scholarly journals Geometry of Nonholonomic Kenmotsu Manifolds

2021 ◽  
pp. 84-87
Author(s):  
A.V. Bukusheva
Keyword(s):  
Curvature Tensor ◽  
Einstein Manifold ◽  
Lie Derivative ◽  
Riemann Curvature Tensor ◽  
Intrinsic Geometry ◽  
Kenmotsu Manifold ◽  
Metric Tensor ◽  
Schouten Tensor ◽  
Kenmotsu Manifolds ◽  
Formal Properties

The concept of the intrinsic geometry of a nonholonomic Kenmotsu manifold M is introduced. It is understood as the set of those properties of the manifold that depend only on the framing  of the D^ distribution D of the manifold M, on the parallel transformation of vectors belonging to the distribution D along curves tangent to this distribution. The invariants of the intrinsic geometry of the nonholonomic Kenmotsu manifold are: the Schouten curvature tensor; 1-form η generating the distribution D; the Lie derivative  of the metric tensor g along the vector field ; Schouten — Wagner tensor field P, whose components in adapted coordinates are expressed using the equalities . It is proved that, as in the case of the Kenmotsu manifold, the Schouten — Wagner tensor of the manifold M vanishes. It follows that the Schouten tensor of a nonholonomic Kenmotsu manifold has the same formal properties as the Riemann curvature tensor. It is proved that the alternation of the Ricci — Schouten tensor coincides with the differential of the structural form. This property of the Ricci — Schouten tensor is used in the proof of the main result of the article: a nonholonomic Kenmotsu manifold cannot carry the structure of an η-Einstein manifold.

Kenmotsu Manifolds with Zero Ricci-Schouten Tensor

2020 ◽  
pp. 10-16
Author(s):  
Aliya V. Bukusheva
Keyword(s):  
Vector Field ◽  
Vector Bundle ◽  
Einstein Manifold ◽  
Parallel Transport ◽  
Lie Derivative ◽  
Tensor Fields ◽  
Kenmotsu Manifold ◽  
Metric Tensor ◽  
Schouten Tensor ◽  
Kenmotsu Manifolds

The paper is dedicated to the investigation of the interior geometry of the Kenmotsu manifolds M. By the interior geometry of the manifold M we mean the aggregate of the properties of the manifold that depend only on the closing of the distribution D of the Kenmotsu manifold as well as on the parallel transport of the vectors from the distribution D along arbitrary curves of the manifold. The invariants of the interior geometry of a Kenmotsu manifold are the following: the Schouten curvature tensor; the 1-form η generating the distribution D; the Lie derivative of the metric tensor g along the structure vector field ; the Schouten-Wagner admissible tensor fields with the components with respect to adapted coordinates; the structural endomorphism φ; the endomorphism N that allows to prolong the interior connection to a connection in a vector bundle. A special attention is payed to the Ricci-Schouten tensor. In particular, it is stated that a Kenmotsu manifold with zero Ricci-Schouten tensor is an Einstein manifold. Conversely, if M is an η-Einstein Kenmotsu manifold and then M is an Einstein manifold with zero Ricci-Schouten tensor. It is proved that the Ricci-Schouten tensor is zero if and only if the Kenmotsu manifold M is locally Ricci-symmetric. This implies the following well-known result: a Kenmotsu manifold is an Einstein manifold if and only if it is locally Ricci-symmetric. An N-connection with torsion, is introduced; this connection is Ricci-flat if and only if M is an Einstein manifold.

On geometry of Kenmotsu manifolds with N-connection

2019 ◽  
pp. 48-60
Author(s):  
A. Bukusheva
Keyword(s):  
Vector Field ◽  
Coordinate System ◽  
Curvature Tensor ◽  
Tensor Field ◽  
Lie Derivative ◽  
Kenmotsu Manifold ◽  
Metric Tensor ◽  
Civita Connection ◽  
Kenmotsu Manifolds ◽  
Levi Civita Connection

