Some remarks on uniformly regular Riemannian manifolds (Q2792251)

In two papers [``Anisotropic function spaces on singular manifolds'', Preprint, \url{arXiv:1204.0606}; Math. Nachr. 286, No. 5--6, 436--475 (2013; Zbl 1280.46022)], \textit{H. Amann} introduced a class of (possibly noncompact) manifolds, which he termed uniformly regular Riemannian manifolds. Loosely speaking, an \(m\)-dimensional Riemannian manifold \((M,g)\) is \textit{uniformly regular} if its differentiable structure is induced by an atlas in which all the local charts are approximately of the same size, all derivatives of coordinate changes are bounded, and the pull-back metric of \(g\) in every local chart is comparable to the Euclidean metric \(g_m\). In the present note, it is proved that the family of uniformly regular Riemannian manifolds coincides with the class of manifolds with bounded geometry, i.e., Riemannian manifolds which have positive injectivity radius, and where all covariant derivatives of the curvature tensor are bounded.