2017
Том 69
№ 9

All Issues

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

Nguyen Doan Tuan, Si Duc Quang

Full text (.pdf)


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 \(d(m,N) \leqslant \frac{\pi }{{2\sqrt k }}\) 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) ≤ \(\frac{{2l}}{{\sqrt k }}\) if k ≠ 0 and Vol (M, g ≤ 2ldiam (M) if k = 0.

English version (Springer): Ukrainian Mathematical Journal 56 (2004), no. 11, pp 1873-1883.

Citation Example: Nguyen Doan Tuan, Si Duc Quang On the Relation between Curvature, Diameter, and Volume of a Complete Riemannian Manifold // Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1576–1583.

Full text