Global Lipschitz geometry of conic singular sub-manifolds with applications to algebraic sets

Author:

André Costa, Vincent Grandjean, Maria Michalska

Keyword:

Mathematics, Differential Geometry, Differential Geometry (math.DG), Algebraic Geometry (math.AG), Geometric Topology (math.GT)

journal:

date:

2023-06-25 16:00:00

Abstract

The main result states that a connected conic singular sub-manifold of a Riemannian manifold, compact when the ambient manifold is non-Euclidean, is Lipschitz Normally Embedded: the outer and inner metric space structures are metrically equivalent. We also show that a closed subset of $\mathbb{R}^n$ is a conic singular sub-manifold if and only if its closure in the one point compactification ${\bf S}^n =\mathbb{R}^n\cup \infty$ is a conic singular sub-manifold. Consequently the connected components of generic affine real and complex algebraic sets are conic at infinity, thus are Lipschitz Normally Embedded.

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