Hopf–Rinow theorem

en

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Hopf–Rinow theorem

Quality:

Hopf–Rinow theorem - theorem that, for a Riemannian manifold, geodesic completeness is equivalent to completeness as a metric space. Article “Hopf–Rinow theorem” in English Wikipedia has 18.8 points for quality (as of July 1, 2025). The article contains 7 references and 5 sections.

This article has the best quality in Russian Wikipedia. However, this article is the most popular in English version.

Since the creation of article “Hopf–Rinow theorem”, its content was written by 34 registered users of English Wikipedia and edited by 125 registered Wikipedia users in all languages.

The article is cited 180 times in English Wikipedia and cited 287 times in all languages.

The highest Authors Interest rank from 2001:

  • Local (English): #73307 in March 2005
  • Global: #129642 in June 2007

The highest popularity rank from 2008:

  • Local (English): #702404 in March 2014
  • Global: #861658 in May 2018

There are 12 language versions for this article in the WikiRank database (of the considered 55 Wikipedia language editions).

The quality and popularity assessment was based on Wikipédia dumps from July 1, 2025 (including revision history and pageviews for previous years).

The table below shows the language versions of the article with the highest quality.

Languages with the highest quality

#LanguageQuality gradeQuality score
1Russian (ru)
Теорема Хопфа — Ринова
21.7734
2Catalan (ca)
Teorema de Hopf-Rinow
20.5532
3English (en)
Hopf–Rinow theorem
18.7801
4Ukrainian (uk)
Теорема Гопфа — Рінова
14.3878
5Vietnamese (vi)
Định lý Hopf–Rinow
12.6765
6Korean (ko)
호프-리노우 정리
12.3284
7Italian (it)
Teorema di Hopf-Rinow
7.7266
8German (de)
Satz von Hopf-Rinow
6.0273
9Chinese (zh)
霍普夫-里诺定理
3.9678
10French (fr)
Théorème de Hopf-Rinow
2.1701
More...

The following table shows the most popular language versions of the article.

Most popular in all the time

The most popular language versions of the article "Hopf–Rinow theorem" in all the time
#LanguagePopularity awardRelative popularity
1English (en)
Hopf–Rinow theorem
118 557
2German (de)
Satz von Hopf-Rinow
22 041
3Italian (it)
Teorema di Hopf-Rinow
10 408
4Russian (ru)
Теорема Хопфа — Ринова
10 342
5French (fr)
Théorème de Hopf-Rinow
9 004
6Chinese (zh)
霍普夫-里诺定理
6 987
7Finnish (fi)
Hopfin–Rinowin lause
2 639
8Dutch (nl)
Stelling van Hopf-Rinow
728
9Ukrainian (uk)
Теорема Гопфа — Рінова
330
10Vietnamese (vi)
Định lý Hopf–Rinow
151
More...

The following table shows the language versions of the article with the highest popularity in the last month.

Most popular in June 2025

The most popular language versions of the article "Hopf–Rinow theorem" in June 2025
#LanguagePopularity awardRelative popularity
1English (en)
Hopf–Rinow theorem
1 119
2German (de)
Satz von Hopf-Rinow
126
3Chinese (zh)
霍普夫-里诺定理
42
4Italian (it)
Teorema di Hopf-Rinow
35
5Russian (ru)
Теорема Хопфа — Ринова
35
6French (fr)
Théorème de Hopf-Rinow
24
7Vietnamese (vi)
Định lý Hopf–Rinow
12
8Catalan (ca)
Teorema de Hopf-Rinow
10
9Finnish (fi)
Hopfin–Rinowin lause
10
10Ukrainian (uk)
Теорема Гопфа — Рінова
9
More...

The following table shows the language versions of the article with the highest Authors’ Interest.

The highest AI

Language versions of the article "Hopf–Rinow theorem" with the highest Authors Interest (number of authors). Only registered Wikipedia users were taken into account.
#LanguageAI awardRelative AI
1English (en)
Hopf–Rinow theorem
34
2French (fr)
Théorème de Hopf-Rinow
20
3Russian (ru)
Теорема Хопфа — Ринова
15
4German (de)
Satz von Hopf-Rinow
13
5Italian (it)
Teorema di Hopf-Rinow
13
6Ukrainian (uk)
Теорема Гопфа — Рінова
8
7Chinese (zh)
霍普夫-里诺定理
7
8Finnish (fi)
Hopfin–Rinowin lause
6
9Dutch (nl)
Stelling van Hopf-Rinow
3
10Vietnamese (vi)
Định lý Hopf–Rinow
3
More...

