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Collapse of warped foliations

机译：弯曲的叶子塌陷

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摘要

The aim of this note is to generalize the concept of warped product to a foliated manifold (M, F, g) as follows: If f:M→(0,∞) is a smooth function constant along the leaves of the foliation F then new metric structure gf on the manifold M is constructed as follows: gf(v,w)=f2g(v,w) if v,w are tangent to F and gf(v,w)=g(v,w) if v or w is perpendicular to F. A foliated manifold (M, F, gf) is called warped foliation while f is called warping function. Next, if (f_n:M→(0,∞))_(n∈N) is a sequence of warping functions on M, the question of the existence of the limit in Gromov–Hausdorff of a sequence ((M, F, gf_n))_(n∈N) warped foliation is asked. A number of examples is considered such foliations with dense leaf or foliations consisting of finite number of Reeb components. Next, sufficient and necessary condition of converging in Gromov–Hausdorff sense of a Riemannian foliation with all leaves compact to the space of leaves with a metric defined by Hausdorff distance of leaves is developed. Finally some results on Hausdorff foliations with all leaves compact are shown.
机译：该注释的目的是将翘曲积的概念推广到叶形歧管（M，F，g），如下所示：如果f：M→（0，∞）是沿着叶形F的叶子的光滑函数常数，则流水M上的新度量结构gf构造如下：如果v，w与F相切，则gf（v，w）= f2g（v，w）;如果v与gf（v，w）= g（v，w）或w垂直于F。叶状流形（M，F，gf）被称为扭曲叶面，而f被称为扭曲函数。接下来，如果（f_n：M→（0，∞））_（n∈N）是一个在M上的翘曲函数序列，则该序列在Gromov–Hausdorff中存在极限（（M，F， gf_n））_（n∈N）弯曲的叶问。许多示例被认为是具有密集叶的叶或由有限数量的Reeb组成的叶。接下来，开发了一个满足Gromov–Hausdorff黎曼叶面意义的收敛的充要条件，其中所有叶子都紧紧压缩到叶子的空间，其度量由叶子的Hausdorff距离定义。最后，显示了所有叶片紧密的Hausdorff叶面的一些结果。

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