Torsionless module - Revision history

2001:62A:4:2F00:6EF0:49FF:FE25:F3FC: /* Properties and examples */ Corrected the link

2014-12-18T16:02:22Z

Properties and examples: Corrected the link

← Older revision		Revision as of 18:02, 18 December 2014
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	In [[mathematics]], '''Reeb stability theorem''', named after [[Georges Reeb]], asserts that if one leaf of a [[codimension]]-one [[foliation]] is [[Closed manifold\|closed]] and has finite [[fundamental group]], then all the leaves are closed and have finite fundamental group.		I am Junior from Alt Kali? studying Economics. I did my schooling, secured 87% and hope to find someone with same interests in Target Shooting.<br><br>my web site: [http://bikeracinggames.co.in/profile/besaenz.html Cannondale mountain bike sizing.]

	~~== Reeb local stability theorem ==~~

	~~Theorem:<ref name="Reeb">G. Reeb~~, {{cite book \| author=G. Reeb \| title=Sur certaines propriétés toplogiques des variétés feuillétées \| series=Actualités Sci. Indust. \| volume=1183 \| publisher=Hermann \| location=Paris \| year=1952 }}</ref> ''Let <math>F</math> be a <math>C^1</math>, codimension <math>k</math> [[foliation]] of a [[manifold]] <math>M</math> and <math>L</math> a [[compact space\|compact]] leaf with finite [[holonomy\|holonomy group]]. There exists a [[Neighbourhood (mathematics)\|neighborhood]] <math>U</math> of <math>L</math>, saturated in <math>F</math> (also called invariant), in which all the leaves are compact with finite holonomy groups. Further, we can define a [[Deformation retract#\|retraction]] <math>\pi: U\to ~~L</math> such that, for every leaf <math>L'\subset U</math>, <math>\pi\|_{L'}:L'\to L</math> is a [[covering]]~~ with ~~a finite number of sheets and, for each <math>y\~~in ~~L</math>, <math>\pi^{-1}(y)</math> is [[homeomorphism\|homeomorphic]] to a [[disk (mathematics)\|disk]] of [[dimension]] k and is [[Transversality (mathematics)\|transverse]] to <math>F</math>~~. ~~The neighborhood~~ <~~math~~>~~U</math> can be taken to be arbitrarily small.''~~

	~~The last statement means in particular that, in a neighborhood of the point corresponding to a compact leaf~~
	~~with finite holonomy, the space of leaves is [[Hausdorff space\|Hausdorff]].~~
	~~Under certain conditions the Reeb Local Stability Theorem may replace the [[Poincaré–Bendixson theorem]] in higher dimensions.~~<~~ref~~>~~J. Palis, jr., W. de Melo, ''Geometric theory of dinamical systems~~: ~~an introduction'', — New-York,~~
	~~Springer,1982.</ref> This is the case of codimension one, singular foliations <math>(M^n,F)</math>, with <math>n\ge 3</math>, and some center-type singularity in <math>Sing(F)</math>.~~

	The Reeb Local Stability Theorem also has a version for a noncompact leaf.<ref>T.Inaba, ''<math>C^2</math> Reeb stability of noncompact leaves of foliations,''— Proc. Japan Acad. Ser. A Math. Sci., 59:158{160, 1983 [http://~~projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid~~.~~pja/1195515640]</ref><ref>J~~. ~~Cantwell and L. Conlon, ''Reeb stability for noncompact leaves~~ in ~~foliated 3-manifolds,'' — Proc. Amer.Math.Soc. 33 (1981), no. 2, 408–410.[http://www.ams.org~~/~~journals~~/~~proc/1981-083-02/S0002-9939-1981-0624942-5/S0002-9939-1981-0624942-5.pdf]</ref>~~

	~~== Reeb global stability theorem ==~~

	~~An important problem in foliation theory is the study of the influence exerted by a compact leaf upon the global structure of a [[foliation]]~~. ~~For certain classes of foliations, this influence is considerable~~.

