DEFORMATIONS OF THE VERONESE EMBEDDING AND FINSLER $2$ -SPHERES OF CONSTANT CURVATURE. (20th November 2022)

• Record Type: Journal Article
• Authors: Lange, Christian; Mettler, Thomas
• Is Part Of: Journal of the Institute of Mathematics of Jussieu. Volume 21:Number 6(2022)
• Page Start: 2103
• Page End: 2134
• Publication Date: 2022-11-20
• Subjects: Finsler structures -- constant curvature -- Zoll metrics -- spindle orbifolds -- holomorphic curves; 53B40 -- 53A20 -- 58C10; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S1474748021000153
• Languages: English
• ISSNs: 1474-7480

Abstract

Abstract: We establish a one-to-one correspondence between, on the one hand, Finsler structures on the $2$ -sphere with constant curvature $1$ and all geodesics closed, and on the other hand, Weyl connections on certain spindle orbifolds whose symmetric Ricci curvature is positive definite and whose geodesics are all closed. As an application of our duality result, we show that suitable holomorphic deformations of the Veronese embedding $\mathbb {CP}(a_1, a_2)\rightarrow \mathbb {CP}(a_1, (a_1+a_2)/2, a_2)$ of weighted projective spaces provide examples of Finsler $2$ -spheres of constant curvature whose geodesics are all closed.