Berger–Kazdan comparison theorem

In mathematics, the BergerKazdan 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.

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