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خمینههای تخت با ابعاد پایین با ردههایی از مترهای فینسلری
Low Dimensional Flat Manifolds with Some Elasses of Finsler Metric
جبر
alg
مقاله استخراج شده از پایان نامه
Research Paper
<span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">منیفلدهای ریمانی مسطح، تا حد ایزومتری، خارج قسمت فضای اقلیدسی </span></span><span style="position:relative;top:5.5pt;"><span style="font-family:Calibri,sans-serif;"><span style="font-size:11.0pt;"> <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image001.png" > </span></span></span> <span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">بر یک گروه بیبرباخ </span></span><span style="position:relative;top:4.0pt;"><span style="font-family:Calibri,sans-serif;"><span style="font-size:11.0pt;"> <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.png" > </span></span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"> است و ردهبندی دقیقی از آنها در ابعاد 2 و 3 وجود دارد. در این مقاله، دو رده از خمینههای فینسلری مسطح دوبعدی و سهبعدی را بررسی کرده و ردهبندی میکنیم.</span></span><strong><span style="font-family:B Nazanin;"><span style="font-size:14.0pt;"></span></span></strong><br>
<span dir="LTR"><span style="font-family:Times New Roman,serif;"><span style="font-size:10.0pt;"></span></span></span>
<strong>Introduction</strong><br>
An <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="9" >-dimensional Riemannian manifold <img alt="" height="37" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="131" > is said to be <em>flat</em> (or locally Euclidean) if <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image006.gif" width="40" > locally isometric with the Euclidean space, that is, <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="13" > admits a covering of coordinates neighborhoods each of which is isometric with a Euclidean domain. A Riemannian manifold <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image006.gif" width="40" > is flat if and only if <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="13" > admits a covering of coordinates neighborhoods on each of, the function <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="44" > is independent of <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image012.gif" width="8" >. A classical result affirms that a Riemannian manifold is flat if and only if its Riemann curvature vanishes (equivalently, the sectional curvature<img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image014.gif" width="17" >; This is usually taken as the definition of a flat Riemannian manifold in the contexts. The universal Riemannian covering space of a complete and flat Riemannian manifold is the Euclidean space <img alt="" height="37" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image016.gif" width="179" >. Up to local isometry, Bieberbach proved that any compact flat Riemannian manifold, is realized as a quotient space <img alt="" height="28" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="15" >, where <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image020.gif" width="8" > is a discrete, co-compact and torsion free subgroup of the Euclidean group <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image022.gif" width="207" > , cf. [2]. The only 1 dimensional complete, flat and connected manifolds are <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image024.gif" width="11" > and <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image026.gif" width="15" >. In 2 dimensions, the only complete, flat and connected manifolds are cylinder, Mӧbius strip, Torus and Klein bottle. In 3 dimensions, there are only 10 complete, flat and connected manifolds including 6 oriented and 4 non-oriented manifolds, cf. [7].<br>
Likewise the Riemannian case, a Finslerian manifold <img alt="" height="37" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image028.gif" width="152" > is said to be <em>flat</em> (or locally Minkowskian) if, <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="13" > admits a covering of coordinates neighborhoods each of which isometric with a single Minkowski normed domain. A Finslerian manifold <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image030.gif" width="40" > is flat if and only if it admits a covering of coordinates neighborhoods on each of, the function <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image032.gif" width="44" > is independent of <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image012.gif" width="8" >. The flag curvature <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image034.gif" width="11" > of any flat Finsler manifold vanishes identically.<br>
<strong>Material and methods</strong><br>
Thanks to the works of Bieberbach and Schoenflies, we apply an group theoretic approach to classify flat Randers manifolds. The key idea is that the isometry group of Randers manifold is a subgroup of the Euclidean group. This fact, may ease our approach to find and count discrete, co-compact and torsion frees subgroups of <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image036.gif" width="30" >. First we find the Bieberbach subgroups and then, we count those that could form an isometry subgroup.<br>
<strong>Results and discussion</strong><br>
Here, flatness of a generic Finsler manifold is aimed to be defined so that it generalizes the flatness for Riemannian manifolds. The following result outcome in dimension 2 and 3:<br>
<strong>Theorem 1. </strong>The only connected and closed <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="9" >-dimensional (<img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image038.gif" width="47" >) closed flat Randers manifolds is the torus <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image040.gif" width="65" > , respectively.<br>
To classify the flat Randers manifolds, we find out that the flat Randers manifolds are flat Riemannian manifolds. Besides, the isometry group of a Randers manifold <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image042.gif" width="95" > is a subgroup of the isometry group <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image044.gif" width="72" >. Our discussion also apply the following results:
<ul>
<li>Every dimensional flat Randers manifold is itself a flat Riemannian manifolds.</li>
<li>Every dimensional flat Randers manifold is orientable.</li>
<li>The non-Riemannian properties for generic Finsler metrics may cause obstructions for a Finsler manifold to be falt.</li>
</ul>
<strong>Conclusion</strong><br>
The following conclusions were drawn from this research.
<ul>
<li>In dimensions 2 and 3, the only connected and closed flat Randers manifolds are the tori <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image040.gif" width="65" > , respectively.</li>
<li>Every dimensional flat Randers manifold is itself a flat Riemannian manifolds.<a href="./files/site1/files/62/11Abstract(1).pdf">./files/site1/files/62/11Abstract(1).pdf</a></li>
</ul>
متر ریشه چهارم, خمینه تخت دوبعدی, ایزومتری, گروه بیبرباخ, متر راندرز
4th-root metric, flat manifold, isometry, Bieberbach group, Randers metric.
261
270
http://mmr.khu.ac.ir/browse.php?a_code=A-10-249-1&slc_lang=fa&sid=1
Sedigheh
Alavi Endrajemi
صدیقه
علوی اندراجمی
salavi@umz.ac.ir
10031947532846003735
10031947532846003735
No
University of Mazandaran
دانشگاه مازندران، دانشکد، علوم ریاضی، گروه ریاضی
Mehdi
Rafie-Rad
مهدی
رفیعی راد
rafie-rad@umz.ac.ir
10031947532846003736
10031947532846003736
Yes
University of Mazandaran
دانشگاه مازندران، دانشکد، علوم ریاضی، گروه ریاضی