Finsler Metrics of Constant Positive Curvature on the Lie Group $s^3$

Details Finsler Metrics of Constant Positive Curvature on the Lie Group $s^3$ Finsler Metrics of Constant Positive Curvature on the Lie Group $s^3$

2000-11-12

arxiv.org/abs/math/0011071v1

Mathematics

Differential Geometry

35 pages with a Maple program

Scientific paper

Guided by the Hopf fibration, we single out a family (indexed by a positive constant K) of right invariant Riemannian metrics on the Lie group $S^3$. Using the Yasuda-Shimada theorem as an inspiration, we determine for each K>1 a privileged right invariant Killing field of constant length. Each such Riemannian metric pairs with the corresponding Killing field to produce a y-global and {\it explicit} Randers metric on $S^3$. Using the machinery of spray curvature and Berwald's formula for it, we prove directly that the said Randers metric has constant positive flag curvature K, as predicted by the Yasuda-Shimada theorem. We also explain why this family of Finslerian space forms is NOT projectively flat.

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