EXCEPTIONAL ORBITS OF HIGHEST DIMENSION
EXCEPTIONAL ORBITS OF HIGHEST DIMENSION
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DOI：
10.2307/1969951
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发表时间：
1956-07
影响因子：
4.9
通讯作者：
D. Montgomery;H. Samelson;C. T. Yang
中科院分区：
文献类型：
--
作者：
D. Montgomery;H. Samelson;C. T. Yang
When a compact connected Lie group G acts as a transformation group on a manifold M of dimension n it is known [3] that the points on orbits of highest dimension form a connected open set whose complement, call it F, has dimension at most n - 2. Among the highest dimensional orbits there may be points forming an exceptional set E where the group of stability Gx is not continuous. An orbit in E is such that some nearby orbits "wrap around" it more than once. We consider here the case where G acts differentiably and show for a class of manifolds including the n-sphere and euclidean n-space that dim (E u F) < n-2. We do not know how to prove this result without differentiability. For the case mentioned the orbits of highest dimension at which G. is continuous form an open connected set; the set is clearly fibred by the orbits so that E u F is the complement of an open connected dense set which is fibred. We now give in detail some of the facts and definitions which will be used. It is always assumed that a compact Lie group G is given as acting on a manifold M of dimension n. For the preliminaries M is not necessarily differentiable.
