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In mathematics, a tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle. The idea behind a tubular neighborhood can be explained in a simple example. Consider a smooth curve in the plane without self-intersections. On each point on the curve draw a line perpendicular to the curve. Unless the curve is straight, these lines will intersect among themselves in a rather complicated fashion. However, if one looks only in a narrow band around the curve, the portions of the lines in that band will not intersect, and will cover the entire band without gaps. This band is a tubular neighborhood. In general, let ''S'' be a submanifold of a manifold ''M'', and let ''N'' be the normal bundle of ''S'' in ''M''. Here ''S'' plays the role of the curve and ''M'' the role of the plane containing the curve. Consider the natural map : which establishes a bijective correspondence between the zero section ''N''0 of ''N'' and the submanifold ''S'' of ''M''. An extension ''j'' of this map to the entire normal bundle ''N'' with values in ''M'' such that ''j''(''N'') is an open set in ''M'' and ''j'' is a homeomorphism between ''N'' and ''j''(''N'') is called a tubular neighbourhood. Often one calls the open set ''T''=''j''(''N''), rather than ''j'' itself, a tubular neighbourhood of ''S'', it is assumed implicitly that the homeomorphism ''j'' mapping ''N'' to ''T'' exists. ==Normal tube == A normal tube to a smooth curve is a manifold defined as the union of all discs such that * all the discs have the same fixed radius; * the center of each disc lies on the curve; and * each disc lies in a plane normal to the curve where the curve passes through that disc's center. 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「Tubular neighborhood」の詳細全文を読む スポンサード リンク
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