Vanishing bach-like tensors on complete gradient shrinking ricci solitons

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 $\mathfrak{B} = \alpha U + \beta V = 0$.