|
In mathematics, an I-bundle is a fiber bundle whose fiber is an interval and whose base is a manifold. Any kind of interval, open, closed, semi-open, semi-closed, open-bounded, compact, even rays, can be the fiber. Two simple examples of I-bundles are the annulus and the Möbius band, the only two possible I-bundles over the circle . The annulus is a trivial or untwisted bundle because it corresponds to the Cartesian product , and the Möbius band is a non-trivial or twisted bundle. Both bundles are 2-manifolds, but the annulus is an orientable manifold while the Möbius band is a non-orientable manifold. Curiously, there are only two kinds of I-bundles when the base manifold is any surface but the Klein bottle . That surface has three I-bundles: the trivial bundle and two twisted bundles. Together with the Seifert fiber spaces, I-bundles are fundamental elementary building blocks for the description of three-dimensional spaces. These observations are simple well known facts on elementary 3-manifolds. Line bundles are both I-bundles and vector bundles of rank one. When considering I-bundles, one is interested mostly in their topological properties and not their possible vector properties, as we might be for line bundles. ==References== * Scott, Peter, "The geometries of 3-manifolds". ''Bulletin of the London Mathematical Society'' 15 (1983), number 5, 401–487. * Hempel, John, "3-manifolds", ''Annals of Mathematics Studies'', number 86, Princeton University Press (1976). 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「I-bundle」の詳細全文を読む スポンサード リンク
|