关于实解析巴拿赫流形
On real analytic Banach manifolds
摘要：
假设$X$是一个具有无条件基的实Banach空间（例如，$X=\ell_2$ Hilbert空间），$Ω\subset X$开，$M\subsetΩ$是$Ω$的可裂闭实解析Banach子流形，$E\to M$是实解析Banach向量丛，${\Cal A}^E\to M$是$E\to M$的实解析截面的芽层.我们证明了层上同调群$H^q(M,{\Cal A}^E)$对所有$q\ge1$为零，并且存在一个从开集$U$的实解析收缩$r:U\to M$，其中$M\subset U\subsetΩ$使得$r(x)=x$对所有$x\in M$ .还给出了一些应用，例如，证明了可分仿射或射影Hilbert空间的任意无穷维实解析Hilbert子流形是实解析可平行化的。
英文摘要：
Let $X$ be a real Banach space with an unconditional basis (e.g., $X=\ell_2$ Hilbert space), $Ω\subset X$ open, $M\subsetΩ$ a closed split real analytic Banach submanifold of $Ω$, $E\to M$ a real analytic Banach vector bundle, and ${\Cal A}^E\to M$ the sheaf of germs of real analytic sections of $E\to M$. We show that the sheaf cohomology groups $H^q(M,{\Cal A}^E)$ vanish for all $q\ge1$, and there is a real analytic retraction $r:U\to M$ from an open set $U$ with $M\subset U\subsetΩ$ such that $r(x)=x$ for all $x\in M$. Some applications are also given, e.g., we show that any infinite dimensional real analytic Hilbert submanifold of separable affine or projective Hilbert space is real analytically parallelizable.
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