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.