Sobolev $H^1$ Geometry of the Symplectomorphism Group

TL;DR

This paper investigates the geometric structure of the symplectomorphism group on a closed symplectic manifold using Sobolev $H^1$ metrics, revealing properties like global geodesics, Fredholm exponential maps, and conjugate points.

Contribution

It establishes the existence of global geodesics and analyzes the exponential map as a non-linear Fredholm map of index zero for the $H^1$ metric on symplectomorphism groups.

Findings

Existence of globally defined geodesics for large $s$

Exponential map is a non-linear Fredholm map of index zero

Presence of conjugate points in the $H^1$ metric geometry

Abstract

For a closed symplectic manifold $(M,\omega)$ with compatible Riemannian metric $g$ we study the Sobolev $H^1$ geometry of the group of all $H^s$ diffeomorphisms on $M$ which preserve the symplectic structure. We show that, for sufficiently large $s$, the $H^1$ metric admits globally defined geodesics and the corresponding exponential map is a non-linear Fredholm map of index zero. Finally, we show that the $H^1$ metric carries conjugate points via some simple examples.