Almost quasi-Yamabe solitons on almost cosymplectic manifolds

2020 ◽  
Vol 17 (05) ◽  
pp. 2050070
Author(s):  
Xiaomin Chen
Keyword(s):  
Vector Field ◽  
Riemannian Product ◽  
Reeb Vector ◽  
Potential Vector ◽  
Cosymplectic Manifolds ◽  
Almost Kähler Manifold ◽  
Almost Kähler ◽  
Yamabe Solitons ◽  
Yamabe Soliton ◽  
Potential Vector Field

In this paper, we study almost cosymplectic manifolds admitting almost quasi-Yamabe solitons [Formula: see text]. First, we prove that an almost cosymplectic [Formula: see text]-manifold is locally isomorphic to a Lie group if [Formula: see text] is a nontrivial closed quasi-Yamabe soliton. Next, we consider an almost [Formula: see text]-cosymplectic manifold admitting a nontrivial almost quasi-Yamabe soliton and prove that it is locally the Riemannian product of an almost Kähler manifold with the real line if the potential vector field [Formula: see text] is collinear with the Reeb vector filed. For the potential vector field [Formula: see text] being orthogonal to the Reeb vector filed, we also obtain two results. Finally, for a closed almost quasi-Yamabe soliton on compact [Formula: see text]-cosymplectic manifolds, we prove that it is trivial if [Formula: see text] is nonnegative, where [Formula: see text] is the scalar curvature.

The k-almost Yamabe solitons and almost coKähler manifolds

2021 ◽  
pp. 2150179
Author(s):  
Xiaomin Chen ◽  
Uday Chand De
Keyword(s):  
Vector Field ◽  
Riemannian Product ◽  
Reeb Vector ◽  
Potential Vector ◽  
Reeb Vector Field ◽  
Almost Kähler Manifold ◽  
Almost Kähler ◽  
Yamabe Solitons ◽  
Potential Vector Field

In this paper, we study almost coKähler manifolds admitting [Formula: see text]-almost Yamabe solitons [Formula: see text]. First, we obtain a classification of almost coKähler [Formula: see text]-manifolds admitting nontrivial closed [Formula: see text]-almost Yamabe solitons. Next, we consider an almost [Formula: see text]-coKähler manifold admitting a nontrivial [Formula: see text]-almost Yamabe soliton and prove that it is locally the Riemannian product of an almost Kähler manifold with the real line if the potential vector field [Formula: see text] is collinear with the Reeb vector field. For the potential vector field [Formula: see text] being orthogonal to the Reeb vector field, we also obtain two results.

scholarly journals Geometry of α-Cosymplectic Metric as ∗-Conformal η-Ricci–Yamabe Solitons Admitting Quarter-Symmetric Metric Connection

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2189
Author(s):  
Pengfei Zhang ◽  
Yanlin Li ◽  
Soumendu Roy ◽  
Santu Dey
Keyword(s):  
Conformal Killing ◽  
Soliton Equation ◽  
Potential Vector ◽  
Research Article ◽  
Cosymplectic Manifolds ◽  
Metric Connection ◽  
Gradient Type ◽  
Yamabe Solitons ◽  
Yamabe Soliton ◽  
Potential Vector Field

The outline of this research article is to initiate the development of a ∗-conformal η-Ricci–Yamabe soliton in α-Cosymplectic manifolds according to the quarter-symmetric metric connection. Here, we have established some curvature properties of α-Cosymplectic manifolds in regard to the quarter-symmetric metric connection. Further, the attributes of the soliton when the manifold gratifies a quarter-symmetric metric connection have been displayed in this article. Later, we picked up the Laplace equation from ∗-conformal η-Ricci–Yamabe soliton equation when the potential vector field ξ of the soliton is of gradient type, admitting quarter-symmetric metric connection. Next, we evolved the nature of the soliton when the vector field’s conformal killing reveals a quarter-symmetric metric connection. We show an example of a 5-dimensional α-cosymplectic metric as a ∗-conformal η-Ricci–Yamabe soliton acknowledges quarter-symmetric metric connection to prove our results.

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.

scholarly journals Yamabe Solitons with potential vector field as torse forming

Cubo (Temuco) ◽  
2018 ◽  
Vol 20 (3) ◽  
pp. 37-47
Author(s):  
Yadab ChandraMandal ◽  
Shyamal Kumar Hui
Keyword(s):  
Vector Field ◽  
Potential Vector ◽  
Yamabe Solitons ◽  
Potential Vector Field

Ricci Solitons on Almost Co-Kähler Manifolds

2018 ◽  
Vol 62 (4) ◽  
pp. 912-922 ◽  
Author(s):  
Yaning Wang
Keyword(s):  
Vector Field ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Constant Length ◽  
Reeb Vector ◽  
Potential Vector ◽  
Reeb Vector Field ◽  
Constant Coefficients ◽  
Rigid Motions ◽  
Potential Vector Field

