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

Abstract: We study the isometry groups of manifolds $\overline {M}_p$, $p\in\mathbb{Z}$, which are closed contact $(4n+1)$-manifolds with closed Reeb orbits. Equivalently, $\overline{M}_p$ is a circle bundle over a closed $4n$-dimensional integral symplectic manifold. We use Wodzicki-Chern-Simons forms on the loop space $L\overline{M}_p$ to prove that $\pi_1({\rm Isom}(\overline{M}_p))$ is infinite for $|p| \gg 0.$ We also give the first high dimensional examples of nonvanishing Wodzicki-Pontryagin forms.