شار ریچی-بورگویگنون روی منیفلدهای سایا

نویسندگان
قدرت اله فصیحی رامندی
شاهرود اعظمی
دانشگاه بین‌المللی امام خمینی (ره)، دانشکدۀ علوم پایه، گروه ریاضی محض
10.52547/mmr.6.4.645
چکیده
در این مقاله ابتدا مفاهیم مقدماتی منیفلد سایا را یاد آوری می‌کنیم بعد شار ریچی-بورگویگنون که تعمیمی از شار ریچی و شار یامابه است را روی منیفلدهای سایا معرفی می‌کنیم. سپس با استفاده از میدان برداری دیتورک معادله شار ریچی-بورگویگنون 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 ریچی-بورگویگنون 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.
کلیدواژه‌ها
شار هندسی
سالیتون
منیفلد سایا
خود-متشابه

عنوان مقاله English

Ricci-Bourgoignon Flow on Contact Manifolds

نویسندگان English

Ghodratallah Fasihi-Ramandi
Shahroud Azami
Faculty of Science, Department of pure mathematics, Imam Khomeini International University, Qazvin, Iran
چکیده English

Introduction

After 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 Ricci-Bourgoignon flow as a generalization of Ricci and Yamabe flows is introduced. Using De Turck vector field, the equation of Ricci-Bourgoignon 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 ρ< and a given initial condition the Ricci-Bourgoignon flow has a unique solution for a short time. Finally, we show that every solution of Ricci-Bourgoignon flow on a closed (compact without boundary) contact manifold is self-similar and the corresponding soliton is steady.

Material and methods

In this scheme, first we summarized some basic concepts on contact manifolds. Then, equation of Ricci-Bourgoignon flow on contact manifolds is introduced. Using De Turck vector filed and theory of PDE’s, short time existence and uniqueness solution for such equation is obtained.

Results and discussion

We obtained a condition for which Ricci-Bourgoignon flows with initial condition have a unique solution for a short time. Also, our results show that every solution of Ricci-Bourgoignon flow on a closed contact manifold is self-similar and the corresponding soliton is steady.

Conclusion

The following conclusions were drawn from this research.

Short time existence and uniqueness theorem for Ricci-Bourgoignon flow examined in this paper.
Our results showed that solutions of this equation on a closed contact manifold are self-similar and their corresponding solitons are steady.
Regardless of the dimension of underlying contact manifold, we showed that for ρ< the Ricci-Bourgoignon flow with given initial condition has a unique solution for a short time../files/site1/files/64/14.pdf


کلیدواژه‌ها English

Geometric flow- Soliton- Contact manifold- self-similar solution.n
1. Azami S., Razavi A., "Existence and uniqueness for solution of Ricci flow on Finsler manifolds", Int. J. of Geom. Meth. in Mod. Phy., Vol. 10, No. 3 (2013) 1-21.## 2. Bidabad B., Sedaghat M. K., "Ricci flow on Finsler surfaces", to appear in Journal of Geometry and Physics, 129 (2018) 238-254. ## 3. Blair D. E., "Riemannian geometry of contact and symplectic manifolds", Birkhäuser, Boston (2010). ## 4. Bourguignon J. P., "Ricci curvature and Einstein metrics, Global differential geometry and global analysis", Lecture nots in Math., 838 (1981) 42-63. ## 5. Catino G., Cremaschi L., Djadli Z., Mantegazza C., Mazzieri L., "The Ricci-Bourguignon flow, Pacific J. Math. (2015). ## 6. Chen Y. G., Giga Y., Goto S., "Uniqueness and Existence of Viscosity Solutions of Generalized Mean Curvature Flow Equations", J. Diff. Geom. 33 (1991) 749-786. ## 7. Dai W. R., Kong D. X., Liu K., "Hyperbolic gometric flow (I): short-time existen and nonlinear stability K., Pure and applide mathematics quarterly, 6 (2010) 331-359. ## 8. Eells Jr J., Sampson J. H., "Harmonic mappings of Riemannian manifolds", American Journal of Mathematics, 86, No. 1, (1964) 109-160. ## 9. Hamiton R. S., "Three-manifolds with positive Ricci Curvature", J. Differential Geomety, 17, No. 2 (1982) 255-306. ## 10. Kholodenko A. L., "Applications of contact geometry and topology in physics", World scientific (2013). ## 11. Müller R., "Ricci flow coupled with harmonic map flow", Annales scientifiques de l’École Normale Supérieure, 45 (2012) 101-142. ## 12. Pirhadi V., Razavi A., "Ricci Flow on Contact Manifolds", Siberian athematical Journal, Vol 56, No. 5, (2015) 912-921. ##