# 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.

## Statement of the theoremEdit

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

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

## ReferencesEdit

- 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. pp. 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.

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