On the level set of a function with degenerate minimum point
On the level set of a function with degenerate minimum point
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关于具有退化极小值点的函数的水平集
DOI：
10.1155/2015/493217
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发表时间：
2015
影响因子：
1.2
通讯作者：
Yasuhiko Kamiyama
中科院分区：
文献类型：
--
作者：
Sakuma Makoto;Yokota Yoshiyuki;Yasuhiko Kamiyama
对于n ≥ 2，设M是n维光滑闭流形和光滑函数.设minf（M）=mand，假设mis由唯一点tp ∈ M达到，使得p是非退化临界点.然后，莫尔斯引理告诉我们，如果f α略大于m，则f−1（a）与Sn −1同构。本文将p从非退化到孤立临界点的条件放宽，得到了同样的结果。一些应用的拓扑结构的多边形空间也包括在内。
Forn≥ 2, letMbe ann‐dimensional smooth closed manifold anda smooth function. We set minf(M) =mand assume thatmis attained by unique pointp∈Msuch thatpis a nondegenerate critical point. Then the Morse lemma tells us that ifais slightly bigger thanm,f−1(a) is diffeomorphic toSn−1. In this paper, we relax the condition onpfrom being nondegenerate to being an isolated critical point and obtain the same consequence. Some application to the topology of polygon spaces is also included.