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In mathematics, in particular in differential geometry, the minimal volume is a number that describes one aspect of a smooth manifold's topology. This topological invariant was introduced by Mikhael Gromov. Given a smooth Riemannian manifold (M, g), one may consider its volume vol(M, g) and sectional curvature Kg. The minimal volume of a smooth manifold M is defined to be Gromov, in 1982, showed that the volume of a complete Riemannian metric on a smooth manifold can always be estimated by the size of its curvature and by the simplicial volume of the manifold, via the inequality

Attributes	Values
rdfs:label	Minimal volume (en)
rdfs:comment	In mathematics, in particular in differential geometry, the minimal volume is a number that describes one aspect of a smooth manifold's topology. This topological invariant was introduced by Mikhael Gromov. Given a smooth Riemannian manifold (M, g), one may consider its volume vol(M, g) and sectional curvature Kg. The minimal volume of a smooth manifold M is defined to be Gromov, in 1982, showed that the volume of a complete Riemannian metric on a smooth manifold can always be estimated by the size of its curvature and by the simplicial volume of the manifold, via the inequality (en)
dcterms:subject	Riemannian geometry
Wikipage page ID	11009703 (xsd:integer)
Wikipage revision ID	982774580 (xsd:integer)
Link from a Wikipage to another Wikipage	Metric Structures for Riemannian and Non-Riemannian Spaces Topological invariant Mathematics Gauss-Bonnet theorem Berger's sphere Closed manifold Topology Simply connected space Riemannian manifold Riemannian geometry Chern-Gauss-Bonnet formula Collapsing manifold Differential geometry Sphere Smooth manifold Uniformization theorem Flat manifold Simplicial volume Sectional curvature Mikhail Gromov (mathematician)
Link from a Wikipage to an external page	http://www.numdam.org/item/PMIHES_1982__56__5_0/
sameAs	Minimal volume Minimal volume Minimal volume
dbp:wikiPageUsesTemplate	dbt:Free_access dbt:! dbt:= dbt:Closed_access dbt:Doi dbt:ISBN dbt:Math dbt:Mvar dbt:No_footnotes
has abstract	In mathematics, in particular in differential geometry, the minimal volume is a number that describes one aspect of a smooth manifold's topology. This topological invariant was introduced by Mikhael Gromov. Given a smooth Riemannian manifold (M, g), one may consider its volume vol(M, g) and sectional curvature Kg. The minimal volume of a smooth manifold M is defined to be Any closed manifold can be given an arbitrarily small volume by scaling any choice of a Riemannian metric. The minimal volume removes the possibility of such scaling by the constraint on sectional curvatures. So, if the minimal volume of M is zero, then a certain kind of nontrivial collapsing phenomena can be exhibited by Riemannian metrics on M. A trivial example, the only in which the possibility of scaling is present, is a closed flat manifold. The Berger spheres show that the minimal volume of the three-dimensional sphere is also zero. Gromov has conjectured that every closed simply connected odd-dimensional manifold has zero minimal volume. By contrast, a positive lower bound for the minimal volume of M amounts to some (usually nontrivial) geometric inequality for the volume of an arbitrary complete Riemannian metric on M in terms of the size of its curvature. According to the Gauss-Bonnet theorem, if M is a closed and connected two-dimensional manifold, then MinVol(M) = 2π\|χ(M)\|. The infimum in the definition of minimal volume is realized by the metrics appearing from the uniformization theorem. More generally, according to the Chern-Gauss-Bonnet formula, if M is a closed and connected manifold then Gromov, in 1982, showed that the volume of a complete Riemannian metric on a smooth manifold can always be estimated by the size of its curvature and by the simplicial volume of the manifold, via the inequality (en)
gold:hypernym	Number
prov:wasDerivedFrom	wikipedia-en:Minimal_volume?oldid=982774580&ns=0
page length (characters) of wiki page	3138 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Minimal_volume
is Link from a Wikipage to another Wikipage of	Mikhael Gromov (mathematician)
is known for of	Mikhael Gromov (mathematician)
is known for of	Mikhael Gromov (mathematician)
is foaf:primaryTopic of	wikipedia-en:Minimal_volume

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