fa شار گرادیان-بورگویگنون هذلولوی هندسه دیفرانسیل Differential Geometry مقاله استخراج شده از پایان نامه Research Paper <span style="font-size:11pt"><span style="line-height:21.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="text-autospace:none"><span style="direction:rtl"><span style="unicode-bidi:embed"><span style="font-family:Calibri,sans-serif"><span lang="AR-SA" style="font-size:10.0pt"><span b="" nazanin="" style="font-family:">در این مقاله، شار گرادیان-بورگویگنون هذلولوی را روی منیفلد فشرده</span></span> <m:omath><i><span cambria="" dir="LTR" math="" style="font-family:"><m:r>M</m:r></span></i></m:omath>&nbsp;<span lang="AR-SA" style="font-size:10.0pt"><span b="" nazanin="" style="font-family:">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. </span></span><span dir="LTR" style="font-size:10.0pt"></span></span></span></span></span></span></span></span><br> &nbsp; <span style="line-height:23.0pt"><b><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">Introduction</span></span></b></span><br> <span style="font-size:11pt"><span style="line-height:23.0pt"><span style="font-family:Calibri,sans-serif"><span lang="FA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">Ricci solitons as a generalization of Einstein manifolds introduced by Hamilton in mid 1980s&lrm;. &lrm;In the last two decades&lrm;, &lrm;a lot of researchers have been done on Ricci solitons&lrm;. &lrm;Currently&lrm;, &lrm;Ricci solitons have became a crucial tool in studding Riemannian manifolds&lrm;, &lrm;especially for manifolds with positive urvature&lrm;. &lrm;Ricci &lrm;solitons &lrm;also &lrm;serve &lrm;as &lrm;similar&lrm; &lrm;solutions &lrm;for&lrm; &lrm;the &lrm;Ricci &lrm;flow &lrm;which &lrm;is &lrm;an &lrm;evolutionary &lrm;equation &lrm;for &lrm;the&lrm; &lrm;metric&lrm;s &lrm;of a&lrm; &lrm;Riemannian &lrm;manifold. &lrm;It &lrm;is &lrm;clear &lrm;that &lrm;the &lrm;Ricci &lrm;flow &lrm;describes &lrm;the &lrm;heat &lrm;character &lrm;of &lrm;the &lrm;metrics &lrm;and &lrm;curvatures &lrm;of &lrm;manifolds.</span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:23.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">On &lrm;the &lrm;other &lrm;hand, &lrm;hyperbolic &lrm;Ricci &lrm;flow &lrm;was &lrm;first &lrm;study &lrm;by &lrm;Kong &lrm;and &lrm;Liu. This &lrm;flow &lrm;is a&lrm; &lrm;system &lrm;of &lrm;non-linear &lrm;evolution &lrm;partial &lrm;differential &lrm;equation&lrm;s of second order.</span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:23.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span lang="FA" style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&lrm;</span></span><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">The &lrm;short &lrm;time &lrm;existence &lrm;and &lrm;uniqueness&lrm; &lrm;theorem &lrm;of &lrm;hyperbolic &lrm;geometric &lrm;flow &lrm;has &lrm;been &lrm;proved &lrm;in. &lrm;It &lrm;is &lrm;s&lrm;hown &lrm;that &lrm;the &lrm;hyperbolic &lrm;Ricci &lrm;flow &lrm;carries &lrm;many &lrm;interesting&lrm; &lrm;properties &lrm;of &lrm;both &lrm;Ricci &lrm;flow &lrm;as &lrm;well &lrm;as &lrm;the &lrm;Einstein &lrm;equation. &lrm;&lrm;</span></span><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times=""><span style="color:black"></span></span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:23.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times=""><span style="color:black">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.</span></span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:23.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times=""><span style="color:black">Results</span></span></span></b></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:23.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times=""><span style="color:black">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.</span></span></span></span></span></span></span><br> &nbsp; شار ریچی, معادلات تکاملی, منیفلد فشرده Ricci Flow, Evolution Equation, Compact Manifold 165 183 http://mmr.khu.ac.ir/browse.php?a_code=A-13-454-2&slc_lang=fa&sid=1 Hamed Faraji حامد فرجی h.faraji@edu.ikiu.ac.ir 10031947532846005475 10031947532846005475 No Imam Khomeini international university دانشگاه بین المللی امام خمینی Shahroud Azami شاهرود اعظمی azami@sci.ikiu.ac.ir 10031947532846005476 10031947532846005476 No Imam Khomeini international university دانشگاه بین المللی امام خمینی Ghodratallah Fasihi-Ramandi قدرت اله فصیحی رامندی fasihi@sci.ikiu.ac.ir 10031947532846005477 10031947532846005477 Yes Imam Khomeini international university دانشگاه بین المللی امام خمینی