Foliations on 3-manifolds
Foliations on 3-manifolds
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3 流形上的叶状结构
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发表时间：
1969
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通讯作者：
John W. Wood
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作者：
John W. Wood
设M是具有切束TM的光滑流形。M上的k平面场（或k分布）是TM的k维子束A。等价地，设a表示与TM相关的k个平面的Grassmann束Gk(M)的截面，其在x e M处的值为k平面ax c TM，。如果两个k平面场作为Gk(M)的截面是同伦的，则它们是同伦的。同伦k-平面场等价于M上的k-平面束，但不是相反。如果L是M的内浸光滑子流形，使得TLX = ax c TMx对所有x e L，则L称为a的积分子流形。如果满足以下三个等价条件，则称为平面域a的完全可积。a . M是由开放与当地坐标集U (x1, *, x”这样定义的流形xk + l =常数,* * *,xm =子流形的一个积分常数。b .光滑,通过每一个点x e M不可或缺的子流形l。C .光滑,如果x和Y是向量场在M Xx, Yx C所有x e M ax的支架(x, Y) x e ax。这些条件的等价性就是Frobenius定理。一个可积的k平面域也被称为叶形（这与其他定义等效），最大的连通积分子流形被称为叶。叶状叶的叶子分割成流形。常微分方程的存在性定理表明光滑线场总是可积的。一般来说，对于k > 1，不可积的k平面场集合在Gk(M)的截面空间中是开密的。G. Reeb[16]提出了流形上k平面场的存在是否意味着叶理的存在。他给出了S3上余维为1的叶化的一个例子。W. B. R. Lickorish bbb， S. P. Novikov和H. Zeischang分别在任何封闭的可定向的3-流形上展示了余维1的叶状。在吗?我们认为无方向性
Let M be a smooth manifold with tangent bundle TM. A k-plane field (or k-distribution) on M is a k-dimensional subbundle a of TM. Equivalently let a denote the section of the Grassmann bundle Gk(M) of k-planes associated to TM whose value at x e M is the k-plane ax c TM,. Two k-plane fields are homotopic if they are homotopic as sections of Gk(M). Homotopic k-plane fields are equivalent as k-plane bundles over M, but not conversely. If L is an injectively immersed, smooth submanifold of M such that TLX = ax c TMx for all x e L, L is called an integral submanifold of a. A kplane field a is called completely integrable if the following three equivalent conditions are satisfied. A. M is covered by open sets U with local coordinates x1, * , x" such that the submanifolds defined by xk+l = constant, * * *, xm = constant are integral submanifolds of a. B. a is smooth and through every point x e M there is an integral submanifold L of a. C. a is smooth and if X and Y are vector fields on M with Xx, Yx C ax for all x e M then the bracket [X, Y]x e ax. The equivalence of these conditions is the Frobenius theorem. An integrable k-plane field is also called a foliation (this is equivalent to other definitions) and the maximal connected integral submanifolds are called leaves. The leaves of a foliation partition the manifold. The existence theorem for ordinary differential equations says that smooth line-fields are always integrable. In general for k > 1 the set of kplane fields which are not integrable is open and dense in the space of sections of Gk(M). G. Reeb [16] has asked if the existence of a k-plane field on a manifold implies the existence of a foliation. He has given an example of a foliation of codimension one on S3. W. B. R. Lickorish [10] and, independently, S. P. Novikov and H. Zeischang have exhibited foliations of codimension one on any closed, orientable 3-manifold. In ? 4 we consider the unorientable