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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| Datum: | 2004 |
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| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2004
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/3867 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | 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. |
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