The Schwarzian derivative on Finsler manifolds of constant curvature

DOI10.1007/S10998-021-00411-ZarXiv2304.00480MaRDI QIDQ6431734

F. Sedighi, Behroz Bidabad

Publication date: 2 April 2023

Abstract: Lagrange introduced the notion of Schwarzian derivative and Thurston discovered its mysterious properties playing a role similar to that of curvature on Riemannian manifolds. Here we continue our studies on the development of the Schwarzian derivative on Finsler manifolds. First, we obtain an integrability condition for the M"{o}bius equations. Then we obtain a rigidity result as follows; Let $(M,F)$ be a connected complete Finsler manifold of positive constant Ricci curvature. If it admits non-trivial M"{o}bius mapping, then $M$ is homeomorphic to the $n$-sphere. Finally, we reconfirm Thurston's hypothesis for complete Finsler manifolds and show that the Schwarzian derivative of a projective parameter plays the same role as the Ricci curvature on theses manifolds and could characterize a Bonnet-Mayer-type theorem.

Mathematics Subject Classification ID

Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds (58B20) Global differential geometry of Finsler spaces and generalizations (areal metrics) (53C60)

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