A generalized almost flat manifold under a bounded weak \(C^{0,\alpha}\)-norm (Q5953411)

The weak norms of \(n\)-dimensional Riemannian manifolds \((M,g)\) on scale \(r>0\), denoted by \(\|(M,g)\|_{C^{k,\alpha},r}\) and \(\|(M,g)\|^W_{C^{k,\alpha},r}\), were introduced in Comment. Math. Helv. 74, No. 3, 345-363 (1999; Zbl 0994.53019) by \textit{P. Petersen}, \textit{G. Wei} and \textit{R. Ye}. Let \({\mathcal M}(n,\alpha,Q)\) denote the following class of \(n\)-dimensional complete Riemannian manifolds: \[ {\mathcal M}(n,\alpha,Q)=\{(M,g)\mid \|(M,g)\|^W_{C^{0,\alpha},r}\leq Q(r)\text{ for } 0<r\leq 1\}, \] where \(Q(r)\) stands for a positive function satisfying \(Q(r)\to 0\) as \(r\to 0\) and \(\alpha>0\). The main theorem of the paper is the following. There exists an \(\varepsilon(n,\alpha,Q)>0\) depending on \(n\), \(\alpha\) and \(Q\) such that if \(M\in{\mathcal M}(n,\alpha,Q)\) and \(\text{diam}(M)\leq\varepsilon(n,\alpha,Q)\), then a finite covering space of \(M\) is a nilmanifold. This result is proved by using geometric methods similar as Gromov's original proof, so that the author does not need the harmonic coordinate.