Generalised Ricci Solitons on Sasakian Manifolds
01 Dec 2017-
About: The article was published on 2017-12-01 and is currently open access. It has received 6 citations till now.
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01 Aug 2019
TL;DR: In this paper, a generic submanifold admitting a Ricci soliton in Sasakian manifold endowed with concurrent vector field was studied. But it was shown that there exists no concurrent vector fields on the invariant distribution of generic sub-manifolds.
Abstract: In the present paper, we deal with the generic submanifold admitting a Ricci soliton in Sasakian manifold endowed with concurrent vector field. Here, we find that there exists never any concurrent vector field on the invariant distribution D of generic submanifold M. Also, we provide a necessary and sufficient condition for which the invariant distribution D and anti-invariant distribution D^{⊥} of M are Einstein. Finally, we give a characterization for a generic submanifold of Sasakian manifold to be a gradient Ricci soliton.
6 citations
Cites background from "Generalised Ricci Solitons on Sasak..."
...Ricci solitons have been studied in some different classes of contact geometry ([3], [4], [15], [16], [18], [20])....
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15 Oct 2020
TL;DR: In this article, it was shown that a K-contact manifold admitting generalised Ricci solitons is an Einstein one, and it was further shown that such a manifold is an Euler manifold.
Abstract: The object of the present paper is to study K-contact manifold admitting generalised Ricci solitons. We prove that a $K$-contact manifold of dimension $(2n+1)$ satisfying the generalised Ricci soliton equation is an Einstein one. Finally, we obtain several remarks.
4 citations
Cites background from "Generalised Ricci Solitons on Sasak..."
...[12] Let (M, g) be a Riemannian manifold and let f be a smooth function....
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...For a smooth vector field X , we have ([12], [13]) X(Y ) = g(X,Y )....
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TL;DR: In this article , a necessary condition on odd-dimensional Riemannian manifold under which both Sasakian structure and the generalised Ricci soliton equation are satisfied is given.
Abstract: In this note, we find a necessary condition on odd-dimensional Riemannian
manifolds under which both of Sasakian structure and the generalised Ricci
soliton equation are satisfied, and we give some examples.
2 citations
01 Dec 2020
TL;DR: In this paper, the authors studied the properties of the Ricci soliton on the Kenmotsu manifold and also analyzed the generalized gradient RICCI soliton equation satisfying some conditions.
Abstract: In the present frame work, we study the properties of $$\eta$$
-Ricci soliton on Kenmotsu manifold and also analysed the generalized gradient Ricci soliton equation satisfying some conditions.
2 citations
1 citations
References
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Book•
01 Jan 1976
TL;DR: In this paper, the tangent sphere bundle is shown to be a contact manifold, and the contact condition is interpreted in terms of contact condition and k-contact and sasakian structures.
Abstract: Contact manifolds.- Almost contact manifolds.- Geometric interpretation of the contact condition.- K-contact and sasakian structures.- Sasakian space forms.- Non-existence of flat contact metric structures.- The tangent sphere bundle.
1,259 citations
Posted Content•
TL;DR: Ricci solitons are natural generalizations of Einstein metrics and play important roles in the singularity study of the Ricci flow as discussed by the authors, and they are also special solutions to Hamilton's Ricci Flow.
Abstract: Ricci solitons are natural generalizations of Einstein metrics. They are also special solutions to Hamilton's Ricci flow and play important roles in the singularity study of the Ricci flow. In this paper, we survey some of the recent progress on Ricci solitons.
306 citations
TL;DR: In this paper, a simple proof of the non-existence of degenerate components of the event horizon in static, vacuum, regular, four-dimensional black hole spacetimes is presented.
Abstract: We present a simple proof of the non-existence of degenerate components of the event horizon in static, vacuum, regular, four-dimensional black hole spacetimes. We discuss the generalization to higher dimensions and the inclusion of a cosmological constant.
101 citations
TL;DR: The generalized Ricci soliton equations (GRSE) as mentioned in this paper are a class of partial differential equations of finite type on (pseudo-) Riemannian manifolds that depend on three real parameters.
Abstract: We introduce a class of overdetermined systems of partial differential equations of finite type on (pseudo-) Riemannian manifolds that we call the generalized Ricci soliton equations. These equations depend on three real parameters. For special values of the parameters they specialize to various important classes of equations in differential geometry. Among them there are: the Ricci soliton equations, the vacuum near-horizon geometry equations in general relativity, special cases of Einstein–Weyl equations and their projective counterparts, equations for homotheties and Killing’s equation. We also prolong the generalized Ricci soliton equations and, by computing differential constraints, we find a number of necessary conditions for a (pseudo-) Riemannian manifold $$(M, g)$$
to locally admit non-trivial solutions to the generalized Ricci soliton equations in dimensions 2 and 3. The paper provides also a collection of explicit examples of generalized Ricci solitons in dimensions 2 and 3 (in some cases).
30 citations