math.DG — Differential Geometry

EVERY CLOSED SMOOTH $n$-MANIFOLD EMBEDS ISOMETRICALLY IN $\mathbb{R}^{\,n(n+1)/2+28}$

Contributed by Yi Wang, Wenchang Yang

We prove that every closed smooth Riemannian $n$-manifold admits a smooth isometric embedding into $\mathbb{R}^{s_n+28}$, where $s_n=n(n+1)/2$. This improves the previous upper bound $s_n+\max\{n+5,\,2n\}$ to a dimension-independent surplus of $28$.