	Theorem:<ref name="Reeb"/> ''Let <math>F</math> be a <math>C^1</math>, codimension one foliation of a closed manifold <math>M</math>. If <math>F</math> contains a [[Compact space\|compact]] leaf <math>L</math> with finite [[fundamental group]], then all the leaves of <math>F</math> are compact, with finite fundamental group. If <math>F</math> is transversely [[orientability\|orientable]], then every leaf of <math>F</math> is [[diffeomorphism\|diffeomorphic]] to <math>L</math>; <math>M</math> is the [[fiber bundle\|total space]] of a [[fibration]] <math>f:M\to S^1</math> over <math>S^1</math>, with [[Fiber (mathematics)\|fibre]] <math>L</math>, and <math>F</math> is the fibre foliation, <math>\{f^{-1}(\theta)\|\theta\in S^1\}</math>.''

	~~This theorem holds true even when <math>F</math> is a foliation of a [[Manifold#Manifold with boundary\|manifold with boundary]], which is, a priori, [[Tangent space\|tangent]]~~
	on certain components of the [[Boundary (topology)\|boundary]] and [[Transversality (mathematics)\|transverse]] on other components.<ref>C. Godbillon, ''Feuilletages, etudies geometriques,'' — Basel, Birkhauser, 1991</ref> In this case it implies [[Reeb sphere theorem]].

	Reeb Global Stability Theorem is false for foliations of codimension greater than one.<ref>W.T.Wu and G.Reeb, ''Sur les éspaces fibres et les variétés feuillitées'', — Hermann, 1952.</ref> However, for some special kinds of foliations one has the following global stability results:

	* In the presence of a certain transverse geometric structure:

	Theorem:<ref>R.A. Blumenthal, ''Stability theorems for conformal foliations'', — Proc. AMS. 91, 1984, p. 55–63. [http://www.ams.org/journals/proc/1984-091-03/S0002-9939-1984-0744654-X/S0002-9939-1984-0744654-X.pdf]</ref> ''Let <math>F</math> be a [[Complete metric space\|complete]] [[Conformal geometry\|conformal]] foliation of codimension <math>k\ge 3</math> of a [[Connectedness\|connected]] manifold <math>M</math>. If <math>F</math> has a compact leaf with finite [[holonomy\|holonomy group]], then all the leaves of <math>F</math> are compact with finite holonomy group.''

	* For [[Complex manifold\|holomorphic]] foliations in complex [[Kähler manifold]]:

	Theorem:<ref>J.V. Pereira, ''Global stability for holomorphic foliations on Kaehler manifolds'', — Qual. Theory Dyn. Syst. 2 (2001), 381–384. {{arxiv\|math/0002086v2}}</ref> ''Let <math>F</math> be a holomorphic foliation of codimension <math>k</math> in a compact complex [[Kähler manifold]]. If <math>F</math> has a [[Compact space\|compact]] leaf with finite [[holonomy\|holonomy group]] then every leaf of <math>F</math> is compact with finite holonomy group.''

	~~== References ==~~

	* C. Camacho, A. Lins Neto: Geometric theory of foliations, Boston, Birkhauser, 1985
	* I. Tamura, Topology of foliations: an introduction, Transl. of Math. Monographs, AMS, v.97, 2006, 193 p.

	~~== Notes ==~~
	~~{{reflist}}~~

	~~[[Category:Foliations]~~]

en>Yobot: /* Relation with semihereditary rings */Reference before punctuation detected and fixed using AWB (9585)

2013-11-08T08:35:45Z

Relation with semihereditary rings: Reference before punctuation detected and fixed using AWB (9585)

← Older revision		Revision as of 10:35, 8 November 2013
Line 1:		Line 1:
			In [[mathematics]], '''Reeb stability theorem''', named after [[Georges Reeb]], asserts that if one leaf of a [[codimension]]-one [[foliation]] is [[Closed manifold\|closed]] and has finite [[fundamental group]], then all the leaves are closed and have finite fundamental group.

