Almost flat manifold

In mathematics, a smooth compact manifold M is called almost flat if for any epsilon>0 there is a Riemannian metric g_epsilon on M such that mbox{diam}(M,g_epsilon)le 1 and

Related Topics:
Mathematics - Compact - Manifold

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g_epsilon is epsilon-flat, i.e. for sectional curvature of K_{g_epsilon} we have |K_{g_epsilon}|

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In fact, given n, there is a positive number epsilon_n>0 such that if a n-dimensional manifold admits an epsilon_n-flat metric with diameter le 1 then it is almost flat.

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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 nill manifold, i.e. a total space of a oriented circle bundle over a oriented circle bundle over ... over a circle.

Related Topics:
Infranil - Nill manifold - Bundle

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