A direct proof of one Gromov's theorem

Abstract

We give a new proof of the Gromov theorem: For any $C>0$ and integer $n>1$ there exists a function ${\Delta }_{C,n}$ such that if the Gromov--Hausdorff distance between complete Riemannian $n$-manifolds $V$ and $W$ is not greater than $\delta$, absolute values of their sectional curvatures $|K_{\sigma }|\le C$, and their injectivity radii $\ge 1/C$, then the Lipschitz distance between $V$ and $W$ is less than ${\Delta }_{C,n}(\delta )$ and ${\Delta }_{C,n}\to 0$ as $\delta \to 0$.