			== Reeb local stability theorem ==

	~~If youve already purchased a Trek bike or any other brand of bike and you prefer to bring your own bike on your tour~~, ~~bring it along~~. ~~Normal pedaling motion disengages the brake cane, allowing for normal pedaling~~. ~~This temporarily blocks penile blood flow causing impotence~~. ~~Alison Addy is the author~~ of ~~many articles on subjects like mountain bike crashes~~ and ~~published at~~. ~~The Shimano shifters~~ which we'~~ve talked about above increase the cyclist~~'~~s capability~~ of ~~moving~~ in ~~and out~~ of ~~gear quickly~~ and ~~without any kind of delay~~. <br><br>~~Once you are sure~~ that ~~this is what you want~~ to do, it is ~~now time for you to decide on~~ the ~~bike~~. ~~General Use~~ (~~Commuting~~, ~~Mountain Bike~~, ~~Road Bike, BMX~~, and ~~Universal Fit~~) ~~General use helmets are usually about~~. The ~~owner~~'~~s son is a mountain biker~~, and ~~very friendly~~, ~~but little gestures of courtesy go a long way~~ in ~~ensuring goodwill between the mountain biking and non~~-~~mountain biking communities~~. ~~You can travel on the equivalent of 10 cents a gallon of gas~~. ~~When you have any queries concerning where in addition to tips on how to use~~ [http://www.~~highdefwallpaper~~.~~net~~/~~profile~~/~~taziesemer Transfering to mountain bike sizing~~.]~~, you can contact us on our website. Now the most successful business gurus are looking to the interns for lessons.~~ <br>~~<br>If you are looking for~~ the ~~best safety mountain bike accessories then~~ the ~~first thing you need to look into is a helmet. It is just much easier to be a participant in~~ a ~~team and enter~~ the ~~race as~~ a ~~whole unit~~. ~~Guided by the dolphin trainer, they will perform a series~~ of ~~tricks such as clapping~~, ~~synchronized jumping, etc~~. ~~If you can~~'~~t ride from home~~, ~~park on either side~~ of ~~Galvin Parkway Road at the first intersection north of Van Buren Street (~~a. If ~~you happen to crash~~, then ~~you will see that your hands~~ are ~~the first thing that will probably come into contact~~ with ~~the ground~~. <br><br>~~When discount sales are on~~, ~~you need~~ to ~~be sure you understand~~ the ~~terms and conditions~~ of ~~your purchase~~, ~~including warranty information. Then~~, ~~the shorter~~ and ~~wider teeth as well as~~ the ~~ramps and pegs~~ in a ~~bike are used in sprocket technology to ease the transition~~ of ~~the gear. Serious head injuries can be prevented~~ with a ~~helmet and today’s styles and designs make them more comfortable to wear. I should have had that Bicycle repair tool with me~~, ~~becuase when I got home and lowered~~ the ~~seat. I've never been into cycling~~ and ~~wanted to give it a try for exercise~~. <br><br>~~No matter what type of bike, no matter how old~~ it is, ~~no matter how much money you~~'~~d like to spend. The correct quill stems are sized down with the inside diameter of the fork~~'~~s steer tube~~. ~~Usually~~, ~~alloy stems tend to be the more cost-effective~~ of the ~~two, however~~ the ~~high end~~ of ~~the level~~, ~~aluminum stems which have been produced from high~~-~~grade alloys could~~ be ~~more expensive than carbon counterparts because they are lighter plus more responsive~~. ~~It is also easier to hold~~ the ~~handlebars if they~~ are ~~covered~~ with ~~rubber than exposed handlebars~~, ~~especially if it is too hot because steel absorbs heat and makes it difficult~~ for ~~you to hold~~. If ~~they connect~~ with ~~the wrong part~~, ~~it could cause you to stop on a dime~~, ~~which will most likely result in you taking a tumble over the handlebars~~.		Theorem:<ref name="Reeb">G. Reeb, {{cite book \| author=G. Reeb \| title=Sur certaines propriétés toplogiques des variétés feuillétées \| series=Actualités Sci. Indust. \| volume=1183 \| publisher=Hermann \| location=Paris \| year=1952 }}</ref> ''Let <math>F</math> be a <math>C^1</math>, codimension <math>k</math> [[foliation]] of a [[manifold]] <math>M</math> and <math>L</math> a [[compact space\|compact]] leaf with finite [[holonomy\|holonomy group]]. There exists a [[Neighbourhood (mathematics)\|neighborhood]] <math>U</math> of <math>L</math>, saturated in <math>F</math> (also called invariant), in which all the leaves are compact with finite holonomy groups. Further, we can define a [[Deformation retract#\|retraction]] <math>\pi: U\to L</math> such that, for every leaf <math>L'\subset U</math>, <math>\pi\|_{L'}:L'\to L</math> is a [[covering]] with a finite number of sheets and, for each <math>y\in L</math>, <math>\pi^{-1}(y)</math> is [[homeomorphism\|homeomorphic]] to a [[disk (mathematics)\|disk]] of [[dimension]] k and is [[Transversality (mathematics)\|transverse]] to <math>F</math>. The neighborhood <math>U</math> can be taken to be arbitrarily small.''

