Kählerian geometry of the infinite-dimensional homogeneous manifold \(M=Diff_ +(S^ 1)/Rot(S^ 1)\) (Q1102562)

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.