Sectional curvature and Weitzenb\"ock formulae

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