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| {{context|date=January 2013}}
| | Hello. Allow me introduce the writer. Her title is Emilia Shroyer but it's not the most feminine name out there. Her [http://homestdtests.org/ husband] and her reside in Puerto Rico but she will have to move one day or another. For years I've been operating as a payroll clerk. The favorite pastime for my children and me is to [http://health.state.Tn.us/hotlines/hotlines.htm play baseball] home std test kit but I haven't made a dime with it.<br><br>Also visit my web site - at home at [http://www.dev.adaptationlearning.net/solid-advice-regards-yeast-infection home std test] [http://xrambo.com/blog/192034 std testing at home] testing, [https://healthcoachmarketing.zendesk.com/entries/53672184-Candida-Tips-And-Cures-That-May-Meet-Your-Needs More inspiring ideas], |
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| In [[mathematics]], a '''complete manifold''' (or '''geodesically complete manifold''') is a ([[Pseudo-Riemannian manifold|pseudo]]-) [[Riemannian manifold]] for which every maximal (inextendible) [[geodesic]] is defined on <math>\mathbb{R}</math>.
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| ==Examples==
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| All [[compact space|compact]] manifolds and all [[homogeneous space|homogeneous]] manifolds are geodesically complete.
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| [[Euclidean space]] <math>\mathbb{R}^{n}</math>, the [[sphere]]s <math>\mathbb{S}^{n}</math> and the [[torus|tori]] <math>\mathbb{T}^{n}</math> (with their natural [[Riemannian metric]]s) are all complete manifolds.
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| A simple example of a non-complete manifold is given by the punctured plane <math>M := \mathbb{R}^{2} \setminus \{ 0 \}</math> (with its induced metric). Geodesics going to the origin cannot be defined on the entire real line.
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| ==Path-connectedness, completeness and geodesic completeness==
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| It can be shown that a finite dimensional [[Connected space#Path connectedness|path-connected]] Riemannian manifold is a [[complete metric space]] (with respect to the [[Riemannian_manifold#Riemannian_manifolds_as_metric_spaces_2|Riemannian distance]]) if and only if it is geodesically complete. This is the [[Hopf–Rinow theorem]]. This theorem does not hold for infinite dimensional manifolds. The example of a non-complete manifold (the punctured plane) given above fails to be geodesically complete because, although it is path-connected, it is not a complete metric space: any sequence in the plane converging to the origin is a non-converging Cauchy sequence in the punctured plane.
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| ==References==
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| * {{Citation | last1=O'Neill | first1=Barrett | title=Semi-Riemannian Geometry | publisher=[[Academic Press]] | isbn=0-12-526740-1 | year=1983}}. ''See chapter 3, pp. 68''.
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| {{DEFAULTSORT:Complete Manifold}}
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| [[Category:Riemannian geometry]]
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| [[Category:Manifolds]]
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Latest revision as of 12:41, 10 April 2014
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