Collapsing, solvmanifolds and infrahomogeneous spaces
Collapsing, solvmanifolds and infrahomogeneous spaces
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DOI：
10.1016/s0926-2245(96)00054-x
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发表时间：
1997-09
影响因子：
0.5
通讯作者：
W. Tuschmann
中科院分区：
文献类型：
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作者：
W. Tuschmann
当用Hausdorff收敛来表述时，M.格罗莫夫的几乎平坦流形定理指出，如果紧致流形M允许有界曲率坍缩到一点，则M的有限覆盖必然是一个零流形。于是，我们不禁要问，更一般的豪斯多夫极限的规定是否仍然会对流形施加某种齐性条件或其他严格的限制，或者，事实上，紧致流形是否甚至可以通过它允许的有界曲率坍缩来拓扑地表征。本文主要在可解范畴中研究这类问题。本文改进了前人的工作，解决了福谷的一个猜想，首先证明了：如果紧致流形M允许有界曲率坍缩为任意维数的紧致平坦orbifold，则M的有限覆盖是解流形的复纯复盖.作为这个结果的部分匡威，我们得到定理，任何紧的infrasolvmanifold M承认一个广义的Seifert纤维化在一个紧的平坦orbifold，和一个序列的局部齐次度量，使M允许有界曲率崩溃到这个orbifold。局部齐次塌缩度量的曲率和直径有界也被构造在任何亚齐次空间，这是仿照一个李群，其根是非平凡的。在拓扑范畴中，上述结果已经在Farrell和Hsiang的精神中导致了下solvmanifold的度量刻画。然而，在光滑设置中，情况完全不同，对于类似的光滑刻画，我们的第一个定理必须适当地尖锐化。事实上，我们表明，类的紧凑光滑流形，承认有限覆盖的齐次空间一般严格大于相应的类的亚齐次空间。在关于几乎平坦流形定理的标准文献中，几乎从未做出过这种关键的区分。最后，我们研究了一类下解流形在Hausdorff收敛下的稳定性和闭性，给出了本文结果的历史，并讨论了一些相关的公开问题和结果。
When phrased in terms of Hausdorff convergence, M. Gromov's almost flat manifold theorem states that if a compact manifold M admits a bounded curvature collapse to a point, then a finite cover of M is necessarily diffeomorphic to a nilmanifold. It is then tempting to ask whether the prescription of more general Hausdorff limits will still place some kind of homogeneity conditions or other severe restrictions on a manifold, or, in fact, whether a compact manifold can even be topologically characterized by the bounded curvature collapses it allows. In this paper, we study these questions in mainly the solvable category. Improving earlier works on this problem, and solving a conjecture of Fukaya, we first show that if a compact manifold M admits a bounded curvature collapse to a compact flat orbifold of arbitrary dimension, then a finite cover of M is diffeomorphic to a solvmanifold. As a partial converse to this result, we obtain the theorem that any compact infrasolvmanifold M admits a generalized Seifert fibration over a compact flat orbifold, and a sequence of locally homogeneous metrics such that M allows a bounded curvature collapse to this orbifold. Locally homogeneous collapsing metrics with bounded curvature and diameter are also constructed on any infrahomogeneous space which is modelled on a Lie goup whose radical is nontrivial. In the topological category, the above results already lead to a metrical characterization of infrasolvmanifolds in the spirit of Farrell and Hsiang. However, in the smooth setting, the situation is quite different, and for an analogous smooth characterization our first theorem would have to be appropiately sharpened. Indeed, we show that the classes of compact smooth manifolds that admit finite coverings by homogeneous spaces are in general strictly bigger than the corresponding classes of infrahomogeneous spaces. In the standard literature on the almost flat manifold theorem, this crucial distinction has nearly never been made. Finally, we investigate some stability and closedness properties of classes of infrasolvmanifolds under Hausdorff convergence, give an account of the history of the results presented here, and discuss some related open questions and conjectures.