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A (pseudo-)Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation. More formally, let (''M'', ''g'') be a pseudo-Riemannian manifold. Then (''M'', ''g'') is conformally flat if for each point ''x'' in ''M'', there exists a neighborhood ''U'' of ''x'' and a smooth function ''f'' defined on ''U'' such that (''U'', ''e''2''f''''g'') is flat (i.e. the curvature of ''e''2''f''''g'' vanishes on ''U''). The function ''f'' need not be defined on all of ''M''. Some authors use ''locally conformally flat'' to describe the above notion and reserve ''conformally flat'' for the case in which the function ''f'' is defined on all of ''M''. ==Examples== *Every manifold with constant sectional curvature is conformally flat. *Every 2-dimensional pseudo-Riemannian manifold is conformally flat. *A 3-dimensional pseudo-Riemannian manifold is conformally flat if and only if the Cotton tensor vanishes. *An ''n''-dimensional pseudo-Riemannian manifold for ''n'' ≥ 4 is conformally flat if and only if the Weyl tensor vanishes. *Every compact, simply connected, conformally flat Riemannian manifold is conformally equivalent to the round sphere. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Conformally flat manifold」の詳細全文を読む スポンサード リンク
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