Abstract:

A Riemannian manifold $ (M^n, g) $ is called $ B^t $-flat if its generalized Bach tensor $ B^t_{ij} \equiv 0 $ for some parameter $ t $. In this paper, we show that a four-dimensional compact $ B^t $-flat Riemannian manifold with $t < 1, t \neq 0$ and satisfying a pointwise inequality must be Einstein. In particular, under the same assumption and $t \ge -\frac{1}{3}$, we conclude that it must be isometric to either a quotient of the round $\mathbb{S}^4$ or a $\mathbb{C P}^2$ with the Fubini-Study metric. This extends the result of Huang-Ma-Li [Huang G, Ma B, Li X. J Geom Phys, 2021, 170: Art 104380].

Key words: Einstein manifold, rigidity theorem, generalized Bach-flat.