The following table shows the language versions of the article with the highest Authors’ Interest in the last month.

The highest AI in June 2025

Language versions of the article "Hopf–Rinow theorem" with the highest AI in June 2025
#LanguageAI awardRelative AI
1Catalan (ca)
Teorema de Hopf-Rinow
0
2German (de)
Satz von Hopf-Rinow
0
3English (en)
Hopf–Rinow theorem
0
4Finnish (fi)
Hopfin–Rinowin lause
0
5French (fr)
Théorème de Hopf-Rinow
0
6Italian (it)
Teorema di Hopf-Rinow
0
7Korean (ko)
호프-리노우 정리
0
8Dutch (nl)
Stelling van Hopf-Rinow
0
9Russian (ru)
Теорема Хопфа — Ринова
0
10Ukrainian (uk)
Теорема Гопфа — Рінова
0
More...

The following table shows the language versions of the article with the highest number of citations.

The highest CI

Language versions of the article "Hopf–Rinow theorem" with the highest Citation Index (CI)
#LanguageCI awardRelative CI
1English (en)
Hopf–Rinow theorem
180
2German (de)
Satz von Hopf-Rinow
54
3French (fr)
Théorème de Hopf-Rinow
15
4Russian (ru)
Теорема Хопфа — Ринова
9
5Chinese (zh)
霍普夫-里诺定理
9
6Ukrainian (uk)
Теорема Гопфа — Рінова
6
7Italian (it)
Teorema di Hopf-Rinow
5
8Catalan (ca)
Teorema de Hopf-Rinow
3
9Korean (ko)
호프-리노우 정리
3
10Dutch (nl)
Stelling van Hopf-Rinow
3
More...

Scores

Estimated value for Wikipedia:
English:
Global:
Popularity in June 2025:
English:
Global:
Popularity in all years:
English:
Global:
Authors in June 2025:
English:
Global:
Registered authors in all years:
English:
Global:
Citations:
English:
Global:

Quality measures

Interwikis

#LanguageValue
caCatalan
Teorema de Hopf-Rinow
deGerman
Satz von Hopf-Rinow
enEnglish
Hopf–Rinow theorem
fiFinnish
Hopfin–Rinowin lause
frFrench
Théorème de Hopf-Rinow
itItalian
Teorema di Hopf-Rinow
koKorean
호프-리노우 정리
nlDutch
Stelling van Hopf-Rinow
ruRussian
Теорема Хопфа — Ринова
ukUkrainian
Теорема Гопфа — Рінова
viVietnamese
Định lý Hopf–Rinow
zhChinese
霍普夫-里诺定理

Popularity rank trends

Best Rank English:
#702404
03.2014
Global:
#861658
05.2018

AI rank trends

Best Rank English:
#73307
03.2005
Global:
#129642
06.2007

Languages comparison

Important global interconnections (July 2024 – June 2025)

Wikipedia readers most often find their way to information on Hopf–Rinow theorem from Wikipedia articles about Geodesic curve, Geodesically complete manifold, Pseudo-Riemannian manifold, Riemannian manifold and Complete metric space. Whereas reading the article about Hopf–Rinow theorem people most often go to Wikipedia articles on Geodesically complete manifold, Clifton–Pohl torus, Exponential map, Intrinsic metric and Willi Rinow.

Cumulative results of quality and popularity of the Wikipedia article

List of Wikipedia articles in different languages (starting with the most popular):

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About WikiRank

The WikiRank project is intended for automatic relative evaluation of the articles in the various language versions of Wikipedia. At the moment the service allows to compare over 44 million Wikipedia articles in 55 languages. Quality scores of articles are based on Wikipedia dumps from July, 2025. When calculating current popularity and AI of articles data from June 2025 was taken into account. For historical values of popularity and AI WikiRank used data from 2001 to 2025... More information