PreprintCC BY 4.0arXiv
Klein bottle and optimal systolic inequality for nonpositively curved surfaces
Abstract
We show that the systolic area of every nonpositively curved closed surface M other than the torus is at least $1$, with equality if and only if M is isometric to a square flat Klein bottle. The proof focuses on the Klein-double 4ℝP2 and exploits Weil's isoperimetric inequality and a comparison theorem involving a new kind of exponential-type map.
Versions
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BibTeX
@misc{katz2026klein,
title = {{Klein bottle and optimal systolic inequality for nonpositively curved surfaces}},
author = {Mikhail G. Katz and Stephane Sabourau},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06705},
eprint = {2609.06705},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06705},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Klein bottle and optimal systolic inequality for nonpositively curved surfaces
AU - Mikhail G. Katz
AU - Stephane Sabourau
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06705
UR - https://arxiv.org/abs/2609.06705
AB - We show that the systolic area of every nonpositively curved closed surface M other than the torus is at least $1$, with equality if and only if M is isometric to a square flat Klein bottle. The proof focuses on the Klein-double 4ℝP^2 and exploits Weil's isoperimetric inequality and a comparison theorem involving a new kind of exponential-type map.
ER -