Deformation of Sasakian metrics

TL;DR

This paper investigates conditions under which small deformations of Reeb flows in Sasakian manifolds preserve the existence of compatible Sasakian metrics, linking this to the basic Euler class and establishing a vanishing theorem.

Contribution

It characterizes the existence of compatible Sasakian metrics under deformations via the triviality of the basic Euler class component and proves a vanishing theorem for basic Dolbeault cohomology.

Findings

Small deformations of Reeb flows in positive Sasakian manifolds admit compatible Sasakian metrics.

The triviality of the (0,2)-component of the basic Euler class characterizes metric compatibility.

A Kodaira-Akizuki-Nakano type vanishing theorem for basic Dolbeault cohomology is established.

Abstract

Deformations of the Reeb flow of a Sasakian manifold as transversely K\"ahler flows may not admit compatible Sasakian metrics anymore. We show that the triviality of the (0,2)-component of the basic Euler class characterizes the existence of compatible Sasakian metrics for given small deformations of the Reeb flow as transversely holomorphic Riemannian flows. We also prove a Kodaira-Akizuki-Nakano type vanishing theorem for basic Dolbeault cohomology of homologically orientable transversely K\"ahler foliations. As a consequence of these results, we show that any small deformations of the Reeb flow of a positive Sasakian manifold admit compatible Sasakian metrics.