论文标题
希尔伯特空间和载体图中的等式次要次曼群
Isoparametric submanifolds in Hilbert spaces and holonomy maps
论文作者
论文摘要
让$π：p \ t to b $是一个平滑的$ g $ - 捆绑物，而紧凑的riemannian歧管$ b $和$ c $ a平滑环的$ b $ constant seed $ a（> 0）$，其中$ g $是紧凑的半简单谎言组。在本文中，我们证明了整体映射$ {\ rm hol} _c：\ Mathcal a_p^{h^s} \ to G $是系数$ a $ a $的同质淹没，其中$ s $是一个非内在整数，$ \ ntecal a_p^s $捆绑$ P $。特别是，我们证明，如果$ s = 0 $，则$ {\ rm hol} _c $具有最小的正规纤维。从这个事实来看，我们可以得出所有等式submanifold的倒数submanifold的每个组件，由$ g $ in $ g $ by holomony map $ {\ rm hol} _c：\ mathcal a_p a_p^{h^0} \ g $ to g $是iSoparametric submanifold in $ \ m nathcal a__p^0}。结果，我们获得了希尔伯特空间中等距亚曼群的新系统构造。
Let $π:P\to B$ be a smooth $G$-bundle over a compact Riemannian manifold $B$ and $c$ a smooth loop in $B$ of constant seed $a(>0)$, where $G$ is compact semi-simple Lie group. In this paper, we prove that the holonomy map ${\rm hol}_c:\mathcal A_P^{H^s}\to G$ is a homothetic submersion of coefficient $a$, where $s$ is a non-negative integer, $\mathcal A_P^{H^s}$ is the Hilbert space of all $H^s$-connections of the bundle $P$. In particular, we prove that, if $s=0$, then ${\rm hol}_c$ has minimal regularizable fibres. From this fact, we can derive that each component of the inverse image of any equifocal submanifold in $G$ by the holonomy map ${\rm hol}_c:\mathcal A_P^{H^0}\to G$ is an isoparametric submanifold in $\mathcal A_P^{H^0}$. As the result, we obtain a new systematic construction of isoparametric submanifolds in a Hilbert space.
