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Sectional curvature and Weitzenb\"ock formulae

Abstract

We establish a new algebraic characterization of sectional curvature bounds seck\sec\geq k and seck\sec\leq k using only curvature terms in the Weitzenb\"ock formulae for symmetric pp-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 44-manifolds with indefinite intersection form and sec>0\sec>0 or sec0\sec\geq0, obtaining new insights into the Hopf Conjecture, without any symmetry assumptions.Comment: LaTeX2e, 25 pages, final version. To appear in Indiana Univ. Math.

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