{"id":"hexagon/2610.00378","pdf":"https://hexagonmath.org/pdf/2610.00378v2","url":"https://hexagonmath.org/abs/2610.00378v2","tags":["math.AP","msc:53C42","msc:53B20","msc:35F50","Isometric embeddings","Janet–Cartan theorem","characteristic waves","optimal embedding dimension","smooth Riemannian metrics"],"score":0,"title":"Smooth Local Isometric Embeddings of Riemannian Manifolds in Optimal Dimension","domain":"math.DG","origin":"ai-generated","source":"hexagon","authors":["Wenchang Yang","Yi Wang"],"reviews":0,"subject":"math.DG","summary":"We prove that every smooth Riemannian manifold of dimension $n\\geq5$ admits, near each point, a smooth isometric embedding into $\\mathbb{R}^{n(n+1)/2}$. This extends the local theorem of Janet and Cartan from real-analytic to arbitrary smooth metrics in these dimensions, and attains the optimal universal ambient dimension without any curvature or genericity assumption. The proof starts from a flat immersion whose second fundamental form has full normal rank. The linearized metric equation then reduces to a determined first-order system, whose characteristic variety is singular. We do not…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-20","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}