Author

# H. Aruna Kumara

Bio: H. Aruna Kumara is an academic researcher from Kuvempu University. The author has contributed to research in topic(s): Soliton & Scalar curvature. The author has an hindex of 5, co-authored 15 publication(s) receiving 68 citation(s).

##### Papers

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TL;DR: In this paper, the authors considered the case of *-Ricci soliton in the framework of a Kenmotsu manifold and proved that soliton constant λ is zero.

Abstract: Abstract In this paper, we consider *-Ricci soliton in the frame-work of Kenmotsu manifolds. First, we prove that if (M, g) is a Kenmotsu manifold and g is a *-Ricci soliton, then soliton constant λ is zero. For 3-dimensional case, if M admits a *-Ricci soliton, then we show that M is of constant sectional curvature –1. Next, we show that if M admits a *-Ricci soliton whose potential vector field is collinear with the characteristic vector field ξ, then M is Einstein and soliton vector field is equal to ξ. Finally, we prove that if g is a gradient almost *-Ricci soliton, then either M is Einstein or the potential vector field is collinear with the characteristic vector field on an open set of M. We verify our result by constructing examples for both *-Ricci soliton and gradient almost *-Ricci soliton.

13 citations

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TL;DR: In this paper, a contact metric manifold whose metric is a Riemann soliton was studied and it was shown that the manifold is either of constant curvature + 1 (and V is Killing) or D-homothetically invariant.

Abstract: In this paper, we study contact metric manifold whose metric is a Riemann soliton. First, we consider Riemann soliton (g; V ) with V as contact vector eld on a Sasakian manifold (M; g) and in this case we prove that M is either of constant curvature +1 (and V is Killing) or D-homothetically xed -Einstein manifold (and V leaves the structure tensor φ invariant). Next, we prove that if a compact K-contact manifold whose metric g is a gradient almost Riemann soliton, then it is Sasakian and isometric to a unit sphere S2n+1. Further, we study H-contact manifold admitting a Riemann soliton (g; V ) where V is pointwise collinear with .Key words: Contact metric manifold, Riemann soliton, gradient almost Riemann soliton.

11 citations

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TL;DR: In this paper, the authors consider the case where the potential vector field is collinear with the characteristic vector field on an open set of manifolds and show that the potential field is equal to the soliton vector field.

Abstract: In this paper, we consider $*$-Ricci soliton in the frame-work of Kenmotsu manifolds. First, we prove that if the metric of a Kenmotsu manifold $M$ is a $*$-Ricci soliton, then soliton constant $\lambda$ is zero. For 3-dimensional case, if $M$ admits a $*$-Ricci soliton, then we show that $M$ is of constant sectional curvature -1. Next, we show that if $M$ admits a $*$-Ricci soliton whose potential vector field is collinear with the characteristic vector field $\xi$, then $M$ is Einstein and soliton vector field is equal to $\xi$. Finally, we prove that if $g$ is a gradient almost $*$-Ricci soliton, then either $M$ is Einstein or the potential vector field is collinear with the characteristic vector field on an open set of $M$. We verify our result by constructing examples for both $*$-Ricci soliton and gradient almost $*$-Ricci soliton.

10 citations

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TL;DR: In this paper, geometrical aspects of perfect fluid spacetime with torse-forming vector field are described and conditions for the Ricci soliton to be expanding, steady or shrinking are also given.

Abstract: In this paper geometrical aspects of perfect fluid spacetime with torse-forming vector field $$\xi $$
are described and Ricci soliton in perfect fluid spacetime with torse-forming vector field $$\xi $$
are determined. Conditions for the Ricci soliton to be expanding, steady or shrinking are also given.

9 citations

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TL;DR: In this article, the authors considered the problem of paracontact geometry on a para-Kenmotsu manifold and showed that if the metric g of g of G of σ, σ is a Gaussian, then G is either the potential vector field collinear with Reeb vector field or Ricci soliton.

Abstract: We consider almost
$$*$$
-Ricci solitons in the context of paracontact geometry, precisely, on a paraKenmotsu manifold. First, we prove that if the metric g of
$$\eta $$
-Einstein paraKenmotsu manifold is
$$*$$
Ricci soliton, then M is Einstein. Next, we show that if
$$\eta $$
-Einstein paraKenmotsu manifold admits a gradient almost
$$*$$
-Ricci soliton, then either M is Einstein or the potential vector field collinear with Reeb vector field
$$\xi $$
. Finally, for three-dimensional case we show that paraKenmotsu manifold is of constant curvature
$$-1$$
. An illustrative example is given to support the obtained results.

5 citations

##### Cited by

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01 Jan 1970

294 citations

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Jazan University

^{1}TL;DR: In this paper, the geometrical aspects of a perfect fluid spacetime in terms of conformal Ricci soliton and conformal η-Ricci solitons with torse-forming vector field were studied.

Abstract: In this paper, we studied the geometrical aspects of a perfect fluid spacetime in terms of conformal Ricci soliton and conformal η-Ricci soliton with torse-forming vector field ξ. Condition for the...

5 citations

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TL;DR: In this paper, the authors studied the Riemann soliton and gradient almost-Riemann-soliton on a certain class of almost Kenmotsu manifolds.

Abstract: The aim of this paper, is to study the Riemann soliton and gradient almost Riemann soliton on certain class of almost Kenmotsu manifolds. Also, some suitable examples of Kenmotsu and (κ,μ)′-almost ...

5 citations

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TL;DR: In this paper, it was shown that if the metric of a non-cosymplectic normal almost contact metric manifold is Riemann soliton with divergence-free potential vector field (Z), then the manifold is quasi-Sakian and is of constant sectional curvature -$\lambda.

Abstract: The quest of the offering article is to investigate \emph{almost Riemann soliton} and \emph{gradient almost Riemann soliton} in a non-cosymplectic normal almost contact metric manifold $M^3$. Before all else, it is proved that if the metric of $M^3$ is Riemann soliton with divergence-free potential vector field $Z$, then the manifold is quasi-Sasakian and is of constant sectional curvature -$\lambda$, provided $\alpha,\beta =$ constant. Other than this, it is shown that if the metric of $M^3$ is \emph{ARS} and $Z$ is pointwise collinear with $\xi $ and has constant divergence, then $Z$ is a constant multiple of $\xi $ and the \emph{ARS} reduces to a Riemann soliton, provided $\alpha,\;\beta =$constant. Additionally, it is established that if $M^3$ with $\alpha,\; \beta =$ constant admits a gradient \emph{ARS} $(\gamma,\xi,\lambda)$, then the manifold is either quasi-Sasakian or is of constant sectional curvature $-(\alpha^2-\beta^2)$. At long last, we develop an example of $M^3$ conceding a Riemann soliton.

4 citations