scholarly journals Geometry of k-Yamabe Solitons on Euclidean Spaces and Its Applications to Concurrent Vector Fields

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 222
Author(s):  
Akram Ali ◽  
Fatemah Mofarreh ◽  
Pişcoran Laurian-Ioan ◽  
Nadia Alluhaibi
Keyword(s):  
Vector Field ◽  
Euclidean Space ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Point Of View ◽  
Euclidean Spaces ◽  
Field Point ◽  
Yamabe Solitons ◽  
Yamabe Soliton

In this paper, we give some classifications of the k-Yamabe solitons on the hypersurfaces of the Euclidean spaces from the vector field point of view. In several results on k-Yamabe solitons with a concurrent vector field on submanifolds in Riemannian manifolds, is proved that a k-Yamabe soliton (Mn,g,vT,λ) on a hypersurface in the Euclidean space Rn+1 is contained either in a hypersphere or a hyperplane. We provide an example to support this study and all of the results in this paper can be implemented to Yamabe solitons for k-curvature with k=1.

Classifications of quotion Yamabe solitons on Euclidean submanifolds with concurrent vector fields

2021 ◽  
pp. 2150103
Author(s):  
Fatemah Mofarreh ◽  
Akram Ali ◽  
Nasser Bin Turki ◽  
Rifaqat Ali
Keyword(s):  
Euclidean Space ◽  
Vector Fields ◽  
Yamabe Solitons ◽  
Euclidean Submanifolds ◽  
Yamabe Soliton

The aim of this paper is to obtain some results for quotion Yamabe solitons with concurrent vector fields. We prove quotion Yamabe soliton [Formula: see text] on a hypersurface in Euclidean space [Formula: see text] contained either in a hyperplane or in a sphere [Formula: see text].

scholarly journals On the Differential Equation Governing Torqued Vector Fields on a Riemannian Manifold

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 1941
Author(s):  
Sharief Deshmukh ◽  
Nasser Bin Turki ◽  
Haila Alodan
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Euclidean Space ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Compact Riemannian Manifold ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Rigidity Results ◽  
Necessary And Sufficient

In this article, we show that the presence of a torqued vector field on a Riemannian manifold can be used to obtain rigidity results for Riemannian manifolds of constant curvature. More precisely, we show that there is no torqued vector field on n-sphere Sn(c). A nontrivial example of torqued vector field is constructed on an open subset of the Euclidean space En whose torqued function and torqued form are nowhere zero. It is shown that owing to topology of the Euclidean space En, this type of torqued vector fields could not be extended globally to En. Finally, we find a necessary and sufficient condition for a torqued vector field on a compact Riemannian manifold to be a concircular vector field.

scholarly journals Some remarks on Yamabe solitons

2019 ◽  
Vol 13 (06) ◽  
pp. 2050120
Author(s):  
Debabrata Chakraborty ◽  
Shyamal Kumar Hui ◽  
Yadab Chandra Mandal
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Lower Bound ◽  
Vector Fields ◽  
Compact Riemannian Manifold ◽  
Geodesic Vector ◽  
Yamabe Solitons ◽  
Yamabe Soliton

The evolution of some geometric quantities on a compact Riemannian manifold [Formula: see text] whose metric is Yamabe soliton is discussed. Using these quantities, lower bound on the soliton constant is obtained. We discuss about commutator of soliton vector fields. Also, the condition of soliton vector field to be a geodesic vector field is obtained.

scholarly journals Some results about concircular vector fields on Riemannian manifolds

Filomat ◽  
2020 ◽  
Vol 34 (3) ◽  
pp. 835-842
Author(s):  
Bang-Yen Chen ◽  
Sharief Deshmukh
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Euclidean Spaces ◽  
Rigidity Results ◽  
Kaehler Manifolds

In this article, we show that the presence of a concircular vector field on a Riemannian manifold can be used to obtain rigidity results for Riemannian and Kaehler manifolds. More precisely, we find new geometrical characterizations of spheres, Euclidean spaces as well as of complex Euclidean spaces using non-trivial concircular vector fields.

scholarly journals Classification of generalized Yamabe solitons in Euclidean spaces

2021 ◽  
pp. 2150022
Author(s):  
Shunya Fujii ◽  
Shun Maeta
Keyword(s):  
Vector Field ◽  
Position Vector ◽  
Euclidean Spaces ◽  
Yamabe Solitons ◽  
Gradient Solitons

In this paper, we consider generalized Yamabe solitons which include many notions, such as Yamabe solitons, almost Yamabe solitons, [Formula: see text]-almost Yamabe solitons, gradient [Formula: see text]-Yamabe solitons and conformal gradient solitons. We completely classify the generalized Yamabe solitons on hypersurfaces in Euclidean spaces arisen from the position vector field.

scholarly journals Classification of Ricci solitons on Euclidean hypersurfaces

2014 ◽  
Vol 25 (11) ◽  
pp. 1450104 ◽  
Author(s):  
Bang-Yen Chen ◽  
Sharief Deshmukh
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Riemannian Manifolds ◽  
Potential Field ◽  
Vector Fields ◽  
Ricci Soliton ◽  
Position Vector ◽  
Ricci Solitons ◽  
Euclidean Submanifolds

