A direct proof of Gromov's theorem

Yu. D. Burago ; S. G. Malev ; D. I. Novikov

Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 10, Tome 353 (2008), pp. 14-26

Résumé

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.