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 $h_{\rho },\rho \ge 0,$ such that ${\pi }_1(\mathrm{I}\mathrm{s}\mathrm{o}\mathrm{m}(M,h_{\rho }))$, 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 $\rho =0.$