Английская Википедия:Berger's isoembolic inequality
Шаблон: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 Шаблон: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 Шаблон:Math be a closed Шаблон:Mvar-dimensional Riemannian manifold with injectivity radius Шаблон:Math. Let Шаблон:Math denote the Riemannian volume of Шаблон:Mvar and let Шаблон:Math denote the volume of the standard Шаблон:Mvar-dimensional sphere of radius one. Then
- <math>\mathrm{vol} (M) \geq \frac{c_m (\mathrm{inj}(M))^m}{\pi^m},</math>
with equality if and only if Шаблон:Math is isometric to the [[sphere|Шаблон:Mvar-sphere]] with its usual round metric. This result is known as Berger's isoembolic inequality.Шаблон:Sfnm The proof relies upon an analytic inequality proved by Kazdan.Шаблон: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.Шаблон:Sfnm Sometimes Kazdan's inequality is called Berger–Kazdan inequality.Шаблон:Sfnm
References
Шаблон:Reflist Books.
External links
Шаблон:Riemannian-geometry-stub
