Splitting theorem of Gradient $ρ$-Einstein solitons

In this paper, we have proved a weighted Laplacian comparison of distance function for manifolds with Bakry-Émery curvature bounded from below. Next, we have shown that a gradient $ρ$-Einstein soliton with a bounded integral condition on Ricci curvature splits off a line isometrically. Moreover, using this result, we have established some boundedness conditions on scalar curvature of gradient $ρ$-Einstein soliton.

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