Rigidity of noncompact manifolds with cyclic parallel Ricci curvature

Abstract

We prove that if M is a complete noncompact Riemannian manifold whose Ricci tensor is cyclic parallel and whose scalar curvature is nonpositive, then M is Einstein, provided the Sobolev constant is positive and an integral inequality is satisfied.

On the Geometry in the Large of Einstein-like Manifolds

“…On the other hand, Deng (see [22]) studied the rigidity of complete A-manifolds and showed that (M, g) is an n-dimensional complete Einstein-like manifold of type A with a Yamabe constant Q(M, g) > 0 and nonpositive scalar curvature, and is an Einstein manifold if there exists a small number C depending on the dimension n and Q(M, g) such that M ( Ric − s/n • g n/2 + W n/2 )dvol g ≤ C for the Weyl curvature tensor, W. In turn, Chu modernized this result in his article [23]. We remark here that the above results were obtained using the methods of the classical Bochner technique.…”