Hyperbolic Gradient-Bourgoignon Flow

‎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.

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.