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 $\mathfrak{ B} = \alpha U + \beta V = 0$.
Categories: math.DG