On the Relation between Curvature, Diameter, and Volume of a Complete Riemannian Manifold

Authors

  • Doan Tuan Nguyen
  • Duc Quang Si

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 Kk > 0, then d(m,N)π2k for any point mM. In the case where dim M = 2, the Gaussian curvature K satisfies the relation Kk ≥ 0, and γ is of length l, we get Vol (M, g) ≤ 2lk 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.