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

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

2011-10-11

arxiv.org/abs/1110.2449v1

Mathematics

Functional Analysis

arXiv admin note: substantial text overlap with arXiv:0802.1955

Scientific paper

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.

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