شار ریچی-بورگویگنون روی منیفلدهای سایا

در این مقاله ابتدا مفاهیم مقدماتی منیفلد سایا را یاد آوری می‌کنیم بعد شار ریچیبورگویگنون که تعمیمی از شار ریچی و شار یامابه است را روی منیفلدهای سایا معرفی می‌کنیم. سپس با استفاده از میدان برداری دیتورک معادله شار ریچیبورگویگنون روی منیفلدهای سایا را به معادلۀ دیگری تحویل یافته می‌کنیم که خطی‌سازی این معادله دیفرانسیل با مشتقات جزئی اکیداً سهموی است و با قضایای معادلات دیفرانسیل سهموی با مشتقات جزئی نشان می‌دهیم که تحت شرایطی شار ریچیبورگویگنون روی منیفلدهای سایا با شرط آغازین دارای جواب است و این جواب یکتا است. هم‌چنین، در نهایت نشان می‌دهیم که هر جواب از شار ریچی-بورگویگنون روی منیفلدهای سایا بسته (فشرده و بدون مرز) خود متشابه است و سالیتون متناظر با آن سالیتون مانا است.

Ricci-Bourgoignon Flow on Contact Manifolds

IntroductionAfter pioneering work of Hamilton in 1982, Ricci flow and other geometric flows became as one of the most interesting topics in both mathematics and physics. In the present paper, firstly, we summarize some introductory concepts about contact manifolds. Then, the notion of RicciBourgoignon flow as a generalization of Ricci and Yamabe flows is introduced. Using De Turck vector field, the equation of RicciBourgoignon flow has been reduced to another equation which its linearization is a strictly parabolic partial differential equation. According to theory of partial differential equation, we have showed that for rho;< and a given initial condition the RicciBourgoignon flow has a unique solution for a short time. Finally, we show that every solution of RicciBourgoignon flow on a closed (compact without boundary) contact manifold is selfsimilar and the corresponding soliton is steady.Material and methodsIn this scheme, first we summarized some basic concepts on contact manifolds. Then, equation of RicciBourgoignon flow on contact manifolds is introduced. Using De Turck vector filed and theory of PDE rsquo;s, short time existence and uniqueness solution for such equation is obtained.Results and discussionWe obtained a condition for which RicciBourgoignon flows with initial condition have a unique solution for a short time. Also, our results show that every solution of RicciBourgoignon flow on a closed contact manifold is selfsimilar and the corresponding soliton is steady.ConclusionThe following conclusions were drawn from this research.Short time existence and uniqueness theorem for RicciBourgoignon flow examined in this paper.Our results showed that solutions of this equation on a closed contact manifold are selfsimilar and their corresponding solitons are steady.Regardless of the dimension of underlying contact manifold, we showed that for rho;< the RicciBourgoignon flow with given initial condition has a unique solution for a short time../files/site1/files/64/14.pdf