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

Satoshi Egi; Yoshiaki Maeda; Steven Rosenberg

doi:10.3842/SIGMA.2026.054

paper

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

arXiv:2011.01800 · doi:10.3842/SIGMA.2026.054

Abstract

Let $M_{p}$ be a circle bundle with first Chern class $p [ω]$ over a closed $4 n$ -dimensional integral symplectic manifold $(M, ω)$ . Equivalently, $M_{p}$ is a closed contact $(4 n + 1)$ -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 $L M_{p}$ to prove that $π_{1} (Isom (M_{p}, g))$ is infinite for $∣ p ∣ ≫ 0$ . We also give the first high-dimensional examples of nonvanishing Wodzicki-Pontryagin forms.