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Transverse Kähler–Ricci Solitons of Five-Dimensional Sasaki–Einstein Spaces Y^{p,q} and T^{1,1}

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Sasakian Manifolds

**Definition**

**1.**

- g is Einstein with $Ri{c}_{g}=2ng$;
- the metric cone $C(M,\overline{g})$ is a Ricci-flat Kähler manifold (i.e., Calabi–Yau manifold); and
- the transverse Kähler metric ${g}^{T}$ satisfies $Ri{c}_{{g}^{T}}=(2n+2){g}^{T}$.

**Definition**

**2.**

#### 2.2. Deformations of Sasaki Structures

**Lemma**

**1.**

**Definition**

**3.**

- 1.
- $d{\pi}_{\alpha}\left(X\right)$is a holomorphic vector field on${V}_{\alpha}$and
- 2.
- the complex vector field$${u}_{X}=i\eta \left(X\right)$$$${\overline{\partial}}_{B}{u}_{X}=-\frac{i}{2}\iota \left(X\right)d\eta \phantom{\rule{0.166667em}{0ex}}.$$

**Remark**

**1.**

#### 2.3. Sasaki–Ricci Flow

**Remark**

**2.**

## 3. Sasaki–Ricci Flow on Spaces ${\mathsf{T}}^{1,1}$ and ${\mathsf{Y}}^{p,q}$

#### 3.1. Sasaki–Ricci flow on Sasaki–Einstein Space ${T}^{1,1}$

**Proposition**

**1.**

**Remark**

**3.**

**Proposition**

**2.**

#### 3.2. Sasaki–Ricci flow on Sasaki–Einstein Space ${Y}^{p,q}$

**Remark**

**4.**

**Proposition**

**3.**

**Remark**

**5.**

**Proposition**

**4.**

## 4. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Visinescu, M.
Transverse Kähler–Ricci Solitons of Five-Dimensional Sasaki–Einstein Spaces Y^{p,q} and T^{1,1}. *Symmetry* **2020**, *12*, 330.
https://doi.org/10.3390/sym12030330

**AMA Style**

Visinescu M.
Transverse Kähler–Ricci Solitons of Five-Dimensional Sasaki–Einstein Spaces Y^{p,q} and T^{1,1}. *Symmetry*. 2020; 12(3):330.
https://doi.org/10.3390/sym12030330

**Chicago/Turabian Style**

Visinescu, Mihai.
2020. "Transverse Kähler–Ricci Solitons of Five-Dimensional Sasaki–Einstein Spaces Y^{p,q} and T^{1,1}" *Symmetry* 12, no. 3: 330.
https://doi.org/10.3390/sym12030330