Submission date: 7 April 2011.

Abstract:

Let G be a compact affine definable C^r group and let r be \infty or \omega. We prove that the representative definable C^r functions on G is dense in the space of continuous functions on G. Moreover we compare the category of compact affine definable C^r groups D with that of compact real algebraic groups A.

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

Keywords and phrases: O-minimal, definable C^r groups, algebraic groups, definable C^r G manifolds, Equivariant Nash Conjecture.

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