Vanishing bach-like tensors on complete gradient shrinking ricci solitons

TL;DR

This paper extends the understanding of Bach-flat gradient shrinking Ricci solitons by analyzing conditions under which certain Bach-like tensors vanish, broadening previous results in geometric analysis.

Contribution

It generalizes prior work by characterizing vanishing conditions of Bach-like tensors on gradient shrinking Ricci solitons.

Findings

Vanishing of specific Bach-like tensors implies geometric constraints.

Extension of results from Bach-flat to more general tensor conditions.

Provides new insights into the structure of Ricci solitons.

Abstract

The Bach tensor is classically defined in dimension 4, and work from J. Bergman \cite{bergman:2004} and others shows that $B = \frac{1}{2}U + \frac{1}{6}V$ where $U$ and $V$ are more basic 2-tensors, which are symmetric, divergence-free, algebraically independent, and quadratic in the Riemann tensor. In this paper, we extend H.-D. Cao and Q. Chen's results \cite{caochen:2013} for Bach-flat gradient shrinking Ricci solitons to solitons with $\mathfrak{B} = \alpha U + \beta V = 0$.