Equivalent definitions for Lipschitz compact connected manifolds (Q2762842)

Let \(X\) be a real normed space. Assume that \(M\) is a compact, connected, topological space for which there exists an open cover \(\{U_j\}_{j\in \{1,\dots, n\}}\) of \(M\) and homeomorphisms \(h_j : W_j= {\overset \circ W}_j \subseteq X \rightarrow U_j\) such that, whenever \(U _i \cap U _j \not= \emptyset\), \(h _i ^{-1} \circ h _j: h _j^{-1} (U _i \cap U _j) \rightarrow h _i ^{-1} (U _i \cap U _j) \) is Lipschitz. The author proves that this is equivalent to the fact that \( M \) is a compact, connected, metric space for which there exists an open cover \( \{ V _\alpha \} _{\alpha \in \{ 1,\dots ,m \} }\) of \( M \) and bijections \( \varphi _\alpha : W' _\alpha = \overset \circ W _\alpha' \subseteq X \rightarrow V _\alpha\) such that \(\varphi _\alpha \) and \( \varphi _\alpha ^{-1} \) are Lipschitz for all \(\alpha \in \{ 1, \dots , m \} \). For Lipschitz \(n\)-manifolds such a result has been obtained by \textit{J. Luukkainen} and \textit{J. Väisälä} [Ann. Acad. Sci. Fenn., Ser. A. I. Math. 3, 85-122 (1977; Zbl 0397.57011)].