Curvature Sign Rigidity and Sharp Pointwise Pinching Thresholds

Abstract

We study connected Riemannian manifolds on which either the sectional curvature or the Ricci tensor is, at each point, strictly positive, strictly negative, or zero, and ask whether the two signs can coexist under pointwise pinching. A general support-rigidity theorem for positive semidefinite divergence-free symmetric tensors is the common analytic mechanism. For sectional curvature in dimension $n\ge 3$, any locally uniform positive lower bound for the absolute pointwise pinching ratio rules out a change of sign. With a fixed pinching constant $\delta_0$, the flat set has no $C^1$ hypersurface piece; it is empty when $\delta_0>1/2$, and is locally porous when $\delta_0=1/2$. These conclusions are sharp in several senses: there are smooth conformally flat local metrics whose sectional curvature changes sign across a flat hypersurface when the pinching degenerates, and there are closed exactly $1/q$-pinched metrics on spheres with isolated flat points. For Ricci curvature, the sharp threshold is \[ \delta_c=\frac1{n-1}. \] A locally uniform gap $\delta_{\mathrm{Ric}}>\delta_c$ forces one Ricci sign globally. Conversely, for every $0<\delta<\delta_c$ there are local sign-changing metrics with exact pointwise pinching $\delta_{\mathrm{Ric}}\equiv\delta$, and exact local examples also exist at $\delta=\delta_c$. Pointwise strictness $\delta_{\mathrm{Ric}}>\delta_c$ is insufficient if the gap collapses at a Ricci-flat interface. For every subcritical $\delta$ we give explicit closed metrics on $\mathbb{S}^1\times\mathbb{S}^{n-1}$, and complete periodic lifts to $\mathbb{R}\times\mathbb{S}^{n-1}$, whose optimal global lower pinching constant is exactly $\delta$. Finally, we prove two ansatz-specific obstructions to compactifying the exact local constructions. The main content of the proof is generated by ChatGPT 5.6 sol and verified by the authors.