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Rigidity characterizations of complete Riemannian manifolds with \(\alpha \)-Bach-flat. (English) Zbl 1471.53043

Summary: For complete manifolds with flat \(\alpha \)-Bach tensor (which is defined by (1.2)), we provide some rigidity results characterized by some point-wise inequalities involving the Weyl curvature and the traceless Ricci curvature. Moveover, some Einstein metrics are also characterized by some \(L^{\frac{n}{2}} \)-integral inequalities. Furthermore, we give some rigidity characterizations for constant sectional curvature.

MSC:

53C24 Rigidity results
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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