In mathematics, the Berger–Kazdan comparison theorem 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 mathematicians Marcel Berger and Jerry Kazdan.
Statement of the theorem
Let (M, g) be a compact m-dimensional Riemannian manifold with injectivity radius inj(M). Let vol denote the volume form on M and let cm(r) denote the volume of the standard m-dimensional sphere of radius r. Then
with equality if and only if (M, g) is isometric to the m-sphere Sm with its usual round metric.
References
Berger–Kazdan comparison theorem Wikipedia(Text) CC BY-SA