Geodesic
A direct proof of Gromov's theorem
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 10, Tome 353 (2008), pp. 14-26 
Yu. D. Burago; S. G. Malev; D. I. Novikov. A direct proof of Gromov's theorem. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 10, Tome 353 (2008), pp. 14-26. http://geodesic.mathdoc.fr/item/ZNSL_2008_353_a1/
@article{ZNSL_2008_353_a1,
     author = {Yu. D. Burago and S. G. Malev and D. I. Novikov},
     title = {A direct proof of {Gromov's} theorem},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {14--26},
     year = {2008},
     volume = {353},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2008_353_a1/}
}
TY  - JOUR
AU  - Yu. D. Burago
AU  - S. G. Malev
AU  - D. I. Novikov
TI  - A direct proof of Gromov's theorem
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2008
SP  - 14
EP  - 26
VL  - 353
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2008_353_a1/
LA  - en
ID  - ZNSL_2008_353_a1
ER  - 
%0 Journal Article
%A Yu. D. Burago
%A S. G. Malev
%A D. I. Novikov
%T A direct proof of Gromov's theorem
%J Zapiski Nauchnykh Seminarov POMI
%D 2008
%P 14-26
%V 353
%U http://geodesic.mathdoc.fr/item/ZNSL_2008_353_a1/
%G en
%F ZNSL_2008_353_a1

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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.

[1] D. Yu. Burago, Yu. D. Burago, S. V. Ivanov, A course in metric geometry, Graduate Studies in Math., 33, Amer. Math. Soc., Providence, RI, 2001, 415 pp. | MR | Zbl

[2] Yu. D. Burago, V. A. Zalgaller, Introduction to Riemannian geometry, Nauka, Moscow, 1994, 319 pp. (Russian) | MR | Zbl

[3] I. Chavel, Riemannian geometry – a modern introduction, Cambridge Tracts in Mathematics, 108, Cambridge University Press, Cambridge, 1993, 386 pp. | MR | Zbl

[4] M. Gromov, Metric structures for Riemannian and Non-Riemannian spaces, Birkhäuser, 1999 | MR | Zbl

[5] Ju. Jost, Riemannian geometry and geometric analysis, Third edition, Universitext, Springer-Verlag, Berlin, 2002, 532 pp. | MR

[6] H. Karcher, “Riemannian center of mass and mollifier smoothing”, Comm. Pure Appl. Math., 30:5 (1977), 509–541 | DOI | MR | Zbl