Volume 8, Issue 2 (Vol. 8,No. 2, 2022)
mmr 2022, 8(2): 165-183 |
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Faraji H, Azami S, Fasihi-Ramandi G. Hyperbolic Gradient-Bourgoignon Flow. mmr 2022; 8 (2) :165-183
URL: http://mmr.khu.ac.ir/article-1-3144-en.html
URL: http://mmr.khu.ac.ir/article-1-3144-en.html
1- Imam Khomeini international university
2- Imam Khomeini international university ,fasihi@sci.ikiu.ac.ir
2- Imam Khomeini international university ,
Abstract: (968 Views)
Introduction
Ricci solitons as a generalization of Einstein manifolds introduced by Hamilton in mid 1980s. In the last two decades, a lot of researchers have been done on Ricci solitons. Currently, Ricci solitons have became a crucial tool in studding Riemannian manifolds, especially for manifolds with positive urvature. Ricci solitons also serve as similar solutions for the Ricci flow which is an evolutionary equation for the metrics of a Riemannian manifold. It is clear that the Ricci flow describes the heat character of the metrics and curvatures of manifolds.
On the other hand, hyperbolic Ricci flow was first study by Kong and Liu. This flow is a system of non-linear evolution partial differential equations of second order.
The short time existence and uniqueness theorem of hyperbolic geometric flow has been proved in. It is shown that the hyperbolic Ricci flow carries many interesting properties of both Ricci flow as well as the Einstein equation.
According to these notions and their applications in both geometry and physics, in this paper we introduce a new hyperbolic flow and study its geometric quantities along to this flow. Self-similar solution of this flow may create interesting geometries on the underlying manifold.
Results
In this paper, we consider the hyperbolic Gradient-Bourguignon flow on a compact manifold M and show that this flow has a unique solution on short-time with imposing on initial conditions. After then, we find evolution equations for Riemannian curvature tensor, Ricci curvature tensor and scalar curvature of M under this flow. In the final section, we give some examples of this flow on some compact manifolds.
Ricci solitons as a generalization of Einstein manifolds introduced by Hamilton in mid 1980s. In the last two decades, a lot of researchers have been done on Ricci solitons. Currently, Ricci solitons have became a crucial tool in studding Riemannian manifolds, especially for manifolds with positive urvature. Ricci solitons also serve as similar solutions for the Ricci flow which is an evolutionary equation for the metrics of a Riemannian manifold. It is clear that the Ricci flow describes the heat character of the metrics and curvatures of manifolds.
On the other hand, hyperbolic Ricci flow was first study by Kong and Liu. This flow is a system of non-linear evolution partial differential equations of second order.
The short time existence and uniqueness theorem of hyperbolic geometric flow has been proved in. It is shown that the hyperbolic Ricci flow carries many interesting properties of both Ricci flow as well as the Einstein equation.
According to these notions and their applications in both geometry and physics, in this paper we introduce a new hyperbolic flow and study its geometric quantities along to this flow. Self-similar solution of this flow may create interesting geometries on the underlying manifold.
Results
In this paper, we consider the hyperbolic Gradient-Bourguignon flow on a compact manifold M and show that this flow has a unique solution on short-time with imposing on initial conditions. After then, we find evolution equations for Riemannian curvature tensor, Ricci curvature tensor and scalar curvature of M under this flow. In the final section, we give some examples of this flow on some compact manifolds.
Type of Study: Research Paper |
Subject:
Differential Geometry
Received: 2020/10/21 | Revised: 2022/11/16 | Accepted: 2021/01/4 | Published: 2022/05/21 | ePublished: 2022/05/21
Received: 2020/10/21 | Revised: 2022/11/16 | Accepted: 2021/01/4 | Published: 2022/05/21 | ePublished: 2022/05/21
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