The Geometry of Loop Spaces IV: Closed Sasakian Manifolds

We prove that a closed regular $(4k+1)$-Sasakian manifold $(M,h_0)$ admits a family of non-isometric metrics $h_\rho, \rho\geq 0,$ such that $\pi_1({\rm Isom}(M, h_\rho))$, the fundamental group of the isometry group, is infinite for $\rho>0.$ For $M= S^{4k+1}$, this result holds for all $\rho>0$, but fails at $\rho=0.$