Mathematical Researches
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شار گرادیان-بورگویگنون هذلولوی
Hyperbolic Gradient-Bourgoignon Flow
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> <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>
<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:">‎</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. ‎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.</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 ‎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.</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="">‎</span></span><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">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. ‎‎</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>
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