About: Almost flat manifold

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In mathematics, a smooth compact manifold M is called almost flat if for any there is a Riemannian metric on M such that and is -flat, i.e. for the sectional curvature of we have . Given n, there is a positive number such that if an n-dimensional manifold admits an -flat metric with diameter then it is almost flat. On the other hand, one can fix the bound of sectional curvature and get the diameter going to zero, so the almost-flat manifold is a special case of a collapsing manifold, which is collapsing along all directions.

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rdfs:label	Almost flat manifold (en) Почти плоское многообразие (ru) Nästan platt mångfald (sv)
rdfs:comment	Почти плоское многообразие — гладкое компактное многообразие М такое, что для любого на М существует риманова метрика ,такая, что и является -плоской,то есть её секционные кривизны в каждой точке удовлетворяют неравенству (ru) Inom matematiken säges en kompakt mångfald M vara nästan platt om för varje finns det en på M så att och är -platt, d.v.s. vi har olikheten för av . (sv) In mathematics, a smooth compact manifold M is called almost flat if for any there is a Riemannian metric on M such that and is -flat, i.e. for the sectional curvature of we have . Given n, there is a positive number such that if an n-dimensional manifold admits an -flat metric with diameter then it is almost flat. On the other hand, one can fix the bound of sectional curvature and get the diameter going to zero, so the almost-flat manifold is a special case of a collapsing manifold, which is collapsing along all directions. (en)
dcterms:subject	Manifolds Riemannian geometry
Wikipage page ID	933946 (xsd:integer)
Wikipage revision ID	1083533968 (xsd:integer)
Link from a Wikipage to another Wikipage	Compact space Manifolds Glossary of Riemannian and metric geometry Journal of Differential Geometry Riemannian metric Nilmanifold Riemannian geometry Collapsing manifold Manifold Sectional curvature
Link from a Wikipage to an external page	https://projecteuclid.org/journals/journal-of-differential-geometry/volume-13/issue-2/Almost-flat-manifolds/10.4310/jdg/1214434488.full https://projecteuclid.org/journals/journal-of-differential-geometry/volume-17/issue-1/Almost-flat-manifolds/10.4310/jdg/1214436698.full
sameAs	Almost flat manifold Almost flat manifold Almost flat manifold Almost flat manifold Almost flat manifold Almost flat manifold
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Reflist dbt:Differential-geometry-stub
has abstract	In mathematics, a smooth compact manifold M is called almost flat if for any there is a Riemannian metric on M such that and is -flat, i.e. for the sectional curvature of we have . Given n, there is a positive number such that if an n-dimensional manifold admits an -flat metric with diameter then it is almost flat. On the other hand, one can fix the bound of sectional curvature and get the diameter going to zero, so the almost-flat manifold is a special case of a collapsing manifold, which is collapsing along all directions. According to the Gromov–Ruh theorem, M is almost flat if and only if it is infranil. In particular, it is a finite factor of a nilmanifold, which is the total space of a principal torus bundle over a principal torus bundle over a torus. (en) Почти плоское многообразие — гладкое компактное многообразие М такое, что для любого на М существует риманова метрика ,такая, что и является -плоской,то есть её секционные кривизны в каждой точке удовлетворяют неравенству (ru) Inom matematiken säges en kompakt mångfald M vara nästan platt om för varje finns det en på M så att och är -platt, d.v.s. vi har olikheten för av . (sv)
prov:wasDerivedFrom	wikipedia-en:Almost_flat_manifold?oldid=1083533968&ns=0
page length (characters) of wiki page	2715 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Almost_flat_manifold
is Link from a Wikipage to another Wikipage of	Glossary of Riemannian and metric geometry Gromov's theorem Riemannian geometry Nilmanifold Gromov-Ruh theorem Mikhael Gromov (mathematician) Gromov–Ruh theorem
is Wikipage redirect of	Gromov-Ruh theorem Gromov–Ruh theorem
is known for of	Mikhael Gromov (mathematician)
is known for of	Mikhael Gromov (mathematician)
is foaf:primaryTopic of	wikipedia-en:Almost_flat_manifold

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