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Study of gradient solitons in three dimensional Riemannian manifolds

Uday Chand De|Gour Gopal Biswas
대한수학회 (The Korean Mathematical Society)|2022.07|대한수학회논문집 (Communications of the KMS)|제37권 제3호|pp. 825-837|DOI 10.4134/CKMS.c210046

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키워드

Ricci solitongradient Ricci solitongradient $m$-quasi-Einstein metricgradient $\rho$-Einstein soliton$\eta$-Ricci solitongradient $\eta$-Ricci solitonsemi-symmetric metric connectionsemi-symmetric metric $P$-connection
주제분류수학

초록

We characterize a three-dimensional Riemannian manifold endowed with a type of semi-symmetric metric $P$-connection. At first, it is proven that if the metric of such a manifold is a gradient $m$-quasi-Einstein metric, then either the gradient of the potential function $\psi$ is collinear with the vector field $P$ or, $\lambda=-(m+2)$ and the manifold is of constant sectional curvature $-1$, provided $P\psi \neq m$. Next, it is shown that if the metric of the manifold under consideration is a gradient $\rho$-Einstein soliton, then the gradient of the potential function is collinear with the vector field $P$. Also, we prove that if the metric of a 3-dimensional manifold with semi-symmetric metric $P$-connection is a gradient $\omega$-Ricci soliton, then the manifold is of constant sectional curvature $-1$ and $\lambda+\mu=-2$. Finally, we consider an example to verify our results.

Abstract

We characterize a three-dimensional Riemannian manifold endowed with a type of semi-symmetric metric $P$-connection. At first, it is proven that if the metric of such a manifold is a gradient $m$-quasi-Einstein metric, then either the gradient of the potential function $\psi$ is collinear with the vector field $P$ or, $\lambda=-(m+2)$ and the manifold is of constant sectional curvature $-1$, provided $P\psi \neq m$. Next, it is shown that if the metric of the manifold under consideration is a gradient $\rho$-Einstein soliton, then the gradient of the potential function is collinear with the vector field $P$. Also, we prove that if the metric of a 3-dimensional manifold with semi-symmetric metric $P$-connection is a gradient $\omega$-Ricci soliton, then the manifold is of constant sectional curvature $-1$ and $\lambda+\mu=-2$. Finally, we consider an example to verify our results.

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