On gradient $\ddot{I}$-Einstein solitons with Bach tensor radially nonnegative

Abstract

In this paper, we study $n$-dimensional gradient $\ddot{I}$-Einstein solitons whose Bach tensor is radially nonnegative. Under this assumption, we show that such $\ddot{I}$-Einstein solitons are locally warped products of an interval and an Einstein manifold, provided either $\ddot{I}\ne 0$ or $\ddot{I}=0$ and the soliton is rectifiable. We obtain as a consequence that these solitons must have harmonic Weyl tensor and vanishing Bach tensor. We also finish the classification of complete locally conformally flat steady $\ddot{I}$-Einstein solitons and classify these manifolds when their Bach tensor is radially nonnegative and $n\in \{3,4\}$.