Isometric immersions of RCD spaces

  • Shouhei Honda

    Tohoku University, Sendai, Japan
Isometric immersions of RCD spaces cover

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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.

Cite this article

Shouhei Honda, Isometric immersions of RCD spaces. Comment. Math. Helv. 96 (2021), no. 3, pp. 515–559

DOI 10.4171/CMH/519