A Ricci soliton (M, g, v, λ) on a Riemannian manifold (M, g) is said to have concurrent potential field if its potential field v is a concurrent vector field. Ricci solitons arisen from concurrent vector fields on Riemannian manifolds were studied recently in [Ricci solitons and concurrent vector fields, preprint (2014), arXiv:1407.2790]. The most important concurrent vector field is the position vector field on Euclidean submanifolds. In this paper we completely classify Ricci solitons on Euclidean hypersurfaces arisen from the position vector field of the hypersurfaces.

scholarly journals Quotient manifolds of flows

2017 ◽  
Vol 26 (02) ◽  
pp. 1740005 ◽  
Author(s):  
Robert E. Gompf
Keyword(s):  
Euclidean Space ◽  
Vector Fields ◽  
Topological Manifold ◽  
Fundamental Groups ◽  
Smooth Manifolds ◽  
Euclidean Spaces ◽  
Orbit Spaces ◽  
Quotient Maps ◽  
Fixed Dimension

This paper investigates which smooth manifolds arise as quotients (orbit spaces) of flows of vector fields. Such quotient maps were already known to be surjective on fundamental groups, but this paper shows that every epimorphism of countably presented groups is induced by the quotient map of some flow, and that higher homology can also be controlled. Manifolds of fixed dimension arising as quotients of flows on Euclidean space realize all even (and some odd) intersection pairings, and all homotopy spheres of dimension at least two arise in this manner. Most Euclidean spaces of dimensions five and higher have families of topologically equivalent but smoothly inequivalent flows with quotient homeomorphic to a manifold with flexibly chosen homology. For [Formula: see text], there is a topological flow on (ℝ2r+1 − 8 points) × ℝm that is unsmoothable, although smoothable near each orbit, with quotient an unsmoothable topological manifold.

scholarly journals A note on warped product almost quasi-Yamabe solitons

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 2009-2016 ◽  
Author(s):  
Adara Blaga
Keyword(s):  
Riemannian Manifolds ◽  
Warped Product ◽  
Sufficient Conditions ◽  
Constant Scalar Curvature ◽  
Product Manifold ◽  
Base Manifold ◽  
Warped Product Manifold ◽  
Necessary And Sufficient ◽  
Yamabe Solitons ◽  
Yamabe Soliton

We consider almost quasi-Yamabe solitons in Riemannian manifolds, derive a Bochner-type formula in the gradient case and prove that under certain assumptions, the manifold is of constant scalar curvature. We also provide necessary and sufficient conditions for a gradient almost quasi-Yamabe soliton on the base manifold to induce a gradient almost quasi-Yamabe soliton on the warped product manifold.

scholarly journals On Jacobi-Type Vector Fields on Riemannian Manifolds

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1139 ◽  
Author(s):  
Bang-Yen Chen ◽  
Sharief Deshmukh ◽  
Amira A. Ishan
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Compact Riemannian Manifold ◽  
Killing Vector ◽  
Sufficient Conditions ◽  
Killing Vector Field ◽  
Natural Question ◽  
Necessary And Sufficient

In this article, we study Jacobi-type vector fields on Riemannian manifolds. A Killing vector field is a Jacobi-type vector field while the converse is not true, leading to a natural question of finding conditions under which a Jacobi-type vector field is Killing. In this article, we first prove that every Jacobi-type vector field on a compact Riemannian manifold is Killing. Then, we find several necessary and sufficient conditions for a Jacobi-type vector field to be a Killing vector field on non-compact Riemannian manifolds. Further, we derive some characterizations of Euclidean spaces in terms of Jacobi-type vector fields. Finally, we provide examples of Jacobi-type vector fields on non-compact Riemannian manifolds, which are non-Killing.

scholarly journals On Certain Mappings of Riemannian Manifolds

1965 ◽  
Vol 25 ◽  
pp. 121-142
Author(s):  
Minoru Kurita
Keyword(s):  
Differential Geometry ◽  
Euclidean Space ◽  
Conformal Mapping ◽  
Riemannian Manifolds ◽  
Stereographic Projection ◽  
Point Of View ◽  
The Other ◽  
Central Projection ◽  
Special Cases ◽  
Special Case

In this paper we consider certain tensors associated with differentiable mappings of Riemannian manifolds and apply the results to a p-mapping, which is a special case of a subprojective one in affinely connected manifolds (cf. [1], [7]). The p-mapping in Riemannian manifolds is a generalization of a conformal mapping and a projective one. From a point of view of differential geometry an analogy between these mappings is well known. On the other hand it is interesting that a stereographic projection of a sphere onto a plane is conformal, while a central projection is projectve, namely geodesic-preserving. This situation was clarified partly in [6]. A p-mapping defined in this paper gives a precise explanation of this and also affords a certain mapping in the euclidean space which includes a similar mapping and an inversion as special cases.

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