Berger's isoembolic inequality

Short description: Gives a lower bound on the volume of a Riemannian manifold

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 m-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 (M, g) be a closed m-dimensional Riemannian manifold with injectivity radius inj(M). Let vol(M) denote the Riemannian volume of M and let c_m denote the volume of the standard m-dimensional sphere of radius one. Then

[math]\displaystyle{ \mathrm{vol} (M) \geq \frac{c_m (\mathrm{inj}(M))^m}{\pi^m}, }[/math]

with equality if and only if (M, g) is isometric to the m-sphere with its usual round metric. This result is known as Berger's isoembolic inequality.({{{1}}}, {{{2}}}) The proof relies upon an analytic inequality proved by Kazdan.({{{1}}}, {{{2}}}) 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.({{{1}}}, {{{2}}}) Sometimes Kazdan's inequality is called Berger–Kazdan inequality.({{{1}}}, {{{2}}})

References

Books.

• Berger, Marcel (2003). A panoramic view of Riemannian geometry. Berlin: Springer-Verlag. doi:10.1007/978-3-642-18245-7. ISBN 3-540-65317-1. 
• Besse, Arthur L. (1978). Manifolds all of whose geodesics are closed. Ergebnisse der Mathematik und ihrer Grenzgebiete. 93. Appendices by D. B. A. Epstein, J.-P. Bourguignon, L. Bérard-Bergery, M. Berger and J. L. Kazdan. Berlin–New York: Springer-Verlag. doi:10.1007/978-3-642-61876-5. ISBN 3-540-08158-5. 
• Chavel, Isaac (1984). Eigenvalues in Riemannian geometry. Pure and Applied Mathematics. 115. Orlando, FL: Academic Press. doi:10.1016/s0079-8169(08)x6051-9. ISBN 0-12-170640-0. 
• Chavel, Isaac (2006). Riemannian geometry. A modern introduction. Cambridge Studies in Advanced Mathematics. 98 (Second edition of 1993 original ed.). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511616822. ISBN 978-0-521-61954-7. 
• Sakai, Takashi (1996). Riemannian geometry. Translations of Mathematical Monographs. 149. Providence, RI: American Mathematical Society. doi:10.1090/mmono/149. ISBN 0-8218-0284-4. 

External links

• Weisstein, Eric W.. "Berger-Kazdan comparison theorem". http://mathworld.wolfram.com/Berger-KazdanComparisonTheorem.html.