Paper · Sep 12, 2021

Geometry of para-Sasakian metric as an almost conformal $η$ -Ricci soliton

Sumanjit Sarkar, Santu Dey, Arindam Bhattacharyya

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Abstract

In this paper, we initiate the study of conformal $η$ -Ricci soliton and almost conformal $η$ -Ricci soliton within the framework of para-Sasakian manifold. We prove that if para-Sasakian metric admits conformal $η$ -Ricci soliton, then the manifold is $η$ -Einstein and either the soliton vector field $V$ is Killing or it leaves $φ$ invariant. Here, we have shown the characteristics of the soliton vector field $V$ and scalar curvature when the manifold admitting conformal $η$ -Ricci soliton and vector field is pointwise collinear with the characteristic vector field $ξ$ . Next, we show that a para-Sasakian metric endowed an almost conformal $η$ -Ricci soliton is $η$ -Einstein metric if the soliton vector field $V$ is an infnitesimal contact transformation. We have also displayed that the manifold is Einstein if it represents a gradient almost conformal $η$ -Ricci soliton. We have developed an example to display the alive of conformal $η$ -Ricci soliton on 3-dimensional para-Sasakian manifold.

Comments: arXiv admin note: text overlap with arXiv:2106.10632

Ricci solitons Einstein metrics Almost & η-Ricci solitons 3-manifolds Scalar curvature Sasakian & contact geometry

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Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Almost & η-Ricci solitons Variations on the Ricci soliton equation: let the constant λ become a function (almost solitons), add a term along a contact form (η-Ricci… Start with: Ricci almost solitons (Pigola et al., 2010) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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Geometry of para-Sasakian metric as an almost conformal $η$ -Ricci soliton

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Geometry of para-Sasakian metric as an almost conformal η-Ricci soliton

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Geometry of para-Sasakian metric as an almost conformal $η$ -Ricci soliton