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flat manifold
In mathematics, a Riemannian manifold is said to be flat if its curvature is everywhere zero. Intuitively, a flat manifold is one that "locally looks like" Euclidean space in terms of distances and angles, e.g. the interior angles of a triangle add up to 180°.
The universal cover of a complete flat manifold is Euclidean space. This can be used to prove the theorem of
that all compact flat manifolds are finitely covered by tori; the 3-dimensional case was proved earlier by .
==Examples==
The following manifolds can be endowed with a flat metric. Note that this may not be their 'standard' metric (for example, the flat metric on the circle is not the metric induced by its embedding into ).
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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