Berger's isoembolic inequality

Template:Short description In mathematics, Berger's isoembolic inequality 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 Template:Mvar-dimensional sphere with its usual "round" metric. The theorem is named after the mathematician Marcel Berger, who derived it from an inequality proved by Jerry Kazdan.

Statement of the theorem

Let Template:Math be a closed Template:Mvar-dimensional Riemannian manifold with injectivity radius Template:Math. Let Template:Math denote the Riemannian volume of Template:Mvar and let Template:Math denote the volume of the standard Template:Mvar-dimensional sphere of radius one. Then

$\mathrm{v}\mathrm{o}\mathrm{l}(M)\ge c_m{\left(\frac{\mathrm{i}\mathrm{n}\mathrm{j}(M)}{\pi }\right)}^m,$

with equality if and only if Template:Math is isometric to the [[sphere|Template:Mvar-sphere]] with its usual round metric. This result is known as Berger's isoembolic inequality.Template:Sfnm The proof relies upon an analytic inequality proved by Kazdan.Template:Sfnm The original work of Berger and Kazdan appears in the appendices of Arthur Besse's book "Manifolds all of whose geodesics are closed." At this stage, the isoembolic inequality appeared with a non-optimal constant.Template:Sfnm Sometimes Kazdan's inequality is called Berger–Kazdan inequality.Template:Sfnm

References

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External links

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