On the Relation between Curvature, Diameter, and Volume of a Complete Riemannian Manifold
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)⩽π2√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) ≤ 2l√k if k ≠ 0 and Vol (M, g ≤ 2ldiam (M) if k = 0.Published
25.11.2004
Issue
Section
Short communications
How to Cite
Nguyen, Doan Tuan, and Duc Quang Si. “On the Relation Between Curvature, Diameter, and Volume of a Complete Riemannian Manifold”. Ukrains’kyi Matematychnyi Zhurnal, vol. 56, no. 11, Nov. 2004, pp. 1576–1583, https://umj.imath.kiev.ua/index.php/umj/article/view/3867.