شار گرادیان - بورگویگنون هذلولوی

در این مقاله، شار گرادیان بورگویگنون هذلولوی را روی منیفلد فشرده m در نظر گرفته و نشان می‌دهیم که این شار یک جواب یکتای زمان - کوتاه با شرط اولیه دارد. در ادامه تحت این شار، معادلات تکاملی را برای تانسور انحنای ریمانی و تانسور انحنای ریچی ارائه خواهیم داد. در پایان، چند مثال از این شار روی منیفلدهای مختلف ارائه می‌شود.  

hyperbolic gradient-bourgoignon flow

introduction‎ricci 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 ‎nonlinear ‎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. selfsimilar solution of this flow may create interesting geometries on the underlying manifold.resultsin this paper, we consider the hyperbolic gradientbourguignon flow on a compact manifold m and show that this flow has a unique solution on shorttime 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.