On ruled surface in 3-dimensional almost contact metric manifold

2017 ◽  
Vol 14 (05) ◽  
pp. 1750076 ◽  
Author(s):  
Murat Kemal Karacan ◽  
Nural Yuksel ◽  
Hasibe Ikiz
Keyword(s):  
Cross Product ◽  
Ruled Surface ◽  
Contact Metric Manifold ◽  
Almost Contact Metric Manifold ◽  
3 Dimensional ◽  
Contact Metric Manifolds ◽  
Special Cases ◽  
Distribution Parameters ◽  
Almost Contact Metric ◽  
Phd Thesis

In this paper, we study ruled surface in 3-dimensional almost contact metric manifolds by using surface theory defined by Gök [Surfaces theory in contact geometry, PhD thesis (2010)]. We also studied the theory of curves using cross product defined by Camcı. In this study, we obtain the distribution parameters of the ruled surface and then some results and theorems are presented with special cases. Moreover, some relationships among asymptotic curve and striction line of the base curve of the ruled surface have been found.

Lifting semi-invariant submanifolds to distribu­tion of almost contact metric manifolds

2020 ◽  
pp. 39-48
Author(s):  
A. V. Bukusheva
Keyword(s):  
Parallel Transport ◽  
Linear Connection ◽  
Contact Metric Manifold ◽  
Almost Contact Metric Structure ◽  
Almost Contact Metric Manifold ◽  
Contact Metric Manifolds ◽  
Almost Contact Metric ◽  
Invariant Submanifolds ◽  
Contact Metric Structure ◽  
Almost Contact Metric Manifolds

Let M be an almost contact metric manifold of dimension n = 2m + 1. The distribution D of the manifold M admits a natural structure of a smooth manifold of dimension n = 4m + 1. On the manifold M, is defined a linear connection that preserves the distribution D; this connection is determined by the interior connection that allows parallel transport of admissible vectors along admissible curves. The assigment of the linear connection is equivalent to the assignment of a Riemannian metric of the Sasaki type on the distribution D. Certain tensor field of type (1,1) on D defines a so-called prolonged almost contact metric structure. Each section of the distribution D defines a morphism of smooth manifolds. It is proved that if a semi-invariant sub­manifold of the manifold M and is a covariantly constant vec­tor field with respect to the N-connection , then is a semi-invariant submanifold of the manifold D with respect to the prolonged almost contact metric structure.

On η-Einstein Trans-Sasakian Manifolds

2011 ◽  
Vol 57 (2) ◽  
pp. 417-440
Author(s):  
Falleh Al-Solamy ◽  
Jeong-Sik Kim ◽  
Mukut Tripathi
Keyword(s):  
Vector Field ◽  
Sufficient Conditions ◽  
Sasakian Manifold ◽  
Necessary And Sufficient Conditions ◽  
Sasakian Manifolds ◽  
Contact Metric Manifold ◽  
Almost Contact Metric Manifold ◽  
3 Dimensional ◽  
Almost Contact Metric ◽  
Necessary And Sufficient

On η-Einstein Trans-Sasakian ManifoldsA systematic study of η-Einstein trans-Sasakian manifold is performed. We find eight necessary and sufficient conditions for the structure vector field ζ of a trans-Sasakian manifold to be an eigenvector field of the Ricci operator. We show that for a 3-dimensional almost contact metric manifold (M,φ, ζ, η, g), the conditions of being normal, trans-K-contact, trans-Sasakian are all equivalent to ∇ζ ∘ φ = φ ∘ ∇ζ. In particular, the conditions of being quasi-Sasakian, normal with 0 = 2β = divζ, trans-K-contact of type (α, 0), trans-Sasakian of type (α, 0), andC6-class are all equivalent to ∇ ζ = -αφ, where 2α = Trace(φ∇ζ). In last, we give fifteen necessary and sufficient conditions for a 3-dimensional trans-Sasakian manifold to be η-Einstein.

scholarly journals ∇N-EINSTEIN ALMOST CONTACT METRIC MANIFOLDS

2021 ◽  
pp. 5-15
Author(s):  
S.V. Galaev ◽  
Keyword(s):  
Tangent Bundle ◽  
Einstein Manifolds ◽  
Contact Metric Manifold ◽  
Contact Manifolds ◽  
Almost Contact Metric Manifold ◽  
Contact Metric Manifolds ◽  
Metric Connection ◽  
Almost Contact Metric ◽  
Almost Contact Metric Manifolds

