Kenmotsu Manifolds Admitting a Non-Symmetric Non-Metric Connection
KKenmotsu Manifolds Admitting a Non - Symmetric Non - Metric Connection
S. K. Chaubey , S. K. Yadav and Mahesh Garvandha
1, 3
Mathematics Section, Department of Information Technology, Shinas College of Technology, PO Box 77, Postal Code 324 Al Aqur, Shinas, Sultanate of Oman. Department of Mathematics, Poornima college of Engineering, Rajasthan, India. e-mail : [email protected]; e-mail : [email protected]; e-mail : [email protected] Abstract
The aim of the present paper is to study the properties of Kenmotsu manifolds equipped with a non-symmetric non-metric connection. We also establish some curvature properties of Kenmotsu manifolds. It is proved that a Kenmotsu manifold endowed with a non-symmetric non-metric is irregular.
Keywords and phrases:
Kenmotsu manifolds, non-symmetric non-metric connection, Ricci semi-symmetric manifold, Einstein manifolds.
AMS Mathematics Subject Classification (2010): . Literature Review
A contact metric manifold is capable to resolve many issues of sciences, engineering and medical sciences, and hence it is attracting the researchers to work in this area. Boothby & Wang (1958) started the study of a differentiable manifold with contact and almost contact metric structures. Kenmotsu (1971) introduced a class of almost contact metric manifold and named as Kenmotsu manifold. Since then, the properties of Kenmotsu manifolds have studied by several authors such as De & Pathak (2004), Sinha & Srivastava (1991), Jun, De & Pathak (2005), De, Yildiz and Yaliniz (2008), Chaubey et al. (2010, 2012, 2015, 2018), De (2008), Cihan (2006) and many others. Let π be a Riemannian manifold associated with the Riemannian metric π . A linear connection π»Μ on π is said to be symmetric if the torsion tensor πΜ of π»Μ vanishes, otherwise it is non-symmetric. f the torsion tensor πΜ assumes the form πΜ (π, π) = Ο(π)π β Ο(π)π for all vector fields π and π on π , then then linear connection π»Μ is called semi-symmetric connection (Friedmann, & J. A. Schouten, 1924). Moreover, if π»Μπ = 0 on π then π»Μ is said to be metric, otherwise non-metric (Agashe & Chafle, 1992). Chaubey & Ojha (2008) defined and studied the properties of non-symmetric non-metric connection on almost contact metric manifolds. The geometrical properties of the same connection have been studied by several authors. We refer (Chaubey & Kumar (2010); Chaubey & De (2019); Chaubey, & Yildiz (2019); Chaubey et al. (2019)) and their references. Above studies motivate us to study the properties of Kenmotsu manifold equipped with a non-symmetric non-metric connection. Preliminaries:
A smooth manifold π of dimension (2n+1) is said to be an almost contact metric manifold if it admits a (1, 1) tensor field π , (1, 0) type vector field π , (0, 1) type vector field π and a compatible metric π of type (0, 2) satisfies π = βπΌ + π β π , π(π) = 1 and π(ππ, ππ) = π(π, π) β π(π)π(π) (2.1) for all π and π on π (Blair ,1976) . Additionally, if π satisfies π» π π = π β π(π)π βΊ (π» π π)(π) = π(π, π) β π(π)π(π) (2.2) for all π on π , then π is said to be a Kenmotsu manifold (1971). Here π» denotes the Levi-Civita connection of π . It is observed that the manifold holds the following relations, π(π (π, π)π) = π(π)π(π, π) β π(π)π(π, π) , (2.3) π (π, π)π = π(π)π β π(π)π, (2.4)
π (π, π)π = π(π)π β π(π, π)π, (2.5)
π(π, π) = β2π π(π) (2.6) or all π , π and π on π (Kenmotsu (1971), Chaubey & Ojha (2010), Chaubey & R. H. Ojha (2012)). Here π and π denote the Riemannian curvature and Ricci tensors of π , respectively. A Kenmotsu manifold π is said to be π -Einstein if the non-vanishing Ricci tensor π satisfies the relation π(π, π) = ππ(π, π) + ππ(π)π(π) for all π and π on π , where π and π are smooth functions on π (Kenmotsu (1971)). Non-symmetric non-metric connection
