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 ≥ k > 0, then for any point m ∈ M, d(m, N) ≤ π 2 √ k . In the case dim M = 2 with a Gaussian curvature K ≥ k ≥ 0 and γ has the length `, we get V ol(M, g) ≤ 2` √ k if k 6= 0 and V ol(M, g) ≤ 2`diam(M) if k = 0. Source: https://www.math.uaic.ro/~annalsmath/pdf-uri%20anale/F1(2004)/Quang.pdf