Transverse K\"{a}hler-Ricci flow and deformations of the metric on the Sasaki space $T^{1,1}$

TL;DR

This paper explores deformations of the standard metric on the Sasaki space T^{1,1} using the Sasaki-Ricci flow, aiming to produce new Sasaki-Einstein metrics while preserving certain geometric structures.

Contribution

It introduces a method to deform the metric on T^{1,1} that preserves some structures but alters the orthogonal complement of the Reeb vector, linked to solutions of the Sasaki-Ricci flow.

Findings

Deformation preserves transverse and leafwise metrics.

The method yields solutions to the Sasaki-Ricci flow equation.

Potential to generate new Sasaki-Einstein metrics.

Abstract

In this paper we investigate the possibility to obtain locally new Sasaki-Einstein metrics on the space $T^{1,1}$ considering a deformation of the standard metric tensor field. We show that from the geometric point of view this deformation leaves transverse and the leafwise metric intact, but changes the orthogonal complement of the Reeb vector field using a particular basic function. In particular, the family of metric obtained using this method can be regarded as solutions of the equation associated to the Sasaki-Ricci flow on the underlying manifold.