Every sub-Riemannian manifold is the Gromov-Hausdorff limit of a sequence Riemannian manifolds
The main theorem of the paper under review states that, for a compact sub-Riemannian manifold \((M, \Delta, g)\), there exists a family of Riemannian metrics \(G_t\) on the manifold \(M\) such that \((M, d_t)\) converges to \((M, d)\) in the sense of Gromov-Hausdorff as \(t \to 0\), where \(d\) is the sub-Riemannian distance and \(d_t\) is the distance determined by the metric \(G_t\). To construct the family \(G_t\), the author takes a Riemannian metric \(G\) on the manifold \(M\), then uses the splitting \(TM = \Delta \oplus \Delta^\perp\) in order to set the metric \(G_t = g \oplus \frac{1}{t} G\). For the theory of sub-Riemannian manifolds including applications of the Gromov-Hausdorff limit in this theory see, e.g., [\textit{R. Montgomery}, A tour of sub-Riemannian geometries, their geodesics and applications. Providence, RI: American Mathematical Society (2002; Zbl 1044.53022)].
scientific article; zbMATH DE number 6819826