A direct proof of Gromov's theorem (Q844459)

Let \({\mathfrak M}(\rho,C,n)\) be the class of complete \(n\)-dimensional Riemannian manifolds \(V\) with sectional curvature \(\leq C<\infty\) and injectivity radius \(>\rho\), where \(C\) and \(\rho\) are positive constants. As is well known, there are two metrics on \({\mathfrak M}(\rho,C,n)\). They are the Lipschitz distance \(d_{\text{Lip}}(X,Y)\) and the Gromov-Hausdorff's one \(d_{GH}(X,Y)\) between the spaces \(X\) and \(Y\). The purpose of this paper is to give a direct proof of a theorem due to Gromov: There exists a positive function \(\Delta=\Delta(C,\rho,n)\) such that \(\Delta(\delta)\to 0\) as \(\delta\to 0\) and if \(V,W\in{\mathfrak M}(\rho,C,n)\) satisfy the condition \(d_{\text{GH}}(V,W)<\delta\), then it holds that \(d_{\text{Lip}}(V,W)<\Delta(\delta)\). While Gromov's proof involves auxiliary imbedding of \(V\) and \(W\) in a Euclidean space of large dimension, here the author constructs by means of a partition of unity a bi-Lipschitz diffeomorphism \(h(x)\) between \(V\) and \(W\) with bi-Lipschitz number \(\Delta(\delta)\).