scholarly journals On 3-Dimensional Contact Metric Generalized(k,μ)-Space Forms

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
D. G. Prakasha ◽  
Shyamal Kumar Hui ◽  
Kakasab Mirji
Keyword(s):  
Constant Curvature ◽  
Space Form ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Space Forms ◽  
3 Dimensional ◽  
Projectively Flat ◽  
Necessary And Sufficient

The present paper deals with a study of 3-dimensional contact metric generalized(k,μ)-space forms. We obtained necessary and sufficient condition for a 3-dimensional contact metric generalized(k,μ)-space form withQϕ=ϕQto be of constant curvature. We also obtained some conditions of such space forms to be pseudosymmetric andξ-projectively flat, respectively.

scholarly journals On locally conformal Kähler space forms

1985 ◽  
Vol 8 (1) ◽  
pp. 69-74 ◽  
Author(s):  
Koji Matsumoto
Keyword(s):  
Space Form ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Covariant Differentiation ◽  
Space Forms ◽  
Riemannian Curvature Tensor ◽  
Locally Conformal Kähler ◽  
Necessary And Sufficient ◽  
Locally Conformal ◽  
Lee Form

Anm-dimensional locally conformal Kähler manifold (l.c.K-manifold) is characterized as a Hermitian manifold admitting a global closedl-formαλ(called the Lee form) whose structure(Fμλ,gμλ)satisfies∇νFμλ=−βμgνλ+βλgνμ−αμFνλ+αλFνμ, where∇λdenotes the covariant differentiation with respect to the Hermitian metricgμλ,βλ=−Fλϵαϵ,Fμλ=Fμϵgϵλand the indicesν,μ,…,λrun over the range1,2,…,m.For l. c. K-manifolds, I. Vaisman [4] gave a typical example and T. Kashiwada ([1], [2],[3]) gave a lot of interesting properties about such manifolds.In this paper, we shall study certain properties of l. c. K-space forms. In§2, we shall mainly get the necessary and sufficient condition that an l. c. K-space form is an Einstein one and the Riemannian curvature tensor with respect togμλwill be expressed without the tensor fieldPμλ. In§3, we shall get the necessary and sufficient condition that the length of the Lee form is constant and the sufficient condition that a compact l. c. K-space form becomes a complex space form. In the last§4, we shall prove that there does not exist a non-trivial recurrent l. c. K-space form.

scholarly journals f-biharmonic and bi-f-harmonic submanifolds of generalized (k, µ)-space-forms

2019 ◽  
Vol 27 (3) ◽  
pp. 97-112
Author(s):  
Shyamal Kumar Hui ◽  
Daniel Breaz ◽  
Pradip Mandal
Keyword(s):  
Space Form ◽  
Ambient Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Space Forms ◽  
Necessary And Sufficient

AbstractHere we have studied f-biharmonic and bi-f-harmonic submanifolds of generalized (k, µ)-space-forms and obtained a necessary and sufficient condition on a submanifold of generalized (k, µ)-space-form to be f-biharmonic and bi-f-harmonic submanifold. We have also studied f-biharmonic hypersurfaces of said ambient space forms.

scholarly journals Classification of warped product pointwise semi-slant submanifolds in complex space forms

Filomat ◽  
2019 ◽  
Vol 33 (16) ◽  
pp. 5273-5290
Author(s):  
Akram Ali ◽  
Ali Alkhaldi ◽  
Jae Lee ◽  
Wan Othman
Keyword(s):  
Warped Product ◽  
Complex Space ◽  
Space Form ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Slant Submanifold ◽  
Space Forms ◽  
Riemannian Product ◽  
Necessary And Sufficient ◽  
Complex Space Forms

