math.DG — Differential Geometry

Isometric embeddings of parallelizable and stably parallelizable Riemannian manifolds

Contributed by Wenchang Yang, Yi Wang

Let $s_n=n(n+1)/2$. We prove that every smooth parallelizable Riemannian $n$-manifold, $n\ge 3$, admits a smooth isometric embedding into $\mathbb{R}^{s_n+1}$ whose image lies in an arbitrarily small ball. For closed stably parallelizable manifolds, each of whose connected components has vanishing Euler characteristic, we obtain small-image isometric embeddings into $\mathbb{R}^{s_n+5}$. The proof combines explicit partial isometries along commuting Hopf flows, an iterative scheme for exact metric correction using oscillatory perturbations constructed from the second fundamental form, and a transversality argument for metric-preserving perturbations.