Volume 8, Issue 2 (Vol. 8,No. 2, 2022)                   mmr 2022, 8(2): 165-183 | Back to browse issues page

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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
1- Imam Khomeini international university
2- Imam Khomeini international university , fasihi@sci.ikiu.ac.ir
Abstract:   (716 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.
 
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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

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