Sectional curvature and Weitzenbock formulae

We establish a new algebraic characterization of sectional curvature bounds $\sec\geq k$ and $\sec\leq using only terms in the Weitzenb\"ock formulae for symmetric $p$-tensors. By introducing analogue Kulkarni-Nomizu product, we provide simple formula such terms. also give an application Bochner technique to closed $4$-manifolds with indefinite intersection form $\sec>0$ or $\sec\geq0$, obtaining insights into Hopf Conjecture, without any symmetry assumptions.