Rigidity of Riemannian manifolds with vanishing generalized Bach tensor

Abstract

A generalized Bach tensor Bijt with parameter t∈R is introduced. A Riemannian manifold (Mn,g) is called Bt-flat if its generalised Bach tensor Bijt≡0 for some parameter t. In this paper, we first study the rigidity of closed Bt-flat Riemannian manifolds with positive constant scalar curvature. When the dimension n=4, we prove that all Bt-flat manifolds that satisfy a point-wise inequality must be of positive constant sectional curvature. Similar rigidity results are also obtained in terms of the Yamabe invariant. Moreover, using the curvature estimates for the general dimension n≥4, we obtain an integral inequality for Bt-flat manifolds, and prove that the equality occurs if and only if these manifolds are of positive constant sectional curvature. In addition, we also obtain similar rigidity results for complete Bt-flat manifolds.