The Geometry of Loop Spaces IV: Closed Sasakian Manifolds

TL;DR

The authors construct a family of metrics $h_\rho = g + \rho^2 \eta\otimes\eta$ on closed regular Sasakian manifolds $M^{4k+1}$ so that $|\pi_1(\mathrm{Isom}(M,h_\rho))| = \infty$ for $\rho>0$, with a sharpened result $|\pi_1(\mathrm{Isom}(S^{4k+1},h_\rho))| = \infty$ for all $\rho>0$ and $\rho\neq 0$ in the sphere case. The key tool is the Wodzicki-Chern-Simons form $CS^W$ on the loop space $LM$; by pulling back $CS^W$ along the natural loop-map $a^L: M \to LM$ associated to the $S^1$-action, they obtain a nonvanishing top form on $M$ for large $\rho$, which forces the infinite order of $\pi_1$ of the isometry group. The analysis is carried out through explicit curvature computations for $h_\rho$, including special treatment of Sasakian space forms and, in the sphere case, the strict contact diffeomorphism group $\mathrm{Diff}_{\eta,str}(S^{4k+1})$, establishing infinite $\pi_1$ in these contexts. The results reveal rich interactions between Sasakian geometry, loop-space invariants, and symmetry groups, and extend to a broad class of Sasakian space forms with precise curvature regimes.

Abstract

We prove that a closed regular $(4k+1)$-Sasakian manifold $(M,h_0)$ admits a family of non-isometric metrics $h_ρ, ρ\geq 0,$ such that $π_1({\rm Isom}(M, h_ρ))$, the fundamental group of the isometry group, is infinite for $ρ>0.$ For $M= S^{4k+1}$, this result holds for all $ρ>0$, but fails at $ρ=0.$

Theorems & Definitions (52)

• Theorem 5.1
• Theorem 6.1
• Theorem 7.1
• Lemma 2.1
• proof
• Example 2.1
• Proposition 3.1
• Lemma 4.1
• proof
• Lemma 4.2