Berger-Kazdan comparison theorem

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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.

[edit] 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 c_m(r) denote the volume of the standard m-dimensional sphere of radius r. Then

$\mathrm{vol} (M) \geq \frac{c_{m} (\mathrm{inj}(M))}{\pi},$

with equality if and only if (M, g) is isometric to the m-sphere S^m with its usual round metric.

[edit] References

Berger, Marcel; Kazdan, Jerry L. (1980). "A Sturm-Liouville inequality with applications to an isoperimetric inequality for volume in terms of injectivity radius, and to Wiedersehen manifolds", Proceedings of Second International Conference on General Inequalities, 1978. Birkhauser, 367–377.
Kodani, Shigeru (1988). "An Estimate on the Volume of Metric Balls". Kodai Mathematical Journal 11 (2): 300–305. doi:10.2996/kmj/1138038881.