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 $4\mathbb{R}{P}^2$ and exploits Weil's isoperimetric inequality and a comparison theorem involving a new kind of exponential-type map.

17 pages, to appear in Annals of Global Analysis and Geometry

References in corpus (3)

• Systolically extremal nonpositively curved surfaces are flat with finitely many singularities
• Nonpositively curved surfaces are Loewner
• Stability of systolic inequalities for the Möbius strip and Klein bottle