			The last statement means in particular that, in a neighborhood of the point corresponding to a compact leaf
			with finite holonomy, the space of leaves is [[Hausdorff space\|Hausdorff]].
			Under certain conditions the Reeb Local Stability Theorem may replace the [[Poincaré–Bendixson theorem]] in higher dimensions.<ref>J. Palis, jr., W. de Melo, ''Geometric theory of dinamical systems: an introduction'', — New-York,
			Springer,1982.</ref> This is the case of codimension one, singular foliations <math>(M^n,F)</math>, with <math>n\ge 3</math>, and some center-type singularity in <math>Sing(F)</math>.

			The Reeb Local Stability Theorem also has a version for a noncompact leaf.<ref>T.Inaba, ''<math>C^2</math> Reeb stability of noncompact leaves of foliations,''— Proc. Japan Acad. Ser. A Math. Sci., 59:158{160, 1983 [http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.pja/1195515640]</ref><ref>J. Cantwell and L. Conlon, ''Reeb stability for noncompact leaves in foliated 3-manifolds,'' — Proc. Amer.Math.Soc. 33 (1981), no. 2, 408–410.[http://www.ams.org/journals/proc/1981-083-02/S0002-9939-1981-0624942-5/S0002-9939-1981-0624942-5.pdf]</ref>

			== Reeb global stability theorem ==

			An important problem in foliation theory is the study of the influence exerted by a compact leaf upon the global structure of a [[foliation]]. For certain classes of foliations, this influence is considerable.

			Theorem:<ref name="Reeb"/> ''Let <math>F</math> be a <math>C^1</math>, codimension one foliation of a closed manifold <math>M</math>. If <math>F</math> contains a [[Compact space\|compact]] leaf <math>L</math> with finite [[fundamental group]], then all the leaves of <math>F</math> are compact, with finite fundamental group. If <math>F</math> is transversely [[orientability\|orientable]], then every leaf of <math>F</math> is [[diffeomorphism\|diffeomorphic]] to <math>L</math>; <math>M</math> is the [[fiber bundle\|total space]] of a [[fibration]] <math>f:M\to S^1</math> over <math>S^1</math>, with [[Fiber (mathematics)\|fibre]] <math>L</math>, and <math>F</math> is the fibre foliation, <math>\{f^{-1}(\theta)\|\theta\in S^1\}</math>.''

			This theorem holds true even when <math>F</math> is a foliation of a [[Manifold#Manifold with boundary\|manifold with boundary]], which is, a priori, [[Tangent space\|tangent]]
			on certain components of the [[Boundary (topology)\|boundary]] and [[Transversality (mathematics)\|transverse]] on other components.<ref>C. Godbillon, ''Feuilletages, etudies geometriques,'' — Basel, Birkhauser, 1991</ref> In this case it implies [[Reeb sphere theorem]].

			Reeb Global Stability Theorem is false for foliations of codimension greater than one.<ref>W.T.Wu and G.Reeb, ''Sur les éspaces fibres et les variétés feuillitées'', — Hermann, 1952.</ref> However, for some special kinds of foliations one has the following global stability results:

			* In the presence of a certain transverse geometric structure:

			Theorem:<ref>R.A. Blumenthal, ''Stability theorems for conformal foliations'', — Proc. AMS. 91, 1984, p. 55–63. [http://www.ams.org/journals/proc/1984-091-03/S0002-9939-1984-0744654-X/S0002-9939-1984-0744654-X.pdf]</ref> ''Let <math>F</math> be a [[Complete metric space\|complete]] [[Conformal geometry\|conformal]] foliation of codimension <math>k\ge 3</math> of a [[Connectedness\|connected]] manifold <math>M</math>. If <math>F</math> has a compact leaf with finite [[holonomy\|holonomy group]], then all the leaves of <math>F</math> are compact with finite holonomy group.''