A Kenmotsu manifold with a given N-connection is considered. From the integrability of the distribution of a Kenmotsu manifold it follows that the N-connection belongs to the class of the quarter-symmetric connections. Among the N-connections, the class of connections adapted to the structure of the Kenmotsu manifold is specified. In particular, it is proved that an N-connection preserves the structure endomorphism φ of the Kenmotsu manifold if and only if the endomorphisms N and φ commute. A formula expressing the N-connection in terms of the Levi-Civita connection is obtained. The Chrystoffel symbols of the Levi-Civita connection and of the N-connection of the Kenmotsu manifold with respect to the adapted coordinates are computed. The properties of the invariants of the interior geometry of the Kenmotsu manifolds are investigated. The invariants of the interior geometry are the following: the Schouten curvature tensor; the 1-form  defining the distribution D; the Lie derivative 0   L g of the metric tensor g along the vector field ;  the tensor field P with the components given with respect to the adapted coordinate system by the formula Pacd  ncad . The field P is called in the work the Schouten — Wagner tensor. It is proved that the Schouten — Wagner tensor of the interior connection of the Kenmotsu manifold is zero. The conditions that satisfies the endomorphism N defining the metric N-connection are found. At the end of the work, an example of a Kenmotsu manifold with a metric N-connection preserving the structure endomorphism φ is given.

scholarly journals On The Formulation of Bianchi Identity from Action Principle

2021 ◽  
Author(s):  
Shiladittya Debnath
Keyword(s):  
Curvature Tensor ◽  
Basic Property ◽  
Covariant Derivative ◽  
Bianchi Identity ◽  
Action Principle ◽  
Lie Derivative ◽  
General Covariance ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Metric Tensor

Abstract In this letter, we investigate the basic property of the Hilbert-Einstein action principle and its infinitesimal variation under suitable transformation of the metric tensor. We find that for the variation in action to be invariant, it must be a scalar so as to obey the principle of general covariance. From this invariant action principle, we eventually derive the Bianchi identity (where, both the 1st and 2nd forms are been dissolved) by using the Lie derivative and Palatini identity. Finally, from our derived Bianchi identity, splitting it into its components and performing cyclic summation over all the indices, we eventually can derive the covariant derivative of the Riemann curvature tensor. This very formulation was first introduced by S Weinberg in case of a collision less plasma and gravitating system. We derive the Bianchi identity from the action principle via this approach; and hence the name ‘Weinberg formulation of Bianchi identity’.

scholarly journals On the quasi-conformal curvature tensor of an almost Kenmotsu manifold with nullity distributions

2018 ◽  
Vol 33 (2) ◽  
pp. 255
Author(s):  
Dibakar Dey ◽  
Pradip Majhi
Keyword(s):  
Vector Field ◽  
Curvature Tensor ◽  
Conformally Flat ◽  
Kenmotsu Manifold ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Kenmotsu Manifolds ◽  
Almost Kenmotsu Manifolds ◽  
Nullity Distribution ◽  
Characteristic Vector Field

The object of the present paper is to characterize quasi-conformally flat and $\xi$-quasi-conformally flat almost Kenmotsu manifolds with  $(k,\mu)$-nullity and $(k,\mu)'$-nullity distributions respectively. Also we characterize almost Kenmotsu manifolds with vanishing extended quasi-conformal curvature tensor and extended $\xi$-quasi-conformally flat almost Kenmotsu manifolds such that the characteristic vector field $\xi$ belongs to the $(k,\mu)$-nullity distribution.

scholarly journals Certain results on Ricci solitons in α-Kenmotsu manifolds

2018 ◽  
Vol 18 (1) ◽  
pp. 11-15
Author(s):  
Rajesh Kumar ◽  
Ashwamedh Mourya
Keyword(s):  
Vector Field ◽  
Ricci Soliton ◽  
Second Order ◽  
Ricci Solitons ◽  
Kenmotsu Manifold ◽  
Constant Multiple ◽  
Metric Tensor ◽  
Kenmotsu Manifolds

In this paper, we study some curvature problems of Ricci solitons in α-Kenmotsu manifold. It is shown that a symmetric parallel second order-covariant tensor in a α-Kenmotsu manifold is a constant multiple of the metric tensor. Using this result, it is shown that if (Lvg + 2S) is parallel where V is a given vector field, then the structure (g, V, λ) yield a Ricci soliton. Further, by virtue of this result, Ricci solitons for n-dimentional α-Kenmotsu manifolds are obtained. In the last section, we discuss Ricci soliton for 3-dimentional α-Kenmotsu manifolds.

scholarly journals Dislocation Density Tensor of Thin Elastic Shells at Finite Deformation

2018 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Dislocation Density ◽  
Curvature Tensor ◽  
Finite Deformation ◽  
Middle Surface ◽  
Gauss Curvature ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Metric Tensor ◽  
Elastic Shells ◽  
Dislocation Density Tensor

