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 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) ≤ if k ≠ 0 and Vol (M, g ≤ 2ldiam (M) if k = 0.__________Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 11, pp. 1576–1583, November, 2004.