Hyperpolar homogeneous foliations on symmetric spaces of noncompact type - an overview
Hyperpolar homogeneous foliations on symmetric spaces of noncompact type - an overview
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DOI：
10.4310/jdg/1299766787
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发表时间：
2008-07
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影响因子：
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通讯作者：
J. Berndt
中科院分区：
文献类型：
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作者：
J. Berndt
黎曼流形M上的叶层F是齐次的，如果它的叶子与M上的等距作用的轨道重合。如果叶层F允许一个截面，即M的一个连通的全闭测地线子流形与F的每片叶相交，并在每个交点处垂直相交，则它是极的。叶状结构F是超极的，如果它有一个平坦的截面。这些注记与Jose Carlos Diaz-Ramos和Hiroshi Tamaru在非紧型Riemannian对称空间上关于超极齐次叶的共同工作有关。除了我们在Arxiv：0807.3517v2[Math.DG]中证明的分类结果外，本文还给出了一些关于非紧型对称空间的相关材料，并对特殊情况M=SL_{r+1}(R)/SO_{r+1}的分类作了更详细的讨论。
A foliation F on a Riemannian manifold M is homogeneous if its leaves coincide with the orbits of an isometric action on M. A foliation F is polar if it admits a section, that is, a connected closed totally geodesic submanifold of M which intersects each leaf of F, and intersects orthogonally at each point of intersection. A foliation F is hyperpolar if it admits a flat section. These notes are related to joint work with Jose Carlos Diaz-Ramos and Hiroshi Tamaru about hyperpolar homogeneous foliations on Riemannian symmetric spaces of noncompact type. Apart from the classification result which we proved in arXiv:0807.3517v2 [math.DG], we present here in more detail some relevant material about symmetric spaces of noncompact type, and discuss the classification in more detail for the special case M = SL_{r+1}(R)/SO_{r+1}.