On the Relation between Curvature, Diameter, and Volume of a Complete Riemannian Manifold - Nguyen, Doan Tuan; Si, Duc Quang (Ukrains’kyi Matematychnyi Zhurnal, 2004)
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 \(d(m,N) \leqslant \frac{π }{{2\sqrt k }}\) 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.