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- Title
Isometric immersions of RCD spaces.
- Authors
Shouhei Honda
- 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.
- Subjects
RIEMANNIAN manifolds; SET theory; METRIC spaces; IMMERSIONS (Mathematics); FINITE, The; SPHERES; EMBEDDING theorems
- Publication
Commentarii Mathematici Helvetici, 2021, Vol 96, Issue 3, p515
- ISSN
0010-2571
- Publication type
Article
- DOI
10.4171/CMH/519