We establish a new algebraic characterization of sectional curvature bounds
sec≥k and sec≤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≥0, obtaining new
insights into the Hopf Conjecture, without any symmetry assumptions.Comment: LaTeX2e, 25 pages, final version. To appear in Indiana Univ. Math.