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

TL;DR

This paper proves that connected conic singular sub-manifolds are Lipschitz Normally Embedded, meaning their outer and inner metrics are equivalent, with applications showing algebraic sets are conic at infinity and Lipschitz Normally Embedded.

Contribution

It establishes the Lipschitz Normal Embedding property for conic singular sub-manifolds and characterizes algebraic sets as conic at infinity, linking geometric and algebraic structures.

Findings

Connected conic singular sub-manifolds are Lipschitz Normally Embedded.

Closure in the one point compactification characterizes conic singular sub-manifolds.

Generic algebraic sets are conic at infinity and Lipschitz Normally Embedded.

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.