The Geometry of Loop Spaces IV: Closed Sasakian Manifolds

TL;DR

This paper explores the geometric properties of closed regular Sasakian manifolds, demonstrating the existence of a family of metrics with infinite isometry group fundamental groups, revealing new insights into their geometric structure.

Contribution

It introduces a family of metrics on regular Sasakian manifolds with infinite isometry group fundamental groups, expanding understanding of their geometric and symmetry properties.

Findings

Existence of metrics with infinite isometry group fundamental groups for certain Sasakian manifolds.

The result holds for all t ho>0$ on spheres, but not at ho=0.

Provides new geometric insights into the structure of Sasakian manifolds.

Abstract

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 $\rho=0.$