Isometric immersions of RCD spaces

Abstract

We prove that if an RCD space has a regular isometric immersion in a Euclidean space, then the immersion is a locally bi-Lipschitz embedding map. This result leads us to prove that if a compact non-collapsed RCD space has an isometric immersion in a Euclidean space via an eigenmap, then the eigenmap is a locally bi-Lipschitz embedding map to a sphere, which generalizes a fundamental theorem of Takahashi in submanifold theory to a non-smooth setting. Applications of these results include a topological sphere theorem and topological finiteness theorems, which are new even for closed Riemannian manifolds.

arXiv comment36 pages. To appear in Comment. Math. Helv

References in corpus (5)

• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure
• Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in RCD(K,\infty) metric measure spaces
• The splitting theorem in non-smooth context
• A sufficient condition to a regular set of positive measure on RCD spaces