real hypersurface M in a complex two-plane Grassmannian G2() we have the Lie derivation and a differential operator of order one associated with the generalized Tanaka–Webster connection . We give a classification of real hypersurfaces M on G2() satisfying , where ξ is the Reeb vector field on M and S the Ricci tensor of M.

Ricci almost solitons and contact geometry

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Amalendu Ghosh
Keyword(s):  
Vector Field ◽  
Contact Manifold ◽  
Ricci Soliton ◽  
Jacobi Field ◽  
Reeb Vector ◽  
Potential Vector ◽  
Reeb Vector Field ◽  
Rigidity Results ◽  
Parallel Ricci Tensor ◽  
Potential Vector Field

Abstract We prove that on a K-contact manifold, a Ricci almost soliton is a Ricci soliton if and only if the potential vector field V is a Jacobi field along the Reeb vector field ξ. Then we study contact metric as a Ricci almost soliton with parallel Ricci tensor. To this end, we consider Ricci almost solitons whose potential vector field is a contact vector field and prove some rigidity results.

Sign in / Sign up

Export Citation Format

Share Document