Submission date: 18 July 2017

Abstract:

Let G be a compact subgroup of GL_n(R). We prove that every affine definable C^r G manifold admits a unique affine definable C^∞ G manifold structure up to definable C^∞ G diffeomorphism (1 ≤ r < ∞). Moreover we prove that every strongly definable C^r G vector bundle over X admits a unique strongly definable C^∞ G vector bundle structure up to definable C^∞ G vector bundle isomorphism (0 ≤ r < ∞). Furthermore we consider raising differentiability of strong definable C^r fiber bundles (0 ≤ r < ∞).

Mathematics Subject Classification: 57S15, 14P20, 57R35, 58A07, 03C64.

Keywords and phrases: Definable C^∞ G manifolds, definable C^∞ G maps, approximation theorem, definable C^∞ G vector bundles, definable C^∞ fiber bundles, o-minimal.

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