A Weitzenböck-Type Formula for Conformally Flat Manifolds and Applications

Abstract A Weitzenböck-type formula for conformally flat manifolds with nonnegative constant $$\sigma _{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>σ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> -curvature is established. Based on this, we introduce a suitable class of Riemannian manifolds, called $$\mathcal {A}_{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> -manifolds, and prove an appropriate version of the Omori-Yau maximum principle for them. As a consequence, we derive several results concerning complete $$\mathcal {A}_{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> -manifolds and establish a Catino-type integral inequality.