On real analytic Banach manifolds

TL;DR

This paper proves the vanishing of sheaf cohomology for real analytic Banach submanifolds and constructs real analytic retractions, with applications to the parallelizability of infinite-dimensional Hilbert submanifolds.

Contribution

It establishes sheaf cohomology vanishing results and constructs real analytic retractions for Banach submanifolds, extending infinite-dimensional complex analysis techniques.

Findings

Sheaf cohomology groups $H^q(M,\mathcal{A}^E)$ vanish for all $q\ge1$.

Existence of real analytic retraction from an open neighborhood onto the submanifold.

Infinite dimensional real analytic Hilbert submanifolds are real analytically parallelizable.

Abstract

Let $X$ be a real Banach space with an unconditional basis (e.g., $X=\ell_2$ Hilbert space), $\Omega\subset X$ open, $M\subset\Omega$ a closed split real analytic Banach submanifold of $\Omega$, $E\to M$ 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 $M\subset U\subset\Omega$ 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.