In this paper, we have investigated some aspects of gradient \(\rho\)-Einstein Ricci soliton in a complete Riemannian manifold. First, we have proved that the compact gradient \(\rho\)-Einstein soliton satisfying some curvature conditions is isometric to the Euclidean sphere by showing that the scalar curvature becomes constant. Second, we have shown that in a non-compact gradient \(\rho\)-Einstein soliton satisfying an integral condition, the scalar curvature vanishes.

中文翻译：

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紧凑梯度 $$\rho$$ ρ -Einstein 孤子与欧几里德球面等距

在本文中，我们研究了完全黎曼流形中梯度\(\rho\) -Einstein Ricci 孤子的某些方面。首先，我们通过证明标量曲率变为常数，证明了满足某些曲率条件的紧梯度\(\rho\) -爱因斯坦孤子与欧几里德球面等距。其次，我们已经证明，在满足积分条件的非紧梯度\(\rho\) -爱因斯坦孤子中，标量曲率消失。