Classical unified field theories: Difference between revisions

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In mathematics, a smooth [[compact space|compact]] [[manifold]] ''M'' is called '''almost flat''' if for any <math>\varepsilon>0 </math> there is a [[Riemannian metric]] <math>g_\varepsilon </math> on ''M'' such that <math> \mbox{diam}(M,g_\varepsilon)\le 1 </math> and  
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<math> g_\varepsilon </math> is <math>\varepsilon</math>-flat, i.e. for the [[sectional curvature]] of <math> K_{g_\varepsilon} </math> we have <math> |K_{g_\epsilon}| < \varepsilon</math>.


In fact, given ''n'', there is a positive number <math>\varepsilon_n>0 </math> such that if a ''n''-dimensional manifold admits an <math>\varepsilon_n</math>-flat metric with diameter <math>\le 1 </math> 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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According to the '''Gromov—Ruh theorem''', ''M'' is almost flat if and only if it is [[Glossary of Riemannian and metric geometry#I|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.  
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==References==
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*{{citation
  | last = Gromov | first = M. | authorlink = Mikhail Gromov
| issue = 2
| journal = [[Journal of Differential Geometry]]
| mr = 540942
| pages = 231–241
| title = Almost flat manifolds
| url = http://projecteuclid.org/getRecord?id=euclid.jdg/1214434488
| volume = 13
| year = 1978}}.
*{{citation
| last = Ruh | first = Ernst A.
| issue = 1
| journal = [[Journal of Differential Geometry]]
| mr = 658470
| pages = 1–14
| title = Almost flat manifolds
| url = http://projecteuclid.org/getRecord?id=euclid.jdg/1214436698
| volume = 17
| year = 1982}}.


[[Category:Riemannian geometry]]
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[[Category:Manifolds]]


 
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{{differential-geometry-stub}}

Latest revision as of 12:08, 19 August 2014

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