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arXiv · 1708.09033

Sectional curvature and Weitzenb\"ock formulae

Renato G. Bettiol·Ricardo A. E. Mendes
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Abstract

We establish a new algebraic characterization of sectional curvature bounds $\sec\geq k$ and $\sec\leq k$ 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 $\sec\geq0$, obtaining new insights into the Hopf Conjecture, without any symmetry assumptions.

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BibTeXRIS

Renato G. Bettiol, Ricardo A. E. Mendes. 2017-08-29. Sectional curvature and Weitzenb\"ock formulae. https://doi.org/10.1512/iumj.2022.71.8927

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