A direct proof of Gromov's theorem

A new proof of a theorem by Gromov is given: for any $C>0$ and integer $n>1$, there exists a function $\Delta_{C,n}(\delta)$ such that if the Gromov–Hausdorff distance between two complete Riemannian $n$-manifolds $V$ and $W$ is at most $\delta$, their sectional curvatures $|K_\sigma|$ do not exceed $C$, and their injectivity radii are at least $1/C$, then the Lipschitz distance between $V$ and $W$ is less than $\Delta_{C,n}(\delta)$, and $\Delta_{C,n}(\delta)\to0$ as $\delta\to0$. Bibl. – 6 titles.