In mathematics, Berger's isoembolic inequality is a result in Riemannian geometry that gives a lower bound on the volume of a Riemannian manifold and also gives a necessary and sufficient condition for the manifold to be isometric to the m-dimensional sphere with its usual "round" metric.
The theorem is named after the mathematician Marcel Berger, who derived it from an inequality proved by Jerry Kazdan.
Let (M, g) be a closed m-dimensional Riemannian manifold with injectivity radius inj(M).
Then with equality if and only if (M, g) is isometric to the m-sphere with its usual round metric.
[2] The original work of Berger and Kazdan appears in the appendices of Arthur Besse's book "Manifolds all of whose geodesics are closed."