The main principle of this paper is to show that, a warped product pointwise semi-slant submanifold of type Mn = Nn1 T xf Nn2? in a complex space form ?M2m (C) admitting shrinking or steady gradient Ricci soliton, whose potential function is a well-define warped function, is an Einstein warped product pointwise semi-slant submanifold under extrinsic restrictions on the second fundamental form inequality attaining the equality in [4]. Moreover, under some geometric assumption, the connected and compactness with nonempty boundary are treated. In this case, we propose a necessary and sufficient condition in terms of Dirichlet energy function which show that a connected, compact warped product pointwise semi-slant submanifold of complex space forms must be a Riemannian product. As more applications, for the first one, we prove that Mn is a trivial compact warped product, when the warping function exist the solution of PDE such as Euler-Lagrange equation. In the second one, by imposing boundary conditions, we derive a necessary and sufficient condition in terms of Ricci curvature, and prove that, a compact warped product pointwise semi-slant submanifold Mn of a complex space form, is either a CR warped product or just a usual Riemannian product manifold. We also discuss some obstructions to these constructions in more details.

scholarly journals Mannheim Curves in Nonflat 3-Dimensional Space Forms

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Wenjing Zhao ◽  
Donghe Pei ◽  
Xinyu Cao
Keyword(s):  
Linear Relationship ◽  
Dimensional Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
De Sitter ◽  
Space Forms ◽  
3 Dimensional ◽  
Constant Coefficients ◽  
Necessary And Sufficient ◽  
The Given

We consider the Mannheim curves in nonflat 3-dimensional space forms (Riemannian or Lorentzian) and we give the concept of Mannheim curves. In addition, we investigate the properties of nonnull Mannheim curves and their partner curves. We come to the conclusion that a necessary and sufficient condition is that a linear relationship with constant coefficients will exist between the curvature and the torsion of the given original curves. In the case of null curve, we reveal that there are no null Mannheim curves in the 3-dimensional de Sitter space.

scholarly journals Some Curvature Properties on Lorentzian Generalized Sasakian-Space-Forms

2019 ◽  
Vol 2019 ◽  
pp. 1-7 ◽  
Author(s):  
Rongsheng Ma ◽  
Donghe Pei
Keyword(s):  
Space Form ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Conformally Flat ◽  
Sasakian Space Form ◽  
Space Forms ◽  
Sasakian Space Forms ◽  
Projectively Flat ◽  
Generalized Sasakian Space Form ◽  
Necessary And Sufficient

In this paper, we investigate the Lorentzian generalized Sasakian-space-form. We give the necessary and sufficient conditions for the Lorentzian generalized Sasakian-space-form to be projectively flat, conformally flat, conharmonically flat, and Ricci semisymmetric and their relationship between each other. As the application of our theorems, we study the Ricci almost soliton on conformally flat Lorentzian generalized Sasakian-space-form.

scholarly journals Pinching Theorems for Statistical Submanifolds in Sasaki-Like Statistical Space Forms

Entropy ◽  
2018 ◽  
Vol 20 (9) ◽  
pp. 690 ◽  
Author(s):  
Ali Alkhaldi ◽  
Mohd. Aquib ◽  
Aliya Siddiqui ◽  
Mohammad Shahid
Keyword(s):  
Constant Curvature ◽  
Necessary And Sufficient Condition ◽  
Einstein Field Equations ◽  
Statistical Manifold ◽  
Field Equations ◽  
Space Forms ◽  
Equality Case ◽  
Statistical Manifolds ◽  
Necessary And Sufficient ◽  
Statistical Space

In this paper, we obtain the upper bounds for the normalized δ -Casorati curvatures and generalized normalized δ -Casorati curvatures for statistical submanifolds in Sasaki-like statistical manifolds with constant curvature. Further, we discuss the equality case of the inequalities. Moreover, we give the necessary and sufficient condition for a Sasaki-like statistical manifold to be η -Einstein. Finally, we provide the condition under which the metric of Sasaki-like statistical manifolds with constant curvature is a solution of vacuum Einstein field equations.

scholarly journals Properly embedded and immersed minimal surfaces in the Heisenberg group

