Berger–Kazdan comparison theorem
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.Let be a compact m-dimensional Riemannian manifold with injectivity radius inj. Let vol denote the volume form on M and let cm denote the volume of the standard m-dimensional sphere of radius r. Then
with equality if and only if is isometric to the m-sphere Sm with its usual round metric.