On an almost contact metric manifold M, an N-connection ∇N defined by the pair (∇,N), where ∇ is the interior metric connection and N: TМ → TM is an endomorphism of the tangent bundle of the manifold M such that Nξ = 0, 􀁇 􀁇 N (D) ⊂ D , is considered. Special attention is paid to the case of a skew-symmetric N-connection ∇N, which means that the torsion of an N-connection considered as a trivalent covariant tensor is skew-symmetric. Such a connection is uniquely defined and corresponds to the endomorphism N = 2ψ, where the endomorphism ψ is defined by the equality ω( X ,Y ) = g (ψX ,Y ) and is called in this work the second structure endomorphism of an almost contact metric manifold. The notion of a ∇N-Einstein almost contact metric manifold is introduced. For the case N = 2ψ, conditions under which almost contact manifolds are ∇N-Einstein manifolds are found.

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].

CERTAIN RESULTS ON THREE-DIMENSIONAL CONTACT METRIC MANIFOLDS

2010 ◽  
Vol 03 (04) ◽  
pp. 577-591 ◽  
Author(s):  
Amalendu Ghosh
Keyword(s):  
Ricci Tensor ◽  
Three Dimensional ◽  
Tensor Fields ◽  
Contact Metric Manifold ◽  
Ricci Operator ◽  
3 Dimensional ◽  
Contact Metric Manifolds ◽  
Parallel Ricci Tensor

In this paper we study 3-dimensional contact metric manifolds satisfying certain conditions on the tensor fields *-Ricci tensorS*, h(= ½Lξφ), τ(= Lξg = 2hφ) and the Ricci operator Q. First, we prove that a 3-dimensional non-Sasakian contact metric manifold satisfies. [Formula: see text] (where ⊕X,Y,Z denotes the cyclic sum over X,Y,Z) if and only if M is a generalized (κ, μ)-space. Next, we prove that a 3-dimensional contact metric manifold with vanishing *-Ricci tensor is a generalized (κ, μ)-space. Finally, some results on 3-dimensional contact metric manifold with cyclic η-parallel h or cyclic η-parallel τ or η-parallel Ricci tensor are presented.

scholarly journals On a semi-symmetric non-metric connection

Filomat ◽  
2011 ◽  
Vol 25 (4) ◽  
pp. 19-27 ◽  
Author(s):  
S.K. Chaubey ◽  
R.H. Ojha
Keyword(s):  
Riemannian Manifold ◽  
Contact Metric Manifold ◽  
Kenmotsu Manifold ◽  
Almost Contact Metric Manifold ◽  
Metric Connection ◽  
Almost Contact Metric

Yano [1] defined and studied semi-symmetric metric connection in a Riemannian manifold and this was extended by De and Senguta [8] and many other geometers. Recently, the present authors [3], [5] defined semi-symmetric non-metric connections in an almost contact metric manifold. In this paper, we studied some properties of a semi-symmetric non-metric connection in a Kenmotsu manifold.

scholarly journals On a semi-symmetric non-metric connection

Filomat ◽  
2012 ◽  
Vol 26 (2) ◽  
pp. 269-275 ◽  
Author(s):  
S.K. Chaubey ◽  
R.H. Ojha
Keyword(s):  
Riemannian Manifold ◽  
Contact Metric Manifold ◽  
Kenmotsu Manifold ◽  
Almost Contact Metric Manifold ◽  
Metric Connection ◽  
Almost Contact Metric

Yano [10] defined and studied semi-symmetric metric connection in a Riemannian manifold and this was extended by De and Senguta [4] and many other geometers. Recently, the present authors [2], [3] defined semi-symmetric non-metric connections in an almost contact metric manifold. In this paper, we studied some properties of a semi-symmetric non-metric connection in a Kenmotsu manifold.

scholarly journals On the Sign of the Curvature of a Contact Metric Manifold

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 892
Author(s):  
David E. Blair
Keyword(s):  
Positive Curvature ◽  
Contact Manifold ◽  
Contact Metric Manifold ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
3 Dimensional ◽  
Contact Metric Manifolds ◽  
Expository Article ◽  
Standard Contact ◽  
Negative Sectional Curvature

In this expository article, we discuss the author’s conjecture that an associated metric for a given contact form on a contact manifold of dimension ≥5 must have some positive curvature. In dimension 3, the standard contact structure on the 3-torus admits a flat associated metric; we also discuss a local example, due to Krouglov, where there exists a neighborhood of negative curvature on a particular 3-dimensional contact metric manifold. In the last section, we review some results on contact metric manifolds with negative sectional curvature for sections containing the Reeb vector field.

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