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.
Categories: math.DG