Kählerian geometry of the infinite-dimensional homogeneous manifold \(M=Diff_ +(S^ 1)/Rot(S^ 1)\)
type: Item
instance of: scholarly article
title: Kählerian geometry of the infinite-dimensional homogeneous manifold \(M=Diff_ +(S^ 1)/Rot(S^ 1)\)
zbMATH Open document ID: 0644.53060
DOI: 10.1007/BF01083503
publication date: 1986-01-01T00:00:00Z
review text: Let \(G=Diff_+(S^ 1)\) be the group of orientation preserving diffeomorphisms of the circle and \(H=SO(2)\) the subgroup of rotations. Theorem 1 identifies the homogeneous space \(M=G/H\) with the space of holomorphic functions f on \(\{| z| <1\}\) normalized by \(f(0)=0\) and \(f'(0)=1\), that are one-to-one and smooth up to the boundary. Theorem 2 states that all invariant Kähler structures on M form a two- parameter family. Their imaginary parts are spanned by the symplectic forms \(\omega_ 1\) and \(\omega_ 2\) provided by the fact that M is an orbit in the coadjoint action of both G and its central extension. Explicit formulas for the metrics, their curvature tensor and sectional curvature follow. An invariant net of mutually perpendicular 1- dimensional complex foliations on M is introduced. Its basic properties and relation to the group of isometries of M is given.
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 58B20
zbMATH DE Number: 4050490
zbMATH Keywords: homogeneous space
zbMATH Keywords: holomorphic functions
zbMATH Keywords: invariant Kähler structures
zbMATH Keywords: complex foliations
zbMATH Keywords: group of isometries
author: A. A. Kirillov
author: Denis S. Juriev
MaRDI profile type: Publication
cites work: Construction of unitary irreducible representations of Lie groups
cites work: A proof of the Bieberbach conjecture
cites work: Q3720764
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published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 4050490