Geodesics on the Moduli Space of Oriented Circles in \mathbb{S}^{3} |
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Authors: | Linyuan Fan Ying L?? Changping Wang Jingyang Zhong |
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Institution: | 1. School of Mathematical Sciences, LMAM, Peking University, Beijing, 100871, People??s Republic of China 2. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, People??s Republic of China
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Abstract: | Let ${\mathbb{Q}^3}$ be the moduli space of oriented circles in the three dimensional unit sphere ${\mathbb{S}^3}$ . Given a natural complex structure such space becomes a three dimensional complex manifold, with a M?bius invariant Hermitian metric h of type (2, 1). Up to M?bius transformations, all geodesics with respect to the Lorentz metric g = Re(h) on ${\mathbb{Q}^3}$ are determined to form a one-parameter family of circles on a helicoid in a space form ${\mathbb{R}^3, \mathbb{H}^3}$ or ${\mathbb{S}^{3}}$ , resp. We show also that any two oriented circles in ${\mathbb{S}^3}$ are connected by countably infinitely many geodesics in ${\mathbb{Q}^3}$ . |
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