2004 ◽  
Vol 70 (3) ◽  
pp. 507-520 ◽  
Author(s):  
Jih-Hsin Cheng ◽  
Jenn-Fang Hwang
Keyword(s):  
Heisenberg Group ◽  
Minimal Surface ◽  
Explicit Expression ◽  
Minimal Surfaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Minimal Graphs ◽  
3 Dimensional ◽  
Necessary And Sufficient ◽  
Band Type

We study properly embedded and immersed p(pseudohermitian)-minimal surfaces in the 3-dimensional Heisenberg group. From the recent work of Cheng, Hwang, Malchiodi, and Yang, we learn that such surfaces must be ruled surfaces. There are two types of such surfaces: band type and annulus type according to their topology. We givn an explicit expression for these surfaces. Among band types there is a class of properly embedded p-minimal surfaces of so called helicoid type. We classify all the helicoid type p-minimal surfaces. This class of p-minimal surfaces includes all the entire p-minimal graphs (except contact planes) over any plane. Moreover, we give a necessary and sufficient condition for such a p-minimal surface to have no singular points. For general complete immersed p-minimal surfaces, we prove a half space theorem and give a criterion for the properness.

scholarly journals Weighted projective Ricci curvature in Finsler geometry

2021 ◽  
Vol 71 (1) ◽  
pp. 183-198
Author(s):  
Tayebeh Tabatabaeifar ◽  
Behzad Najafi ◽  
Akbar Tayebi
Keyword(s):  
Ricci Curvature ◽  
Finsler Geometry ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Randers Metric ◽  
Ricci Flat ◽  
Projectively Flat ◽  
Flat Metric ◽  
Necessary And Sufficient ◽  
Randers Metrics

Abstract In this paper, we introduce the weighted projective Ricci curvature as an extension of projective Ricci curvature introduced by Z. Shen. We characterize the class of Randers metrics of weighted projective Ricci flat curvature. We find the necessary and sufficient condition under which a Kropina metric has weighted projective Ricci flat curvature. Finally, we show that every projectively flat metric with isotropic weighted projective Ricci and isotropic S-curvature is a Kropina metric or Randers metric.

scholarly journals Some Conditions on Trans-Sasakian Manifolds to Be Homothetic to Sasakian Manifolds

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1887
Author(s):  
Sharief Deshmukh ◽  
Amira Ishan ◽  
Olga Belova ◽  
Suha B. Al-Shaikh
Keyword(s):  
Sufficient Conditions ◽  
Sasakian Manifold ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Sasakian Manifolds ◽  
3 Dimensional ◽  
Necessary And Sufficient ◽  
Simply Connected

In this paper, we study 3-dimensional compact and connected trans-Sasakian manifolds and find necessary and sufficient conditions under which these manifolds are homothetic to Sasakian manifolds. First, four results in this paper deal with finding necessary and sufficient conditions on a compact and connected trans-Sasakian manifold to be homothetic to a compact and connected Sasakian manifold, and the fifth result deals with finding necessary and sufficient condition on a connected trans-Sasakian manifold to be homothetic to a connected Sasakian manifold. Finally, we find necessary and sufficient conditions on a compact and simply connected trans-Sasakian manifold to be homothetic to a compact and simply connected Einstein Sasakian manifold.

On Projectively Flat (α, β)-metrics

2009 ◽  
Vol 52 (1) ◽  
pp. 132-144 ◽  
Author(s):  
Zhongmin Shen
Keyword(s):  
Riemannian Metric ◽  
Finsler Metrics ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Regular Case ◽  
Projectively Flat ◽  
Necessary And Sufficient

AbstractThe solutions to Hilbert's Fourth Problem in the regular case are projectively flat Finsler metrics. In this paper, we consider the so-called (α, β)-metrics defined by a Riemannian metric α and a 1-form β, and find a necessary and sufficient condition for such metrics to be projectively flat in dimension n ≥ 3.

Sign in / Sign up

Export Citation Format

Share Document