			* For [[Complex manifold\|holomorphic]] foliations in complex [[Kähler manifold]]:

			Theorem:<ref>J.V. Pereira, ''Global stability for holomorphic foliations on Kaehler manifolds'', — Qual. Theory Dyn. Syst. 2 (2001), 381–384. {{arxiv\|math/0002086v2}}</ref> ''Let <math>F</math> be a holomorphic foliation of codimension <math>k</math> in a compact complex [[Kähler manifold]]. If <math>F</math> has a [[Compact space\|compact]] leaf with finite [[holonomy\|holonomy group]] then every leaf of <math>F</math> is compact with finite holonomy group.''

			== References ==

			* C. Camacho, A. Lins Neto: Geometric theory of foliations, Boston, Birkhauser, 1985
			* I. Tamura, Topology of foliations: an introduction, Transl. of Math. Monographs, AMS, v.97, 2006, 193 p.

			== Notes ==
			{{reflist}}

			[[Category:Foliations]]

108.75.80.56: Torsionless modules are not always flat

2012-02-29T03:01:59Z

Torsionless modules are not always flat

New page

If youve already purchased a Trek bike or any other brand of bike and you prefer to bring your own bike on your tour, bring it along. Normal pedaling motion disengages the brake cane, allowing for normal pedaling. This temporarily blocks penile blood flow causing impotence. Alison Addy is the author of many articles on subjects like mountain bike crashes and published at. The Shimano shifters which we've talked about above increase the cyclist's capability of moving in and out of gear quickly and without any kind of delay. Once you are sure that this is what you want to do, it is now time for you to decide on the bike. General Use (Commuting, Mountain Bike, Road Bike, BMX, and Universal Fit) General use helmets are usually about. The owner's son is a mountain biker, and very friendly, but little gestures of courtesy go a long way in ensuring goodwill between the mountain biking and non-mountain biking communities. You can travel on the equivalent of 10 cents a gallon of gas. When you have any queries concerning where in addition to tips on how to use [http://www.highdefwallpaper.net/profile/taziesemer Transfering to mountain bike sizing.], you can contact us on our website. Now the most successful business gurus are looking to the interns for lessons. If you are looking for the best safety mountain bike accessories then the first thing you need to look into is a helmet. It is just much easier to be a participant in a team and enter the race as a whole unit. Guided by the dolphin trainer, they will perform a series of tricks such as clapping, synchronized jumping, etc. If you can't ride from home, park on either side of Galvin Parkway Road at the first intersection north of Van Buren Street (a. If you happen to crash, then you will see that your hands are the first thing that will probably come into contact with the ground. When discount sales are on, you need to be sure you understand the terms and conditions of your purchase, including warranty information. Then, the shorter and wider teeth as well as the ramps and pegs in a bike are used in sprocket technology to ease the transition of the gear. Serious head injuries can be prevented with a helmet and today’s styles and designs make them more comfortable to wear. I should have had that Bicycle repair tool with me, becuase when I got home and lowered the seat. I've never been into cycling and wanted to give it a try for exercise. No matter what type of bike, no matter how old it is, no matter how much money you'd like to spend. The correct quill stems are sized down with the inside diameter of the fork's steer tube. Usually, alloy stems tend to be the more cost-effective of the two, however the high end of the level, aluminum stems which have been produced from high-grade alloys could be more expensive than carbon counterparts because they are lighter plus more responsive. It is also easier to hold the handlebars if they are covered with rubber than exposed handlebars, especially if it is too hot because steel absorbs heat and makes it difficult for you to hold. If they connect with the wrong part, it could cause you to stop on a dime, which will most likely result in you taking a tumble over the handlebars.