Lipschitz Normal Embeddings and Determinantal Singularities

TL;DR

This paper proves that model determinantal singularities, specifically spaces of matrices with rank constraints, are Lipschitz normally embedded, establishing a key metric property of these algebraic varieties.

Contribution

It demonstrates that the space of matrices with rank constraints is Lipschitz normally embedded, advancing understanding of metric properties of determinantal singularities.

Findings

Model determinantal singularities are Lipschitz normally embedded.

The proof applies to spaces of matrices with rank constraints.

Discussion on extending results to general determinantal singularities.

Abstract

The germ of an algebraic variety is naturally equipped with two different metrics up to bilipschitz equivalence. The inner metric and the outer metric. One calls a germ of a variety Lipschitz normally embedded if the two metrics are bilipschitz equivalent. In this article we prove that the model determinantal singularity, that is the space of $m\times n$ matrices of rank less than a given number, is Lipschitz normally embedded. We will also discuss some of the difficulties extending this result to the case of general determinantal singularities.