On real analytic Banach manifolds

Abstract

Let X be a real Banach space with an unconditional basis (for example, X = ℓ2 Hilbert space), let Ω ⊂ X be open, let M ⊂ Ω be a closed split real analytic Banach submanifold of Ω, let E → M be a real analytic Banach vector bundle, and let 𝒜 E → M be the sheaf of germs of real analytic sections of E → M. We show that the sheaf cohomology groups Hq(M, 𝒜 E) vanish for all q ⩾ 1, and there is a real analytic retraction r:U → M from an open set U with M ⊂ U ⊂ Ω such that r(x) = x for all x ∈ M. Some applications are also given, for example, we show that any infinite-dimensional real analytic Hilbert submanifold of separable affine or projective Hilbert space is real analytically parallelizable.