Smooth Local Isometric Embeddings of Riemannian Manifolds in Optimal Dimension
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…
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