Determinantal variety and normal embedding

K. Katz, M. Katz, D. Kerner, Y. Liokumovich

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

The space GLn+ of matrices of positive determinant inherits an extrinsic metric space structure from ân2. On the other hand, taking the infimum of the lengths of all paths connecting a pair of points in GLn+ gives an intrinsic metric. We prove bi-Lipschitz equivalence between intrinsic and extrinsic metrics on GLn+, exploiting the conical structure of the stratification of the space of n × n matrices by rank.

Original languageEnglish
Pages (from-to)27-34
Number of pages8
JournalJournal of Topology and Analysis
Volume10
Issue number1
DOIs
StatePublished - 1 Mar 2018

Bibliographical note

Publisher Copyright:
© 2018 World Scientific Publishing Company.

Funding

M.K. was partially funded by the Israel Science Foundation grant No. 1517/12. D.K. was partially supported by the Israel Science Foundation grant 844/14. This paper answers a question posed by Asaf Shachar at MOa and we thank him for posing the question. We are grateful to Yves Cornulier for a helpful comment posted there. We are grateful to Jake Solomon for providing the proof in Sec. 4 of the general case of the bi-Lipschitz property for the determinantal variety. We thank Jason Starr for a helpful comment posted at MO.b We thank Amitai Yuval for pointing out a gap in an earlier version of the article, and Alik Nabutovsky and Kobi Peterzil for useful suggestions.

FundersFunder number
Seventh Framework Programme337560, 334347
Israel Science Foundation1517/12, 844/14

    Keywords

    • Determinantal variety
    • bi-Lipschitz equivalence
    • conical stratification
    • intrinsic metric

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