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| In [[mathematics]], the '''Berger–Kazdan comparison theorem''' is a result in [[Riemannian geometry]] that gives a lower bound on the volume of a [[Riemannian manifold]] and also gives a necessary and sufficient condition for the manifold to be [[Isometry (Riemannian geometry)|isometric]] to the ''m''-[[dimension]]al [[sphere]] with its usual "round" metric. The theorem is named after the [[mathematician]]s [[Marcel Berger]] and [[Jerry Kazdan]].
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| ==Statement of the theorem==
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| Let (''M'', ''g'') be a [[compact space|compact]] ''m''-dimensional Riemannian manifold with [[injectivity radius]] inj(''M''). Let vol denote the volume form on ''M'' and let ''c''<sub>''m''</sub>(''r'') denote the volume of the standard ''m''-dimensional sphere of radius ''r''. Then
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| :<math>\mathrm{vol} (M) \geq \frac{c_{m} (\mathrm{inj}(M))}{\pi},</math>
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| with equality [[if and only if]] (''M'', ''g'') is isometric to the ''m''-sphere '''S'''<sup>''m''</sup> with its usual round metric.
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| ==References==
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| *{{cite book|last = Berger|first = Marcel|authorlink=Marcel Berger|coauthors = [[Jerry Kazdan|Kazdan, Jerry L.]]|chapter = A Sturm–Liouville inequality with applications to an isoperimetric inequality for volume in terms of injectivity radius, and to Wiedersehen manifolds|title = Proceedings of Second International Conference on General Inequalities, 1978|publisher = Birkhauser|year = 1980|pages = 367–377}}
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| *{{cite journal|last = Kodani|first = Shigeru|title = An Estimate on the Volume of Metric Balls|journal = Kodai Mathematical Journal|volume = 11|issue = 2|year = 1988|pages = 300–305|url = http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.kmj/1138038881|doi = 10.2996/kmj/1138038881}}
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| ==External links==
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| * {{MathWorld|urlname=Berger-KazdanComparisonTheorem|title=Berger-Kazdan comparison theorem}}
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| {{DEFAULTSORT:Berger-Kazdan comparison theorem}}
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| [[Category:Geometric inequalities]]
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| [[Category:Theorems in Riemannian geometry]]
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Latest revision as of 04:07, 6 May 2014
Wilber Berryhill is what his wife enjoys to call him and he totally enjoys this title. To perform lacross is the thing I love most of all. Her family members life in Ohio. My working day job is an invoicing officer but I've already utilized for another one.
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