Abstract
In this paper it is shown that a three-dimensional non-cosymplectic quasi-Sasakian manifold admitting Ricci almost soliton is locally \(\phi \)-symmetric. It is proved that a Ricci almost soliton on a three-dimensional quasi-Sasakian manifold reduces to a Ricci soliton. It is also proved that if a three-dimensional non-cosymplectic quasi-Sasakian manifold admits gradient Ricci soliton, then the potential function is invariant in the orthogonal distribution of the Reeb vector field \(\xi\). We also improve some previous results regarding gradient Ricci soliton on three-dimensional quasi-Sasakian manifolds. An illustrative example is given to support the obtained results.
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In this paper we have proved that if the metric of a three-dimensional quasi-Sasakian manifold is Ricci almost soliton, then the manifold is locally \(\phi \)-symmetric. This result establishes the relation between Ricci soliton and symmetry of the manifold. The symmetry of a manifold is important because it is related with the curvature of the manifold. The curvature has important physical significance in view of theory of gravitation. We have characterized nature of the potential function related with gradient Ricci soliton. We have also shown that a three-dimensional quasi-Sasakian manifold admitting gradient Ricci soliton is Einstein manifold. Einstein manifolds are important in the theories of Riemannian geometry, relativity and cosmology.
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Sarkar, A., Sil, A. & Paul, A.K. Ricci Almost Solitons on Three-Dimensional Quasi-Sasakian Manifolds. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 89, 705–710 (2019). https://doi.org/10.1007/s40010-018-0504-8
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DOI: https://doi.org/10.1007/s40010-018-0504-8