Skew flat fibrations
Skew flat fibrations
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偏斜扁平纤维
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发表时间：
2014
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通讯作者：
Michael C. Harrison
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作者：
Michael C. Harrison
如果没有两个纤维相交也不包含平行方向，则 $$mathbb R^n$$Rn 由 $$mathbb R^p$$Rp 的定向副本引起的纤维化称为偏斜。 Ovsienko 和 Tabachnikov 给出了这种纤维化存在的 p 和 n 条件。 Salvai 给出了 $$mathbb R^3$$R3 的光滑纤维振动通过倾斜定向线的分类，类似于 Gluck 和 Warner 对 $$S^3$$S3 的定向大圆纤维振动的分类。我们证明 Salvai 的分类具有拓扑变化，它概括为通过 $$mathbb R^p$$Rp 的倾斜副本来表征 $$mathbb R^n$$Rn 的所有连续纤维。我们证明 $$mathbb R^3$$R3 的纤维化空间通过斜向线变形收缩到 Hopf 纤维化的子空间，因此具有 $$S^2$$S2 的一对不相交副本的同伦类型。我们讨论了复数和四元数环境中的斜纤维化，并通过 $$mathbb C^p$$Cp 的斜向副本（分别为 $$mathbb H^p$$Hp）给出 $$mathbb C^n$$Cn （分别为 $$mathbb H^n$$Hn）纤维化存在的必要条件。
A fibration of $$mathbb R^n$$Rn by oriented copies of $$mathbb R^p$$Rp is called skew if no two fibers intersect nor contain parallel directions. Conditions on p and n for the existence of such a fibration were given by Ovsienko and Tabachnikov. A classification of smooth fibrations of $$mathbb R^3$$R3 by skew oriented lines was given by Salvai, in analogue with the classification of oriented great circle fibrations of $$S^3$$S3 by Gluck and Warner. We show that Salvai’s classification has a topological variation which generalizes to characterize all continuous fibrations of $$mathbb R^n$$Rn by skew oriented copies of $$mathbb R^p$$Rp. We show that the space of fibrations of $$mathbb R^3$$R3 by skew oriented lines deformation retracts to the subspace of Hopf fibration, and therefore has the homotopy type of a pair of disjoint copies of $$S^2$$S2. We discuss skew fibrations in the complex and quaternionic setting and give a necessary condition for the existence of a fibration of $$mathbb C^n$$Cn (respectively, $$mathbb H^n$$Hn) by skew oriented copies of $$mathbb C^p$$Cp (respectively, $$mathbb H^p$$Hp).