Negative sectional curvature in a complex shape forces its internal geometry to follow a strict numerical law.
The Hopf conjecture states that the curvature of a manifold dictates its fundamental topology in a very specific way. Geometry experts previously only knew this rule applied to Kähler manifolds, which are highly rigid and orderly. This proof expands the law to almost Kähler manifolds, which are far more flexible and common in nature. It confirms that even when a shape is warped and stretched, its curvature still exerts a predictable grip on its identity. This result provides a new mathematical foundation for understanding how gravity or energy fields can twist space without breaking its core structure.
Hirzebruch χy-genus of compact almost Kähler manifold with negative sectional curvature
arXiv · 2604.27423
Let \((X,J,\omega)\) be a closed \(2n\)-dimensional almost Kähler manifold with negative sectional curvature. We prove that if the Nijenhuis tensor of the almost complex structure is sufficiently small, then the components of the Hirzebruch \(\chi_{y}\)-genus satisfy the inequality \((-1)^{n-p}\chi_{p}(X)\geq 1\) for all \(p=0,1,\cdots,n\). In particular, this result implies the Hopf conjecture in this setting, namely that the Euler number satisfies \((-1)^{n}\chi(X)\geq n+1\). The proof is base