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| In [[mathematics]], in particular in [[differential geometry]], the '''minimal volume''' is a number that describes one aspect of a [[Riemannian manifold]]'s [[topology]]. This [[topological invariant]] was introduced by [[Mikhail Gromov (mathematician)|Mikhail Gromov]].
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| ==Definition==
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| Consider a [[closed manifold|closed]] [[orientability|orientable]] [[connected space|connected]] smooth [[manifold]] <math>M^n</math> with a smooth [[Riemannian metric]] <math>g</math>, and define <math>Vol({\it M,g})</math> to be the volume of a manifold <math>M</math> with the metric <math>g</math>. Let <math>K_g</math> represent the [[sectional curvature]].
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| The minimal volume of <math>M</math> is a smooth invariant defined as | |
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| :<math>MinVol(M):=\inf_{g}\{Vol(M,g) : |K_{g}|\leq 1\}</math>
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| that is, the infimum of the volume of <math>M</math> over all metrics with bounded sectional curvature.
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| Clearly, any manifold <math>M</math> may be given an arbitrarily small volume by selecting a Riemannian metric <math>g</math> and scaling it down to <math>\lambda g</math>, as <math>Vol(M, \lambda g) =\lambda^{n/2}Vol(M, g)</math>. For a meaningful definition of minimal volume, it is thus necessary to prevent such scaling. The inclusion of bounds on sectional curvature suffices, as <math>\textstyle K_{\lambda g} = \frac{1}{\lambda} K_g</math>. If <math>MinVol(M)=0</math>, then <math>M^n</math> can be "collapsed" to a manifold of lower dimension (and thus one with <math>n</math>-dimensional volume zero) by a series of appropriate metrics; this manifold may be considered the [[Gromov-Hausdorff convergence|Hausdorff limit]] of the related sequence, and the bounds on sectional curvature ensure that this convergence takes place in a topologically meaningful fashion.
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| ==Related topological invariants==
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| The minimal volume invariant is connected to other topological invariants in a fundamental way; via [[Chern-Weil theory]], there are many topological invariants which can be described by integrating polynomials in the curvature over <math>M</math>. In particular, the [[Chern class]]es and [[Pontryagin class]]es are bounded above by the minimal volume.
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| ==Properties==
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| Gromov has conjectured that every closed [[Simply connected space|simply connected]] odd-dimensional manifold has zero minimal volume. This conjecture clearly [[Sphere|does not hold for even-dimensional manifolds]].
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| ==References==
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| {{No footnotes|date=June 2009}}
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| * Gromov, M. ''Metric Structures for Riemannian and Non-Riemannian Spaces'', Birkhäuser (1999) ISBN 0-8176-3898-9.
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| * Gromov, M. ''Volume and bounded cohomology'', Publ. Math. IHES 56 (1982) 1—99.
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| [[Category:Riemannian geometry]]
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