On the Relation between Curvature, Diameter, and Volume of a Complete Riemannian Manifold
In this note, we prove that if N is a compact totally geodesic submanifold of a complete Riemannian manifold M, g whose sectional curvature K satisfies the relation K ≥ k > 0, then \(d(m,N) \leqslant \frac{\pi }{{2\sqrt k }}\) for any point m ∈ M. In the case where dim M = 2, the Gaussian...
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| Date: | 2004 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
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Institute of Mathematics, NAS of Ukraine
2004
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3867 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509998780514304 |
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| author | Nguyen, Doan Tuan Si, Duc Quang Нгуєн, Доан Туан Сі, Дук Куанг |
| author_facet | Nguyen, Doan Tuan Si, Duc Quang Нгуєн, Доан Туан Сі, Дук Куанг |
| author_sort | Nguyen, Doan Tuan |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:12:48Z |
| description | In this note, we prove that if N is a compact totally geodesic submanifold of a complete Riemannian manifold M, g whose sectional curvature K satisfies the relation K ≥ k > 0, then \(d(m,N) \leqslant \frac{\pi }{{2\sqrt k }}\) for any point m ∈ M. In the case where dim M = 2, the Gaussian curvature K satisfies the relation K ≥ k ≥ 0, and γ is of length l, we get Vol (M, g) ≤ \(\frac{{2l}}{{\sqrt k }}\) if k ≠ 0 and Vol (M, g ≤ 2ldiam (M) if k = 0. |
| first_indexed | 2026-03-24T02:50:01Z |
| format | Article |
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| id | umjimathkievua-article-3867 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:50:01Z |
| publishDate | 2004 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/4b/724a15f6f187acfd45d39b58e827ad4b.pdf |
| spelling | umjimathkievua-article-38672020-03-18T20:12:48Z On the Relation between Curvature, Diameter, and Volume of a Complete Riemannian Manifold Про співвідношення між кривизною, діаметром та об'ємом повного ріманового многовиду Nguyen, Doan Tuan Si, Duc Quang Нгуєн, Доан Туан Сі, Дук Куанг In this note, we prove that if N is a compact totally geodesic submanifold of a complete Riemannian manifold M, g whose sectional curvature K satisfies the relation K ≥ k > 0, then \(d(m,N) \leqslant \frac{\pi }{{2\sqrt k }}\) for any point m ∈ M. In the case where dim M = 2, the Gaussian curvature K satisfies the relation K ≥ k ≥ 0, and γ is of length l, we get Vol (M, g) ≤ \(\frac{{2l}}{{\sqrt k }}\) if k ≠ 0 and Vol (M, g ≤ 2ldiam (M) if k = 0. Доведено, що якщо $N$ — компактний цілком геодезичний підмноговид повного ріманового многовиду $(M, g)$ із секційною кривизною $K$, що задовольняє умову $K ≥ k > 0$, то для будь- якої точки $m ∈ M$ виконується нерівність $d(m,N) \leqslant \frac{\pi }{{2\sqrt k }}$. У випадку, коли $\dim M = 2$, гауссова кривизна К многовиду задовольняй умову $K ≥ k > 0$ та $γ$ мак довжину $l$, отримано 21 співвідношення $\text{Vol} (M, g) ≤ \frac{{2l}}{{\sqrt k }}$ для $k ≠ 0$ та $\text{Vol} (M, g) ≤ 2l \text{diam } (M)$ для $k = 0$. Institute of Mathematics, NAS of Ukraine 2004-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3867 Ukrains’kyi Matematychnyi Zhurnal; Vol. 56 No. 11 (2004); 1576–1583 Український математичний журнал; Том 56 № 11 (2004); 1576–1583 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3867/4453 https://umj.imath.kiev.ua/index.php/umj/article/view/3867/4454 Copyright (c) 2004 Nguyen Doan Tuan; Si Duc Quang |
| spellingShingle | Nguyen, Doan Tuan Si, Duc Quang Нгуєн, Доан Туан Сі, Дук Куанг On the Relation between Curvature, Diameter, and Volume of a Complete Riemannian Manifold |
| title | On the Relation between Curvature, Diameter, and Volume of a Complete Riemannian Manifold |
| title_alt | Про співвідношення між кривизною, діаметром
та об'ємом повного ріманового многовиду |
| title_full | On the Relation between Curvature, Diameter, and Volume of a Complete Riemannian Manifold |
| title_fullStr | On the Relation between Curvature, Diameter, and Volume of a Complete Riemannian Manifold |
| title_full_unstemmed | On the Relation between Curvature, Diameter, and Volume of a Complete Riemannian Manifold |
| title_short | On the Relation between Curvature, Diameter, and Volume of a Complete Riemannian Manifold |
| title_sort | on the relation between curvature, diameter, and volume of a complete riemannian manifold |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3867 |
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