Hexagon

math.DG — Differential Geometry

A proof of the second pinching theorem in dimension four

Contributed by JINXI ZHANG

Let M⁴ be a closed connected manifold minimally immersed in the unit sphere S⁵(1), and let h be its second fundamental form. We prove that constant S = |h|² > 4 implies S > 8. No constancy of tr(h³) is assumed. The proof combines integral curvature identities with a joint estimate for two contractions of the symmetrized Hessian. The remaining algebraic inequality is verified in Mathematica using exact rational arithmetic. The code and data are included in the supplementary material.

A mix of human-written and AI-generated textHuman understanding: all partsChern conjectureMinimal hypersurfaceconstant scalar curvaturesecond pinching problem

Oct 20, 2026PDF ↗View work →

hexagon:2610.00378v1

math.DG — Differential Geometry

Smooth Local Isometric Embeddings of Riemannian Manifolds in Optimal Dimension

Contributed by Wenchang Yang, Yi Wang

We prove that every smooth Riemannian manifold of dimension $n\ge 5$ admits a smooth local isometric embedding into $\mathbb{R}^{n(n+1)/2}$ . This extends the classical Janet–Cartan theorem from real-analytic to arbitrary smooth metrics in these dimensions, achieving the optimal universal ambient dimension without curvature or genericity assumptions. The proof constructs characteristic waves around a flat immersion whose second fundamental form has full normal rank. Their Hermitian quadratic mean generates the required metric correction, while linear responses cancel oscillatory quadratic interactions and the remaining linear strain. Exact Fourier localization yields an iteration converging in $C^\infty$ on a fixed neighborhood.The detailed geometric and microlocal constructions are proved in the appendices.

A mix of human-written and AI-generated textHuman understanding: some partsIsometric embeddingsJanet–Cartan theoremcharacteristic wavesoptimal embedding dimensionsmooth Riemannian metrics

Oct 20, 2026PDF ↗View work →

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A proof of the second pinching theorem in dimension four

Smooth Local Isometric Embeddings of Riemannian Manifolds in Optimal Dimension

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