Local quasi-isometries and tangent cones of definable germs

Author:

Nhan Nguyen

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), Algebraic Topology (math.AT)

journal:

date:

2023-05-24 16:00:00

Abstract

In this paper, we introduce the notion of local quasi-isometry for metric germs and prove that two definable germs are quasi-isometric if and only if their tangent cones are bi-Lipschitz homeomorphic. Since bi-Lipschitz equivalence is a particular case of local quasi-isometric equivalence, we obtain Sampaio's tangent cone theorem as a corollary. As an application, we reprove the theorem by Fernandes-Sampaio, which states that the tangent cone of a Lipschitz normally embedded germ is also Lipschitz normally embedded.

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