Sectional curvature and Weitzenb\"ock formulae

TL;DR

This paper introduces a novel algebraic approach to characterize sectional curvature bounds using Weitzenb"ock formulae and applies it to study the Hopf Conjecture in 4-manifolds with positive or nonnegative sectional curvature.

Contribution

It develops a new algebraic characterization of sectional curvature bounds via curvature terms in Weitzenb"ock formulae and introduces a symmetric Kulkarni-Nomizu product for simplified formulas.

Findings

New algebraic characterization of sectional curvature bounds

Application of Bochner technique to 4-manifolds with positive/ nonnegative curvature

Insights into the Hopf Conjecture without symmetry assumptions

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.