Real and bi-Lipschitz versions of the Theorem of Nobile

The renowned Theorem of Nobile, proved by Nobile in 1975, states that a pure dimensional complex analytic set $X$ is analytically smooth if and only if its Nash transformation $\eta: \mathcal{N}(X) \to X$ is an analytic isomorphism. While the Theorem of Nobile was fundamental in complex geometry, it remained an open question for 50 years whether the theorem held for real analytic sets, even more so for cases that demand only $C^{k}$ smoothness. This paper presents a proof for the real version of the Theorem of Nobile, even under $C^{k}$ smoothness conditions. Specifically, we prove that for a pure dimensional real analytic set $X$ the following statements are equivalent: (1) $X$ is a real analytic (resp. $C^{k+1,1}$) submanifold; (2) the mapping $\eta\colon\mathcal{N}(X)\to X$ is a real analytic (resp. $C^{k,1}$) diffeomorphism; (3) the mapping $\eta\colon\mathcal{N}(X)\to X$ is a $C^{\infty}$ (resp. $C^{k,1}$) diffeomorphism; (4) $X$ is a $C^{\infty}$ (resp. $C^{k+1,1}$) submanifold. Consequently, we prove the bi-Lipschitz version of the Theorem of Nobile, demonstrating that a complex analytic set $X$ is analytically smooth if and only if its Nash transformation $\eta\colon \mathcal{N}(X)\to X$ is a homeomorphism that is locally bi-Lipschitz. A sharp version of this theorem, which holds in the much more general setting of locally definable sets in an o-minimal structure, is also presented here.