Splitting theorem of gradient \(\rho \)-Einstein solitons

Abstract

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 \(\rho \)-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 \(\rho \)-Einstein soliton.