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 language | English |
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Pages (from-to) | 27-34 |
Number of pages | 8 |
Journal | Journal of Topology and Analysis |
Volume | 10 |
Issue number | 1 |
DOIs | |
State | Published - 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.
Funders | Funder number |
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Seventh Framework Programme | 337560, 334347 |
Israel Science Foundation | 1517/12, 844/14 |
Keywords
- Determinantal variety
- bi-Lipschitz equivalence
- conical stratification
- intrinsic metric