Let π be a Kenmotsu manifold of dimension (2π + 1). A linear connection π»Μ on π , defined by π»Μ π π = π» π π β π(π)π β π(π, π)π (3.1) for all vector fields π and π on π , is known as a non-symmetric non-metric connection (De & Pathak (2004)) , if the torsion tensor πΜ of π»Μ takes the form πΜ (π, π) = π(π)π β π(π)π and ( π»Μ π π)(π, π) = 2π(Y )g(X, Z) + 2π(Z)g(X, Y ) . (3.2) In consequence of the equations (2.1), (2.3) and (3.1), we get π»Μ π π = π» π π β π β π(π)π = β2π(π)π . (3.3) If π Μ denotes the Riemannian curvature tensor with respect to π»Μ , then it relates to π by the relation π Μ(π, π)π = π (π, π)π β π½(π, π)π + π½(π, π)π β π(π, π)(π» π π β π(π)π) +π(π, π)(π» π π β π(π)π), (3.4) where π½ is tensor field of type (0, 2) and defined as π½(π, π) = (π» π π)(π) + π(π)π(π) + π(π, π) = 2π(π, π) . (3.5) In view of (3.5), equation (3.4) takes the form π Μ(π, π)π = π (π, π)π + π(π, π)π β π(π, π)π + 2[π(π, π)π(π) β π(π, π)π(π)]π . (3.6) ontracting equation (3.6) along the vector field π , we lead πΜ(π, π) = π(π, π) + 2(π + 1)π(π, π) β 2π(π)π(π) , (3.7) which gives
πΜ π = ππ + 2(π + 1)π β 2π(π)π . (3.8) The contraction of (3.8) gives πΜ = π + 2π(2π + 3) . (3.9) Here πΜ and π are the scalar curvatures with respect to π»Μ and π» , respectively. πΜ and π denote the Ricci operators corresponding to the Ricci tensors πΜ and π with respect to π»Μ and π» , respectively. Setting π = π in (3.7) and then using the equations (2.1) and (2.4), we obtain
π Μ(π, π)π = π (π, π)π + π(π)π β π(π)π + 2[π(π)π(π) β π(π)π(π)]π = 0.
This shows that the Kenmotsu manifold M is irregular (
π Μ(π, π)π = 0 ) with respect to π»Μ.
Thus we state:
Theorem 3.1.
Every (2π + 1) -dimensional Kenmotsu manifold equipped with π»Μ is irregular with respect to π»Μ. Ricci semi-symmetric Kenmotsu manifold with a non-symmetric non-metric connection
This section deals with the study of Ricci semi-symmetric Kenmotsu manifold equipped with a non-symmetric non-metric connection π»Μ . It is well known that (π Μ(π, π) β πΜ)(π, π) = βπ Μ (π Μ (π, π)π, π) β π Μ (π, π Μ (π, π)π). In view of (2.1) and (3.7), we obtain
π Μ (π, π) β πΜ)(π, π) = (π (π, π) β π)(π, π) β π(π, π)π(π, π) + π(π, π)π(π, π) β π(π, π)π(π, π) + π(π, π)π(π, π), (4.1) where (π (π, π) β π)(π, π) = βπ(π (π, π)π, π) β π(π, π (π, π)π) for all vector fields
X, Y, Z and U on M . If possible, we suppose that π Μ β πΜ = π β π, then (4.1) gives β π(π, π)π(π, π) + π(π, π)π(π, π) β π(π, π)π(π, π) + π(π, π)π(π, π) = 0 (4.2) Changing Y and U by π in (4.2) and then using the equations (2.1), (2.2), and (2.6), we obtain π(π, π) = β2ππ(π, π), π = β2π(2π + 1), (4.3) which shows that the Kenmotsu manifold endowed with a non -symmetric non-metric connection π»Μ , under assumption is an Einstein manifold. Also, from equations (3.7) and (4.3), we find πΜ(π, π) = 2π(π, π) β 2π(π)π(π). (4.4) This reflects that the Kenmotsu manifold M w.r.t. π»Μ under consideration is an π -Einstein manifold. Thus, we can state: Theorem 4.1.
Let (π, π) be a (2π + 1) -dimensional Kenmotsu manifold equipped with a non-symmetric non-metric connection π»Μ . If π Μ β πΜ = π β π on π , then the manifold to be an Einstein manifold also, π is an π -Einstein with respect to π»Μ . With the help of (4.3), equation (3.9) takes the form πΜ = 4π, which shows that if a Kenmotsu manifold equipped with π»Μ satisfies π Μ β πΜ = π β π , then the scalar curvature with respect to π»Μ is constant. Thus, we state: Corollary.
If a (2π + 1) -dimensional Kenmotsu manifold π endowed with π»Μ satisfies π Μ β πΜ =π β π , then the scalar curvature of π with respect to π»Μ to be constant. Arslan et al. (2014) proved that A semi-Riemannian Einstein manifold M of dimension π , β₯ 4 , satisfies π β πΆ β πΆ β π = ππ(πβ1)
π(π, π ) = ππ(πβ1)
π(π, πΆ) , (4.5) where
π(π, π ) denotes the
Tachibana tensor and πΆ is the conformal curvature tensor .β From (4.3) and (4.5), we have πΆ β π β π β πΆ = π(π, π ) = π(π, πΆ). (4.6) Thus, we state:
Corollary.
If a (2π + 1) -dimensional Kenmotsu manifold π equipped with π»Μ satisfies π Μ β πΜ = π β π , then π satisfies the equation (4.6). References
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