AbstractIn this paper, we prove that if an almost co-Kähler manifold of dimension greater than three satisfying $\unicode[STIX]{x1D702}$-Einstein condition with constant coefficients is a Ricci soliton with potential vector field being of constant length, then either the manifold is Einstein or the Reeb vector field is parallel. Let $M$ be a non-co-Kähler almost co-Kähler 3-manifold such that the Reeb vector field $\unicode[STIX]{x1D709}$ is an eigenvector field of the Ricci operator. If $M$ is a Ricci soliton with transversal potential vector field, then it is locally isometric to Lie group $E(1,1)$ of rigid motions of the Minkowski 2-space.

scholarly journals Ricci solitons on almost Kenmotsu 3-manifolds

2017 ◽  
Vol 15 (1) ◽  
pp. 1236-1243 ◽  
Author(s):  
Yaning Wang
Keyword(s):  
Vector Field ◽  
Hyperbolic Space ◽  
Lie Group ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Reeb Vector ◽  
Potential Vector ◽  
Reeb Vector Field ◽  
Gradient Ricci Soliton ◽  
Potential Vector Field

Abstract Let (M3, g) be an almost Kenmotsu 3-manifold such that the Reeb vector field is an eigenvector field of the Ricci operator. In this paper, we prove that if g represents a Ricci soliton whose potential vector field is orthogonal to the Reeb vector field, then M3 is locally isometric to either the hyperbolic space ℍ3(−1) or a non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure. In particular, when g represents a gradient Ricci soliton whose potential vector field is orthogonal to the Reeb vector field, then M3 is locally isometric to either ℍ3(−1) or ℍ2(−4) × ℝ.

CONTACT GEOMETRY AND RICCI SOLITONS

2010 ◽  
Vol 07 (06) ◽  
pp. 951-960 ◽  
Author(s):  
JONG TAEK CHO ◽  
RAMESH SHARMA
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Ricci Soliton ◽  
Reeb Vector ◽  
Potential Vector ◽  
Reeb Vector Field ◽  
Base Manifold ◽  
Gradient Ricci Soliton ◽  
Unit Tangent Bundle ◽  
Potential Vector Field

We show that a compact contact Ricci soliton with a potential vector field V collinear with the Reeb vector field, is Einstein. We also show that a homogeneous H-contact gradient Ricci soliton is locally isometric to En+1 × Sn(4). Finally we obtain conditions so that the horizontal and tangential lifts of a vector field on the base manifold may be potential vector fields of a Ricci soliton on the unit tangent bundle.

scholarly journals A Note on LP -Sasakian Manifolds with Almost Quasi-Yamabe Solitons

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Sunil Kumar Yadav ◽  
D. L. Suthar ◽  
Biniyam Shimelis
Keyword(s):  
Vector Field ◽  
Sasakian Manifolds ◽  
Potential Vector ◽  
Metric Connection ◽  
Yamabe Solitons ◽  
Potential Vector Field

We categorize almost quasi-Yamabe solitons on LP -Sasakian manifolds and their CR -submanifolds whose potential vector field is torse-forming, admitting a generalized symmetric metric connection of type α , β . Finally, a nontrivial example is provided to confirm some of our results.

Notes on contact Ricci solitons

2010 ◽  
Vol 54 (1) ◽  
pp. 47-53 ◽  
Author(s):  
Jong Taek Cho
Keyword(s):  
Vector Field ◽  
Ricci Soliton ◽  
Homogeneous Manifold ◽  
Ricci Solitons ◽  
Reeb Vector ◽  
Potential Vector ◽  
Reeb Vector Field ◽  
Potential Vector Field

AbstractA compact contact Ricci soliton (whose potential vector field is the Reeb vector field) is Sasaki–Einstein. A compact contact homogeneous manifold with a Ricci soliton is Sasaki–Einstein.

A note on almost co-Kähler manifolds

2020 ◽  
Vol 17 (10) ◽  
pp. 2050153 ◽  
Author(s):  
Uday Chand De ◽  
Sudhakar K. Chaubey ◽  
Young Jin Suh
Keyword(s):  
Vector Field ◽  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Kahler Manifold ◽  
Yamabe Solitons ◽  
Yamabe Soliton

We characterize almost co-Kähler manifolds with gradient Yamabe, gradient Einstein and quasi-Yamabe solitons. It is proved that if the metric of a [Formula: see text]-almost co-Kähler manifold [Formula: see text] is a gradient Yamabe soliton, then [Formula: see text] is either [Formula: see text]-almost co-Kähler or [Formula: see text]-almost co-Kähler or the metric of [Formula: see text] is a trivial gradient Yamabe soliton. A [Formula: see text]-almost co-Kähler manifold with gradient Einstein soliton is [Formula: see text]-almost co-Kähler. Finally, it is shown that an almost co-Kähler manifold admitting a quasi-Yamabe soliton, whose soliton vector is pointwise collinear with the Reeb vector field of the manifold, is [Formula: see text]-almost co-Kähler. Consequently, some results of almost co-Kähler manifolds are deduced.

Sign in / Sign up

Export Citation Format

Share Document