The Geometry of Loop Spaces III: Isometry Groups of Contact Manifolds

TL;DR

The paper studies the isometry groups of certain high-dimensional contact manifolds, showing their fundamental groups are infinite for large parameters, and introduces new examples of nonvanishing Wodzicki-Pontryagin forms.

Contribution

It demonstrates that the fundamental group of the isometry group is infinite for large circle bundle parameters and provides the first high-dimensional examples of nonvanishing Wodzicki-Pontryagin forms.

Findings

undamental group of isometry group is infinite for large p

First high-dimensional examples of nonvanishing Wodzicki-Pontryagin forms

Uses Wodzicki-Chern-Simons forms on loop space to derive results

Abstract

Let $M_p$ be a circle bundle with first Chern class $p[\omega]$ over a closed $4n$-dimensional integral symplectic manifold $(\overline{M}, \omega)$. Equivalently, $M_p$ is a closed contact $(4n+1)$-manifold whose Reeb orbits are all closed and have the same period. For a metric $g$ on $M_p$ compatible with the symplectic structure and the geometry of the circle fiber, we use Wodzicki-Chern-Simons forms on the loop space $LM_p$ to prove that $\pi_1({\rm Isom}(M_p,g))$ is infinite for $|p| \gg 0.$ We also give the first high dimensional examples of nonvanishing Wodzicki-Pontryagin forms.