Manifold - Manifold

A topological space modeled on Euclidean space is called a manifold. The simplest figure is a point, which is called a x-dimensional manifold. Among line figures, lines, half lines, circles, and line segments that extend infinitely to the left and right are one-dimensional manifolds ( Figure A ). In contrast, line figures such as those in Figure B are not manifolds. That is, the set of points in the vicinity of point P in Figure B , that is, near point P, is a cross in (a) and a T-shape in (b), which is not a line segment in either case.

A surface figure is called a two-dimensional manifold if the neighborhood of each point is the same topology as a disk. Planes, spheres, disks, and tori (torus surfaces) as shown in Figure C are manifolds, but a figure like (e), which is a sphere with a rectangle attached to it, is not a manifold because the neighborhood of the attached point P is not a disk. A three-dimensional figure in which the neighborhood of each point is a sphere (i.e. a three-dimensional sphere) is a three-dimensional manifold, and ordinary three-dimensional space and a three-dimensional sphere are themselves three-dimensional manifolds. Similarly, an n-dimensional figure in which the neighborhood of each point is an n-dimensional sphere is called an n-dimensional manifold. n-dimensional space and an n-dimensional sphere are n-dimensional manifolds. Manifolds are divided into those without boundaries, such as planes, spheres, and tori, and those with boundaries, such as disks (the boundary of a disk is its circumference). The neighborhood of each point in a manifold without a boundary is an open sphere obtained by removing the boundary from a sphere. Therefore, if we imagine a nearsighted insect on a two-dimensional manifold without a boundary, wherever the insect is, it will always see the same open disk (a disk without the boundary circumference), i.e., it will always see the same scenery. Furthermore, spheres and tori are closed sets with finite size in space, and are called closed manifolds (closed surfaces in the two-dimensional case).

When a manifold can be triangulated and regarded as a polyhedron, it is called a combinatorial manifold, and when a differential structure can be introduced into the manifold, it is called a differentiable (manifold).

[Hiroshi Noguchi]

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

ユークリッド空間をモデルとした位相空間を多様体という。いちばん単純な図形は点であり、これは〇(れい)次元多様体という。線の図形のうち、左右無限に延びている直線、半直線、円周、線分が一次元多様体である（図A）。これに対して図Bのような線の図形は多様体ではない。すなわち、図Bで点Pの近傍、つまり点Pの近くにある点の集合が、(a)では十字形であり、(b)ではT字形であり、どちらにしろ線分ではないからである。

　面の図形は、その各点で、その近傍が円板と同位相になるものを二次元多様体という。図Cのような、平面や球面や円板やトーラス（輪環面）は様体であるが、球面に矩形(くけい)などを取り付けた(e)のような図形は、取り付けた点Pの近傍が円板でないので多様体ではない。各点の近傍が球体（つまり三次元球体）となるような三次元的図形が三次元多様体で、普通の三次元空間や三次元球体自身はそれぞれ三次元多様体である。同様に、各点の近傍がn次元球体となるようなn次元的図形をn次元多様体という。n次元空間やn次元球体はn次元多様体である。多様体は、平面や球面やトーラスのように境界のないものと、円板のように境界（円板はその円周が境界となる）をもつものとに分かれる。境界のない多様体の各点の近傍は球体からその境界を除いた開球体となる。よって境界のない二次元多様体上に近眼の虫がいると仮定すると、虫はどこにいても同じ開円板（境界の円周を省いた円板）、つまりいつも同じ景色を眺めていることになる。さらに球面やトーラスは空間の中で有限の大きさをもつ閉集合であり、閉じた多様体（二次元の場合、閉曲面）という。

　多様体が三角形分割できて多面体とみなせるとき、組合せ多様体といい、さらに多様体に微分構造が導入できるとき微分（可能）多様体という。

［野口　廣］

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