The dislocation density tensors of thin elastic shells have been formulated explicitly in terms of the Riemann curvature tensor. The formulation reveals that the dislocation density of the shells is  proportional to KA3=2, where K is the Gauss curvature and A is the determinant of metric tensor ofthe middle surface.

scholarly journals Second order parallel tensors on almost Kenmotsu manifolds satisfying the nullity distributions

Filomat ◽  
2014 ◽  
Vol 28 (4) ◽  
pp. 839-847 ◽  
Author(s):  
Yaning Wang ◽  
Ximin Liu
Keyword(s):  
Second Order ◽  
Dimensional Manifold ◽  
Kenmotsu Manifold ◽  
Metric Tensor ◽  
Riemannian Product ◽  
Almost Kenmotsu Manifold ◽  
Kenmotsu Manifolds ◽  
Almost Kenmotsu Manifolds ◽  
Nullity Distribution ◽  
Order Parallel

In this paper, we prove that if there exists a second order symmetric parallel tensor on an almost Kenmotsu manifold (M2n+1, ?, ?, ?, g) whose characteristic vector field ? belongs to the (k,?)'-nullity distribution, then either M2n+1 is locally isometric to the Riemannian product of an (n+1)-dimensional manifold of constant sectional curvature -4 and a flat n-dimensional manifold, or the second order parallel tensor is a constant multiple of the associated metric tensor of M2n+1. Furthermore, some properties of an almost Kenmotsu manifold admitting a second order parallel tensor with ? belonging to the (k,?)-nullity distribution are also obtained.

scholarly journals Ricci Solitons in β-Kenmotsu Manifolds

2018 ◽  
Vol 56 (1) ◽  
pp. 149-163
Author(s):  
Rajesh Kumar
Keyword(s):  
Vector Field ◽  
Ricci Soliton ◽  
Second Order ◽  
Ricci Solitons ◽  
Kenmotsu Manifold ◽  
Constant Multiple ◽  
Metric Tensor ◽  
3 Dimensional ◽  
Kenmotsu Manifolds

Abstract The object of the present paper is to study Ricci soliton in β-Kenmotsu manifolds. Here it is proved that a symmetric parallel second order covariant tensor in a β-Kenmotsu manifold is a constant multiple of the metric tensor. Using this result, it is shown that if (ℒVg +2S)is ∇-parallel where V is a given vector field, then the structure (g, V, λ) yields a Ricci soliton. Further, by virtue of this result, we found the conditions of Ricci soliton in β-Kenmotsu manifold to be shrinking, steady and expending respectively. Next, Ricci soliton for 3-dimensional β-Kenmotsu manifold are discussed with an example.

scholarly journals Some Symmetric Curvature Conditions on Kenmotsu Manifolds

2018 ◽  
Vol 1 (1) ◽  
pp. 50-56
Author(s):  
Riddhi Jung Shah
Keyword(s):  
Negative Curvature ◽  
Einstein Manifold ◽  
Constant Negative Curvature ◽  
Kenmotsu Manifold ◽  
3 Dimensional ◽  
Kenmotsu Manifolds

In this paper, we study locally and globally φ-symmetric Kenmotsu manifolds. In both curvature conditions, it is proved that the manifold is of constant negative curvature - 1 and globally φ-Weyl projectively symmetric Kenmotsu manifold is an Einstein manifold. Finally, we give an example of 3-dimensional Kenmotsu manifold.

η-Einstein nearly Kenmotsu manifolds

2019 ◽  
Vol 12 (06) ◽  
pp. 2040010 ◽  
Author(s):  
Pelin Tekin ◽  
Nesip Aktan
Keyword(s):  
Scalar Curvature ◽  
Curvature Tensor ◽  
Kenmotsu Manifold ◽  
Projectively Flat ◽  
Kenmotsu Manifolds ◽  
Projective Curvature ◽  
Conharmonic Curvature Tensor ◽  
Projective Curvature Tensor ◽  
Tensor Formula

In this paper, we showed that an [Formula: see text]-Einstein nearly Kenmotsu manifold with projective curvature tensor [Formula: see text], and conharmonic curvature tensor [Formula: see text], satisfy the conditions [Formula: see text] and [Formula: see text], respectively. Furthermore, we obtain scalar curvature of a projectively flat and a conharmonically flat [Formula: see text]-Einstein nearly Kenmotsu manifold.

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