Riemannian geometry of Diff(S¹)/S¹ revisited

Abstract

Abstract. A further study of Riemannian geometry Diff(S 1 )/S 1 is presented. We describe Hermitian and Riemannian metrics on the complexification of the homogeneous space, as well as the complexified symplectic form. It is based on the ideas from [12], where instead of using the Kähler structure symmetries to compute the Ricci curvature, the authors rely on classical finite-dimensional results of Nomizu et al on Riemannian geometry of homogeneous spaces.

Quasi-invariance of Brownian measures on the group of circle homeomorphisms and infinite-dimensional Riemannian geometry

“…See [19] for an alternative approach; see [21,22,26] for related results. The function α(k) = k 3 − k satisfies…”

“…This work has been developed for several years, fully indepent from [19]; some identities appearing in this paper appear also in [19]; the interested reader can confront the methodologies of the two articles.…”

Reflection Positive Stochastic Processes Indexed by Lie Groups

“…A different approach also exhibiting the invariance under rigid rotations is described in [18] (cf. also [21,22]).…”

Geometric investigations of a vorticity model equation

“…Formulas for the sectional curvature of Diff(S 1 )/Rot(S 1 ) were computed by Bowick-Rajeev [8] and Zumino [50] using the formula of Freed [24] for Kähler geometry, by Kirillov-Yur'ev [30] using complex normal coordinates, and by Gordina-Lescot [26] directly using the covariant derivative. Here we will use Arnold's curvature formula [1] for right invariant metrics on Lie-groups, which allows us to see the effect of the mean term µ.…”

“…It is related to the space Diff(S 1 )/P SL 2 (R) with theḢ 3/2 -metric which arises in Teichmüller theory [38,25]. Formulas for sectional curvatures of theḢ 1/2 anḋ H 3/2 -metrics were computed using various methods in [8,30,50,26]. See the book of Sergeev [44] for a recent survey of these and related topics.…”