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Sectional curvature and Weitzenb\"ock formulae

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arxiv 1708.09033 v3 pith:M36BNRSM submitted 2017-08-29 math.DG

Renato G. Bettiol , Ricardo A. E. Mendes This is my paper · ORCID
classification math.DG
keywords curvatureformulaesectionalsymmetrictermsweitzenbalgebraicanalogue
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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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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Sharp Weitzenb\"{o}ck and PIC2 Estimates from Sectional-Scalar Curvature Pinching

    math.DG 2026-07 accept novelty 8.0 of 10

    Sharp Bochner-type estimates show that sectional-scalar curvature pinching forces vanishing of the second Betti number, with the equality case forcing a complex projective space.

  2. New Curvature Conditions for the Bochner Technique

    math.DG 2019-08 accept novelty 7.0 of 10

    A closed Riemannian manifold whose curvature operator has positive average of its lowest n-p eigenvalues has zero p-th and (n-p)-th Betti numbers, with quantitative diameter-dependent estimates.

  3. Convex Algebraic Geometry of Curvature Operators

    math.DG 2019-08 conditional novelty 7.0 of 10

    The set of algebraic curvature operators with sectional curvature at least k is a spectrahedron only up to dimension 3, a spectrahedral shadow in dimension 4, and not a spectrahedral shadow from dimension 5 onward, wi...

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