### Abstract

A new proof of a theorem by Gromov is given: for any positive C and any integer n greater than 1, there exists a function Δ_{C,n}(δ) such that if the Gromov-Hausdorff distance between two complete Riemannian n-manifolds V and W is at most δ, their sectional curvatures K_{σ} do not exceed C, and their injectivity radii are at least 1/C, then the Lipschitz distance between V and W is less than Δ_{C,n}(δ), and Δ_{C,n}(δ) → 0 as δ → 0. Bibliography: 6 titles.

Original language | English (US) |
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Pages (from-to) | 361-367 |

Number of pages | 7 |

Journal | Journal of Mathematical Sciences |

Volume | 161 |

Issue number | 3 |

DOIs | |

State | Published - Jul 1 2009 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Mathematics(all)
- Applied Mathematics

### Cite this

*Journal of Mathematical Sciences*,

*161*(3), 361-367. https://doi.org/10.1007/s10958-009-9559-z

}

*Journal of Mathematical Sciences*, vol. 161, no. 3, pp. 361-367. https://doi.org/10.1007/s10958-009-9559-z

**A direct proof of Gromov's theorem.** / Burago, Yury D.; Malev, S. G.; Novikov, D. I.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A direct proof of Gromov's theorem

AU - Burago, Yury D.

AU - Malev, S. G.

AU - Novikov, D. I.

PY - 2009/7/1

Y1 - 2009/7/1

N2 - A new proof of a theorem by Gromov is given: for any positive C and any integer n greater than 1, there exists a function ΔC,n(δ) such that if the Gromov-Hausdorff distance between two complete Riemannian n-manifolds V and W is at most δ, their sectional curvatures Kσ do not exceed C, and their injectivity radii are at least 1/C, then the Lipschitz distance between V and W is less than ΔC,n(δ), and ΔC,n(δ) → 0 as δ → 0. Bibliography: 6 titles.

AB - A new proof of a theorem by Gromov is given: for any positive C and any integer n greater than 1, there exists a function ΔC,n(δ) such that if the Gromov-Hausdorff distance between two complete Riemannian n-manifolds V and W is at most δ, their sectional curvatures Kσ do not exceed C, and their injectivity radii are at least 1/C, then the Lipschitz distance between V and W is less than ΔC,n(δ), and ΔC,n(δ) → 0 as δ → 0. Bibliography: 6 titles.

UR - http://www.scopus.com/inward/record.url?scp=70449526799&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70449526799&partnerID=8YFLogxK

U2 - 10.1007/s10958-009-9559-z

DO - 10.1007/s10958-009-9559-z

M3 - Article

AN - SCOPUS:70449526799

VL - 161

SP - 361

EP - 367

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 3

ER -