Abstract
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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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 11, pp. 1576–1583, November, 2004.
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Si Duc, Q., Nguyen Doan, T. On the Relation between Curvature, Diameter, and Volume of a Complete Riemannian Manifold. Ukr Math J 56, 1873–1883 (2004). https://doi.org/10.1007/s11253-005-0157-z
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DOI: https://doi.org/10.1007/s11253-005-0157-z