Brownian motion and Ricci curvature on an infinite dimensional symplectic group related to the diffeomorphism group of the circle

Abstract

An embedding of the group \(\Diff(S^{1})\) of orientation preserving diffeomorphims of the unit circle \(S^1\) into an infinite-dimensional symplectic group, \(\Sp(\infty)\), is studied. The authors prove that this embedding is not surjective. A Brownian motion is constructed on \(\Sp(\infty)\). This study is motivated by recent work of H. Airault, S. Fang and P. Malliavin. The Ricci curvature of the infinite-dimensional symplectic group is computed. The result shows that in almost all directions, the Ricci curvature is negative infinity.