SMOOTH SU (p,q)-ACTIONS ON THE (2p+2q-1)-SPHERE AND ON THE COMPLEX PROJECTIVE (p+q-1)-SPACE

Mukoyama, Kazuo

doi:10.2206/kyushujm.55.213

Cited by 5 publications

(1 citation statement)

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“…He has proposed certain new idea to investigate the latter case. His idea was improved and used by Uchida [5][6][7] and Mukōyama [2,3]. In the case of these papers, G is a simple Lie group and the singular orbits of given K action are contained in open orbits of extended G action.…”

Section: Introductionmentioning

confidence: 99%

On Smooth C*×SO(n,C)-Actions on the Sphere of Dimension 2n−1

Uchida

2003

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We shall study smooth C Ã Â SOðn; CÞ actions on S 2nÀ1 , each of which is an extension of the standard Uð1Þ Â SOðnÞ action. We shall show such an action is characterized by a smooth R 2 -action on S 3 and a smooth mapping of S 3 onto P 1 ðCÞ. Moreover, we shall show there are uncountably many topologically distinct such extensions.KEYWORDS: non-compact Lie group action, sphere, codimension one IntroductionConsider the standard Uð1Þ Â SOðnÞ action on the ð2n À 1Þ-sphere S 2nÀ1 . Here we regard S 2nÀ1 as the unit sphere in C n , the complex number space of n-dimension. This action has codimension one principal orbits and just two singular orbits. In this paper, we shall study smooth C Ã Â SOðn; CÞ actions on S 2nÀ1 , each of which is an extension of the standard Uð1Þ Â SOðnÞ action. We shall show such an action is characterized by a pair of a smooth R 2 -action on S 3 and a smooth mapping of S 3 onto P 1 ðCÞ. Practically, we shall construct a smooth C Ã Â SOðn; CÞ action on S 2nÀ1 from a pair of a smooth R 2 -action on S 3 and a smooth mapping of S 3 onto P 1 ðCÞ satisfying certain condition. Moreover, we shall show there are uncountably many topologically distinct such extensions. More precisely, we may construct smooth actions È m;c for any positive integer m and real number c, such that if ðm; cÞ 6 ¼ ðm 0 ; c 0 Þ then the corresponding actions are mutually distinct as continuous actions.Let G be a non-compact semi-simple Lie group and K a maximal compact connected subgroup of G. For a smooth K action on the sphere S N with codimension one orbits, we have studied smooth G actions each of which is an extension of the given K action. Asoh [1] has investigated smooth SLð2; CÞ actions on 3-manifolds. The maximal compact subgroup of SLð2; CÞ is SUð2Þ and it is known that the smooth action of SUð2Þ on 3-manifold is transitive or has a codimension one orbit. He has proposed certain new idea to investigate the latter case. His idea was improved and used by Uchida [5][6][7] and Mukōyama [2,3]. In the case of these papers, G is a simple Lie group and the singular orbits of given K action are contained in open orbits of extended G action.Uchida [8] has studied smooth SLðm; RÞ Â SLðn; RÞ actions on S mþnÀ1 each of which is an extension of the standard SOðmÞ Â SOðnÞ action. In that case, SLðm; RÞ Â SLðn; RÞ is not a simple Lie group and the singular orbits of the standard SOðmÞ Â SOðnÞ action on S mþnÀ1 are invariant under any extended smooth SLðm; RÞ Â SLðn; RÞ action. On the contrary, one of the singular orbits of the standard Uð1Þ Â SOðnÞ action is contained in an open orbit of extended action and the other singular orbit is invariant under the extended action, for any smooth C Ã Â SOðn; CÞ actions on S 2nÀ1 , each of which is an extension of the standard Uð1Þ Â SOðnÞ action. The author wishes to express gratitude to the referees and the editors for their advice.

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Section: Introductionmentioning

confidence: 99%