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Journal of the London Mathematical Society Advance Access originally published online on July 9, 2009
Journal of the London Mathematical Society 2009 80(2):375-387; doi:10.1112/jlms/jdp031
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© 2009 London Mathematical Society

On real analytic Banach manifolds

Imre Patyi

Department of Mathematics and Statistics
Georgia State University
Atlanta, GA 30303-3083
USA

Scott Bradford Simon

Department of Mathematics
SUNY Stony Brook
Stony Brook, NY 11794-3651
USA
sbsimon@math.sunysb.edu

Let X be a real Banach space with an unconditional basis (for example, X = {ell}2 Hilbert space), let {Omega} sub X be open, let M sub {Omega} be a closed split real analytic Banach submanifold of {Omega}, 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 sub U sub {Omega} such that r(x) = x for all x isin 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.

2000 Mathematics Subject Classification 32C05 (26E05, 32C35, 32V40, 46G20).

The authors were supported in part by NSF grants DMS 0600059 and DMS 0203072, and a Research Initiation Grant from Georgia State University.

Received October 12, 2007; revised January 27, 2009; published online July 9, 2009.


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